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Visual Analysis in Archaeology. An Artiﬁcial Intelligence Approach
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01 Chapter 5
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Visual Analysis in Archaeology. An Artificial
04 Intelligence Approach
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07 Juan A. Barceló
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13 Idea and Aims

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15 Archaeology is a quintessentially “visual” discipline, because visual perception
16 makes us aware of fundamental properties of objects and allows us to discover how
17 objects were produced and used in the past. The approach I adopt here is to follow
18 current computational theories of visual perception to ameliorate to way archae-
19 ology can deal with the analysis and explanation of the most usual visual marks:
20 shape and texture. In any case, I am not interested in the mere mechanical pro-
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21 cedure of extracting shape information among visual input, but in explaining why
22 archaeological evidences have the shape they have.
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Introduction
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Archaeologists are interested in finding the social cause (production, use, distribu-
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tion) of what they “see” at the archaeological site (or at the museum collection).
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By assuming, that what they perceive in the present is simply the material effects
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of human work made in the past, archaeologists try to understand “archaeological
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percepts” as material things that were products configured through human labor at
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the very beginning of their causal history (Barceló 2007).
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The first we have to take into account when dealing with archaeology is that it is a
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quintessentially “visual” discipline. Among all features that describe archaeological
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evidences, some of them, the most important for the recognition and/or the discov-
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ery of the way the item was produced and or used in the past, have something to do
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with what we have been trained to “see”. Tasks such as identifying the nature of the
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evidence, an artifact type, identifying decorative patterns or use wear in archaeolog-
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ical materials, recognizing archaeological structures in a satellite or aerial image,
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42 J.A. Barceló (B)

43 Departament de Prehistòria, Facultat de Lletres, Universitat Autònoma de Barcelona, Campus
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Bellaterra, 08193 Cerdanyola, Barcelona, España
e-mail: juanantonio.barcelo@uab.es
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A.M.T. Elewa, Morphometrics for Nonmorphometricians, Lecture Notes in Earth 93

Sciences 124, DOI 10.1007/978-3-540-95853-6_5,
C Springer-Verlag Berlin Heidelberg 2010
94 J.A. Barceló
46 identifying activity areas, material accumulations or buildings at the site, interpret-

47 ing burials or settlement patterns can be considered to be within the purview of the
48 analysis of visual marks. There are also non-visual features characterizing ancient
49 objects and materials (i.e., compositional data based on mass spectrometry, chrono-
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50 logical data based on radioactive decay measures, etc.), but visual perception makes
51 us aware of many fundamental properties of material evidences of human action in
52 the past.
53 Unfortunately, there is no universal method of searching for informative visual
54 marks. They can be extracted from any archaeological record almost ad infinitum,
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55 but one usually fails to formalize the significant criterion for what is intrinsically
56 “visual”. An additional difficulty is that different visual features will almost defi-
57 nitely be of importance for different explanations (Shelley 1996). To cope with this
58 problem, archaeologists have traditionally assumed that there is a roughly fixed set
59 or vocabulary of “supposed” descriptive visual regularities shared by a single popu-
lation of objects, which are also distinctive enough. Archaeologists believe that what
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61 they see is a “seed”, a “bone”, a “bowl”, “a knife”, the “wall of a house”, a “prince
62 burial”, etc., and they can distinguish between different kinds of “bowls”, different
63 kinds of “prince burials”, and so on. This way of identification-based explanation
64 seems then a tricky way of solving any archaeological research problem. It pre-
65 tends to explain what has been “seen”, not in terms of their visual characteristics,
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66 but in terms of subjective recognition. Nevertheless, what an archaeologist “sees”
67 at the archaeological site are not stones, walls, pit holes, mounds, buildings, pottery
68 sherds, plants, animal carcasses, or anything like but a hierarchized organization of
69 visual marks and higher level cues to explanatory categories.
The approach we adopt here is to follow current computational theories of visual
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71 perception to ameliorate to way archaeology can deal with the analysis and explana-
72 tion of visual marks. Computer vision has been defined as a process of recognizing
73 elements of interest in an image, and it can be described as the automatic logical
74 deduction of structures or properties of the three-dimensional objects from either
a single image or multiple images and the recognition of objects with the help of
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76 these properties (Kulkarni 2001). Any reasonable sophisticated visual system must
77 involve a set of processes that extract a variety of types of information from the
78 visual input. This information is captured in a variety of internal intermediate-level
79 representations (neural networks, for instance) which form the basis for higher-level
recognition processes.
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81 Following modern studies of computer object-recognition (Grimson 1991;

AQ1 82 Palmer 1999; Bernardini and Rushmeier 2002; Forsyth and Ponce 2003; Carbonetto
83 et al. 2004; Ponce et al. 2007), we should consider specialized archaeological
84 perception essentially as building larger and larger explanatory structures from ele-
85 mentary visual features. Archaeological explanation is then a gradual process that
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86 proceeds from the general to the specific and that overlaps with, guides, and con-
87 strains the derivation of a causal explanation from an image or visual representation
88 of some archaeological evidence (Fig. 5.1).
89 Consequently, it is common to categorize visual process into low, intermediate,
90 and high levels (Marr 1982; Palmer 1999). Low-level information is typically about
5 Visual Analysis in Archaeology. An Artificial Intelligence Approach 95
91 Fig. 5.1 A schema showing Raw visual data

92 the process of visual
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interpretation
Image segmentation
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Low-level interpretation of picture elements
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98 Object recognition
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100 High-level image interpretation
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the spatial relationships among primitive, two-dimensional visual features such as
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observed shape, texture, and composition variability patterns. Intermediate infor-
mation describes the properties arising from forms of organization of the low-level
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primitives, and may include descriptions of the three-dimensional spatial relation-
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ship (location) among visual properties. The overall explanatory process is thus
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broken down into the extraction of a number of different observable physical prop-
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erties (low-level analysis), followed by a final decision based on these properties
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(high-level analysis), what implies breaking down the perception of meaningful
visual marks into different explanatory stages.
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The job of the archaeologist is not to provide with a representation of the past
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in abstract but to look for the information he or she needs to interact with its possi-
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ble explanation. Visual features should be treated as evidence and their estimation
accuracy should be directly correlated to their power to resolve alternative hypothe-
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ses. By this account, hierarchies of feature detectors should be combined together
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in ways given by coactivity of the underlying detectors and necessary knowl-
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edge structures necessary to integrate them. Complex association structures are
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formed when simple feature detectors and prior knowledge structures become asso-
ciated through repeated sequential fixations of the corresponding features (Barceló
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2008).
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124 What is Shape?

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126 The attempts at defining the term shape usually found in the related literature are
127 often based on the concept of all the properties of a configuration of points which
128 are not altered for effects of size, position and orientation, or by translation, rota-
129 tion and scaling (Kendall 1977; Kendall et al. 1999; Bookstein 1991; Small 1996;
130 Palmer 1999; Dryden and Mardia 1998). While such definitions manage to capture
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131 an important property of some visual features as perceived by humans, namely what
132 relates the different appearances of the same object seen from different perspectives,
133 they do not clearly specify what a shape is. An alternative and less conventional def-
134 inition of shape has been advanced by Costa and Cesar (2001, p. 266): a shape can
135 be understood as any “single”, “distinct”, “whole” or “united” visual entity.
96 J.A. Barceló
136 When we see something, we are not seeing an object, but our senses capture sen-
137 sorial information (luminance contrasts), which should then be transformed into an
138 intermediate-level representation of what gives the perceived entity its individuality.
139 Formally speaking, a surface is a boundary of separation between two phases. In its
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140 turn, a phase is a homogenous mass of substance, solid, liquid or gas, possessing
141 a well-defined boundary. When we have two phases in mutual contact, we have an
142 interface. What gives individuality to any solid entity, kept in atmosphere, is in fact
143 its air-solid interface, or in the case of solid entities in contact, a solid-solid inter-
144 face, which are often simply referred to as a solid surface. The surface of solids
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145 plays a significant role to discover the way they have been produced and the way
146 they have been used.
147 Surfaces have two main properties: shape and texture. Shape can be best charac-
148 terized as the perceived interfacial boundaries or discontinuities themselves. In fact,
149 we usually take the geometry of the identified contour or silhouette as a surrogate
of the object’s shape. It will imply essentially the operation of detecting significant
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151 local changes among luminance values in a visually perceived scene and its trans-
152 lation into a geometric language, joining points with lines, fitting surfaces to lines,
153 or “solidifying” connected planes (Barceló 2000). Texture is the definition of sur-
154 face attributes having either visual or actual variety, and defining the appearance of
155 the surface. Any surface has variations in its local properties like albedo and color
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156 variations, uniformity, density, coarseness, roughness, regularity, linearity, direc-
157 tionality, direction, frequency, phase, hardness, brightness, bumpiness, specularity,
158 reflectivity and transparency (Tuceryan and Jain 1993; Fleming 1999).
159 In both cases, geometry is used as a visual language to represent a theoretical
model of the pattern of contrast and luminance, which is the strict equivalent of per-
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161 ceptual models of sensory input in the human brain (Barceló 2001). In fact, shapes
162 are concepts corresponding to geometrical abstractions that may never be perfectly
163 represented in the real world. The constructed geometry of an archaeological artifact
164 refers to the idealized form represented by those portions of the artifact that were
deliberately modified as part of the production of the artifact from raw material.
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166 By idealized geometry (or geometric abstraction) is meant a smoothed form of the
167 interfacial boundary for which variation from the smoothed form appears to simply
168 reflect variation due to the production process (Read 2007).
169 This implies to consider perceived variation in the interfacial boundary of an
artifact to arise from successive modification by the artisan through a sequential,
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171 conceptual process going from an initial abstract ideal form to the final geometry of
172 the set of surfaces defining the finished artifact (Van der Leeuw 2000). The particular
173 morphology of the boundary may be determined from physical constraints acting on
174 the process underlying its formation process (artisan work, user action). An exam-
175 ple would be the distribution of forces acting on the formation of the boundary
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176 of the artifact, as occurs with the hands of the potter making pottery with a pot-
177 tery wheel. On the other hand, the design of a thrusting spear point is likely to be
178 squat and short with a wide tip angle, combining relatively long cutting edges with
179 a short blade and a relatively wide base suitable for hafting with a strong, robust
180 shaft. A throwing spear point, in contrast, needs to optimize the requirements for
aerodynamics, killing power and accuracy. A slim, elongated point combines mass
181 with a relatively acute tip angle and a small presentation area and base. A smaller
182 base means that a smaller shaft leads to a lower overall weapon mass. According to
183 Newtonian mechanics, a lighter missile can be launched at a higher velocity with a
184 flatter trajectory resulting in a faster, more powerful projectile weapon (Christenson
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185 1986; Flenniken and Raymond 1986; Crompton 2007). Nevertheless, in most cases
186 the underlying physics may be too complex to model if there is no single pattern
187 that constrains the interfacial boundaries.
188 By virtue of the properties of the raw material and the features of human labor
189 or action, many objects from the past have a constructed shape. This can be the case
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190 of tools, pottery containers or most built structures (pit holes, graves, walls, build-
191 ings). However, the precise relationship between shape and formation processes is
192 not always direct and easy to explain. In the case of prehistoric stone arrow point
193 made of retouched flint, for instance, its shape of the tool is simply the mechanical
194 consequence of the flake removal, in such a way that the edge of such tools does
not represent necessary a cognized shape on the part of the artisans (Bisson 2000;
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196 Collins 2008).

197 In some other cases, the actual geometry of perceived interfacial boundaries can
198 be the result of taphonomic processes. The actual shape of a wall, as it is perceived
199 in the moment of the archaeological excavation, is the result of the destruction of
200 the original wall, in such a way that the original ordering of building blocks may
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201 be lost. The same is true for a broken pottery vessel, transformed into an amount
202 of fragments whose individual shape is not any more the result of human labor in
203 the past. Mounds resulting from the accumulation of stones, debris or animal bones
204 also can be defined in terms of edges and boundaries explaining the formation (or
deformation) processes involved (Mameli et al. 2002). As a result, the precise shape
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206 of any archaeological deposition should be analyzed to understand the formation

207 process of the archaeological site (Barceló et al. 2003, 2009). At higher perceptual
208 scales, in the case of soil and landscape features, as territories, valleys, drainage
209 basins etc., the geometry of their interfacial boundaries may also be the result of
natural processes or social events having contributed to its actual appearance.
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211 In general, and following Leyton (1992, p. 73) if boundaries (or edges) are
212 understood as perceived discontinuities or asymmetries generated through time, we
213 should be able to recover the history of the perceived (and “differentiated”) archae-
214 ological entity from the perception of change. In other words, archaeologists use
shape information, that is to say data on geometric discontinuities, as memories of
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216 process-history.
217 In many other categories of archaeological evidence, the processes of contour for-
218 mation and transformation result in the essential properties of size, mass and shape
219 changing and reducing with successive re-formation events. However, when dealing
220 with characteristically uneven and asymmetric objects like prehistoric stone tools,
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221 built structures and the like, the very concept of shape regularity acquires another
222 dimension, given the particular way an irregular contour may be the result of a
223 sequence of events, modifying each one a previous shape. Central to this concept
224 is the manner in which irregularly shaped archaeological evidences were designed,
225 reduced, resharpened, recycled, and discarded within its use life. In the case of pre-
historic stone tools, many of them suffered multiple steps or stages of production.
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226 As the flake tool edge becomes dull, it can be resharpened by the removal of minor
227 stone chips from the dulled area. The more a flaked tool is used and subsequently
228 retouched the greater the amount of visible resharpening either in the form of total
229 length of edge resharpening or total surface area with flakes removed. The same is
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230 true for a wall made of building blocks, an irregular mound made of accumulated
231 debris, or an excavated pit hole.
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234 Low Level Visual Analysis
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236 The archaeological record is not made of shapes. It is a series of perceptual infor-
237 mation waiting for an observer. The observer will impose order by recognizing
238 interfacial boundaries between different components with different visual marks
239 and by creating a functional model of them. This key assumption has been tradi-
tionally neglected in archaeology, preferring a subjective approach where shapes
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241 exist as primitives bits of information and may be defined by universal picture
242 stereotypes, e.g., “round” “ovoid”, or even worst by user-defined stereotypes, like
243 “hat-shaped,” “cigar-shaped,” “kidney-shaped”. Such assumption does not take into
244 account that objects such as hats and cigars come in a wide variety of morpholog-
245 ical configurations, making the visualized reference standard and qualitative terms
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246 subject to variations of individual perception. It has been considered an alternative
247 the identification of underlying geometries in qualitative terms, such as irregular,
248 indented, sinuous, etc. to describe it. Even the simple call to standard shapes of
249 Euclidean geometry (rectangle, parallelepiped, circle, sphere, cylinder, cone, etc.) is
a misleading answer. Euclidean geometry with its well-defined and mathematically
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251 tractable curves and lines is usually only found as an approximation over a range of
252 dimensions where human manufacture labor has imposed it, or in limited situations
253 where a single energy or force term dominates (e.g., surface tension). The surface of
254 artisan-made materials, objects, tools, accumulations or built structures may seem
Euclidean only at some particular scale. Magnify the field view and they become
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256 rough or irregular.

257 The result is lack of replication among experts and arguments over the reality
258 of perceived visual information (Dibble and Chase 1981; Whallon 1982; Djindjian
259 1993; Orton et al. 1993; Andrefsky 2005; Read 2007).
Since the observer arbitrarily constructs such configurations, archaeological
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261 objects cannot fulfill the parameters of a prototype as long as material objects are
262 governed by the physical variation intrinsic to the labor process that generated the
263 object in the past, and the remaining variation generated by the post-depositional
264 processes that altered its visual characteristics since then.
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267 Visual Data Acquisition and Encoding

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269 In order to be able to increase the objectivity of visual information, archaeologists

270 need a kind of instrumental “observer” equipped with range and intensity sensors
271 (Barceló 2005). The former acquire range images, in which each pixel encodes the
272 distance between the sensor and a point in the scene. The latter are the familiar
273 digital cameras acquiring grey-level images. That is to say, such an instrumental
274 observer or automated archaeologist may use a CCD camera to observe a pattern
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275 of structured light projected on the scene (e.g., a beam of laser light, or a grid of
276 lines). If the sensor has been calibrated, depth can be inferred by triangulation. In
277 so doing range sensors can measure depth at single points, on lines (acquiring range
278 profiles), or in 2D fields of view (acquiring range images). Most measurements
279 can be directly calculated from a pixel-based representation by simple counting
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280 procedures multiplied by calibrated pixel size with adjustments for specific mea-
281 surements. This measurement is methodologically and fundamentally different from
282 standard measurements based on conventional tools like calipers and tapes.
283 Digital images contain all the useful information to derive geometry and texture
284 for a 3D modeling application. However, the reconstruction of detailed, accurate
and photo-realistic 3D models from images is a difficult task, in particular for
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286 large and complex archaeological evidences (Manferdini et al. 2008). Image-based
287 methods require a mathematical formulation (perspective or projective geometry) to
288 transform two-dimensional image measurements into 3D coordinates.
289 There are three categories of optical 3D data acquisition: (1) image-based meth-
290 ods, e.g. photogrammetry, (2) range-based methods, e.g. laser scanning, and (3)
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291 combinations of both. The choice of the most appropriate technology for a given
292 task depends on the object or area under investigation, the experience of the user,
293 the available budget and time, and further parameters. Techniques applied to the
294 restitution of small archaeological objects are based on the exhausting calculation
of 3D point clouds, which represent the outer surfaces of the objects. The most
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296 popular are:

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298 1. Laser scanning. The measuring of the 3D points coordinates is implemented

299 through a laser beam that is transmitted towards the object and reflected back
to the source. The time that is needed for the beam to travel from the laser beam
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301 source to the object and back, multiplied by the speed of laser light, yields the
302 distance of the points from the source; hence, their location on an arbitrarily
303 defined 3D coordinates system.
304 2. Optical scanning. Special structured light devices and laser diodes producing
straight (horizontal or vertical) line tracks are used for the exact definition of 3D
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306 points on the object. Sophisticated photogrammetric procedures may lead to the
307 calculation of a dense point cloud that describes the outer surfaces of the objects.
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309 These kind of instrumental observers generate as an output detailed point clouds
310 of three-dimensional Cartesian coordinates in a common coordinate system. The
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311 digitized data generated by the scanner is composed of thousands of x, y, z coordi-

312 nates that describe a point cloud that represents the surface of the object scanned.
313 The laser scanner measures, in principle, the distance to a target point and the
314 respective vertical and horizontal angles. Besides target distance, the relative inten-
315 sity of the returned echo signal as well as the true color of the target point should
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316 be recorded in order to obtain an estimation of visual marks variability to be trans-

317 lated into a shape model. A laser digitizer, for instance, captures surface data points
318 that may be less than 300 microns (0.3 mm) apart, producing high-density geomet-
319 ric meshes with an average resolution of over 1,000 points per cm2 . The accuracy
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320 of the measurements derived from the acquired point clouds coordinates usually
321 exceed those possible using traditional 2D tools such as calipers and rulers (Doi
322 and Sato 2005; Tsioukas et al. 2004; Trinkl 2005; Kampel and Sablatnig 2006;
323 Petersen et al. 2006; Lambers et al. 2007; Lambers and Remondino 2007; Farjas and
324 García Lázaro 2008; Karasik and Smilansky 2008; Manferdini et al. 2008; Avern
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325 in press).
326 An alternative approach is computer tomography (Casali 2006; Dimitrov et al.
327 2006; Kardjilov et al. 2006). The word “tomography” derives from the Greek tomos
328 (slice) and graphein (to write). Here, the “instrumental observer” scans thin (vir-
329 tual) slices of the object with a narrow x-ray beam, which rotates around the object,
producing an image of each slice as a cross section of the object and showing each
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331 of the possible internal components in a 10 mm slice. Digital geometry processing

332 is used to generate a three-dimensional image of the inside of an object from a large
333 series of two-dimensional X-ray images taken around a single axis of rotation. The
334 image is made up of a matrix of thousands of tiny squares or pixels (65,000 pixels
335 in a conventional image). Each pixel has an associated measure of how much of
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336 the initial x-ray beam is absorbed by the different components of the object’s fab-
337 ric at each point in its solid body (the computed tomography number, measured in
338 Hounsfield units). This varies according to the density of the component. The denser
339 the component is the higher the computed tomography number, ranging from 1,000
HU (air) to 1,000 HU (bone). The resulting visual model would be displayed with
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341 a different shade of grey for every different computed tomography number. By con-
342 vention, high computed tomography numbers are displayed as white and low as
343 black.
344 3D microscopy is another possibility for encoding the microtopography of a
surface, and hence some details of its shape geometry. This technology creates a
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346 series of individual image planes (up to 200) and overlaps focus levels to construct
347 a three-dimensional composite image (Bello and Soligo 2008).
348 Using any of those technologies of data capture and encoding, the resulting
349 data is only a spatial array of visual bindings which can be subdivided into sets
of marks (points, lines, areas, volumes) that express the position and the geometry
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351 of perceived boundaries (shape), and retinal properties (texture).

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354 Edge Detection

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356 Shapes are not something to be captured using digital cameras, laser scans, or
357 computer tomography equipment because they are not a part of reality. We have
358 to “discover” in some way an explanatory representation of luminance regularities
359 that have been acquired and encoded by the “instrumental observer”. A geometri-
360 cal model showing how interfacial boundaries between luminance areas are related
361 should provide the keys for detecting individual bits of reality in what is apparently
362 a continuous array of visual marks.
363 The method for “finding” the interfacial boundaries that allow the identification
364 of individualized archaeological observables can be approached by calculating the
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365 luminance gradient in the data array – that is, the direction of maximum rate of
366 change of luminance values, and a scalar measurement of this rate. It should coin-
367 cide with the outer frame of the observed object, usually called edge, contour or
368 silhouette. This is a line marking a constant level of luminance. As an interfacial
369 boundary, this line never ends, although it may branch or loop back upon itself. It
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370 is not an intrinsic property of observed objects, but it arises in images in differ-
371 ent contexts: discontinuity in the surface depth, discontinuity in surface orientation,
372 markings on the surfaces, etc. In other words, it is the boundary that delimits distinct
373 spatial areas which appear when visual appearances are “significantly different”
374 from one area to the next.
A contour or edge is simply a linear separation between regions with different
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376 texture (visual or retinal properties) within an image. Shape discovery is then the
377 operation of detecting significant local changes among luminance values in a visual
378 scene. The method for “finding” edges in the images that represent archaeological
379 evidences can be approached by calculating the texture gradient (usually a “lumi-
380 nance gradient) in the data array – that is, the direction of maximum rate of change
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AQ3 381 of luminance values, and a scalar measurement of this rate. Marr and Hildreth (1980)
382 initially defined the procedure, finding the position of maximum variation in the map
383 of luminance (grey or RGB-color levels). First-order differential operators compute
384 the variation levels of such intensity function, and the algorithm finds the edge by
detecting the highest value in the first derivative of the intensity function. A more
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386 economical algorithm for finding edges would be to detect zero-crossings of the
387 second derivative of the intensity function. The second derivative of a function is
388 just the slope of its previously calculated first derivative. The second derivative thus
389 computes “the slope of the slope” of the original luminance function. Notice that in
this second derivative function, the position of the interfacial boundary corresponds
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391 to the zero value in between a highly positive and a highly negative value (Sonka
392 et al. 1984).
393 Many different variants and ameliorations of this primitive procedure have been
394 published. The Canny edge detector first smoothes the image to eliminate and noise;
it then finds the image gradient to highlight regions with high spatial derivatives.
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396 The algorithm then tracks along these regions and suppresses any pixel that is not
397 at the maximum (non-maximum suppression). Finally, the gradient array is fur-
398 ther reduced by hysteresis to track along the remaining pixels that have not been
399 suppressed. Hysteresis uses two thresholds and if the magnitude is below the first
400 threshold, it is set to zero (made a non-edge). If the magnitude is above the high
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401 threshold, the presence of an edge at this point is affirmed. And if the magnitude is
402 between the 2 thresholds, then it is set to zero unless there is a path from this pixel
403 to a pixel with a gradient above this threshold (Russ 2006).
404 This is not the proper place to discuss all approaches to edge detection. There
405 is huge literature, indeed an industry, concerned with such algorithms (Martin et al.
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406 2004; Heideman 2005; Russ 2006; Gonzalez and Woods 2007; O’Gorman et al.
407 2008). Nevertheless, conventional edge extraction techniques, being sensitive to
408 (image) noise and intensity variations, often do not give us the true boundaries of
409 objects in images. On the other hand, their outputs usually contain spurious or weak
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410 edges. It is now generally acknowledged that, without a higher-level information
411 of the object itself (such as the geometry of the surface), such techniques produce
412 erroneous results. Consequently, it seems a good idea to build more optimal edge
413 detector by training a neural network with a certain predefined network structure
414 with examples of edge and non-edge patterns. Good examples of this procedure
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415 have been published by Wang et al. (2000) and by Martin et al. (2004). The idea is
416 to regard edge detection as a form of statistical pattern classification, using features
417 extracted from an image patch to estimate the posterior probability of a boundary
418 passing through the center point. Only two classes of pixels, edge or non-edge, need
419 to be discriminated. The neural edge detector can directly estimate the probability
by training.
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424 Preliminary Approaches to Shape Encoding

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426 Traditionally, archaeologists have referred to diameters and heights when they
427 spoke about shape, forgetting important parameters like surface area or volume.
428 The conventional method for capturing artifact morphology has been to take lin-
429 ear measurements with calipers at fixed loci along an arbitrary line of maximum
bilateral symmetry, generally defined as Length (DeBoer 1980; Pobelome et al.
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431 1997; Lycett et al. 2006; Steine 2005; Rushmeier et al. 2007; Mara and Sablatnig
AQ4 432 2005). Such linear measurements, however, are absolute quantities reflecting only
433 size. No geometric information is provided on the relative position of the var-
434 ious breadth and thickness measurements. Consequently, the variables sampled
constitute an abstract collection of relative size measurements, approximating the
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436 artifact’s morphology (see discussion in Crompton 1995, 2007; Meltzer and Cooper
437 2006).
438 Size is a magnitude causing all the metric variables to increase in dimension as it
439 increases. On the opposite, shape should be dimensionless, that is, size independent.
Obviously, there is no assurance that two archaeological artifacts with identical size
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440
441 values at different parts of their extension will have similar shapes. The shape of
442 every square, for example, is the same whether it is a large square or a small square.
443 An important corollary of this is that attempts to examine shape differences should
444 attempt to account for the effects of isometric size prior to the analysis of shape. To
445 solve this problem, it has been suggested to modify raw size measures that represent
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446 a length or width as a proportion of the length of the artifact (Wynn and Tierson
447 1990; Lycett et al. 2006). An alternative approach would be to average distance or
448 size-based measurements in terms of a global parameter. Feret’s Diameter can also
449 be used for this averaging. Generally, it is the greatest distance possible between
450 any two points along the contour. When such a global measure of size is difficult to
451 calculate, we can estimate a surrogate using a measure of thread length. This gives
452 an estimate as to the true length of a threadlike object. It assumes that the object to
453 be measured is threadlike in form.
454
p + p2 − 16 Area
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455
(5.1)
456 4
457
458
In Eq. (1), p is the perimeter of the contour, and Area is a measure of the sur-
459
face of the object. Note that this is an estimate only. Use this parameter when the
objects are known to be threadlike and bend so that the Diameter parameter is a poor
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460
461
estimate of the true length. Another neglected size parameter is perimeter length.
462
Its measuring, however, often depends to the orientation of the object, and of the
463
kinematics of use of it.
464
However, dividing each length or width measure by a single measurement desig-
nated to represent “size” (such as maximum length, Feret’s diameter or height) not
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465
466
always removes correlations with size. Given the problems associated with these
467
methods, geometric mean size-adjustment has been suggested. It implies the size-
468
adjustment of the data on a specimen-by-specimen basis, dividing each variable
469
in turn by the geometric mean of all variables for that individual specimen. This
470
method isometrically corrects for size enabling direct comparison of allometric
AQ5 shape variation (Lysett et al. 2005).
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471
472
Much more efficient for shape encoding are descriptions based on relational
AQ6 indexes. Russ (2002) gives a preliminary list of the most common and general,
473
474
apparently adapted to describe any kind of shape:
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475
476
(1) Elongation. Perhaps the simplest shape factor to understand is an Aspect
477
Ratio, i.e., length divided by breadth, which measures an aspect of elongation
478
of an object.
479
length MaximumDiameter
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480 or (5.2)
481
width MinimumDiameter
482
(2) Roundness. It measures the degree of departure from a circle of an object’s
483
two-dimensional binary configuration. This is based not on a visual image or
484
an estimate of shape; rather, it is based on the mathematical fact that, in a
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485
circular object with a fixed area, an increase in the length of the object causes
486
the shape to depart from a circle.
487
488 4 Area
(5.3)
489
π p2
490
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491 In the equation, p is the perimeter of the contour, and Area is a measure
492 of the surface of the object. The roundness calculation is constructed so that
493 the value of a circle equals 1.0, while departures from a circle result in values
494 less than 1.0 in direct proportion to the degree of deformation. For instant, a
495 roundness value of 0.492 corresponds approximately to an isosceles triangle.
104 J.A. Barceló
496 (3) Shape Factor (or Formfactor). It is similar to Roundness, but emphasizes the
497 configuration of the perimeter rather than the length relative to object area. It
498 is based on the mathematical fact that a circle (Shape factor value also equal to
499 1.0), compared to all other two-dimensional shapes (regular or irregular), has
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500 the smallest perimeter relative to its area. Since every object has a perimeter
501 length and an area, this mathematical relationship can be used to quantify the
502 degree to which an object’s perimeter departs from that of a smooth circle,
503 resulting in a value less than 1.0. Squares are around 0.78. A thin thread-like
504 object would have the lowest shape factor approaching 0.
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505
506 4π Area
(5.4)
507 p2
508
509 In the equation, p is the perimeter of the contour, and Area is a mea-
sure of the surface of the object. Notice that formfactor varies with surface
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510
511 irregularities, but not with overall elongation.

512 (4) Quadrature: The degree of quadrature of a solid, where 1 is a square and 0.800
513 an isosceles triangle. This shape is expressed by:
514
p
515 √ (5.5)
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516 4 Area
517
In the equation, p is the perimeter of the contour, and Area is a measure of
518
the surface of the object.
519
(5) Curl. It measures the degree of departure of an object from a straight line,
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520
which usually is applied to irregular lines or long, narrow (squiggly) objects.
521
522
length
523 (5.6)
524
skeletonlength
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525
In the equation, maximum length (or Feret’s diameter) should be divided by
526
the object’s symmetry axis length or its center line distance (skeleton length).
527
(6) Solidity. This measure is based on the ratio of the area of the true object to the
528
area of a snug polygonal box fitted around the object. The degree of difference
529
between the object and its fitted box is a quantitative measure of the degree of
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530
irregularity of the object; irregularity itself becomes a quantifiable aspect of
531
morphology.
532
533
Area
534 (5.7)
ConvexArea
535
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536 (7) Convexity. This measure is based on the ratios of the perimeter of the true
537 object to the perimeter of a snug polygonal box fitted around the object. The
538 degree of difference between the object and its fitted box is a quantitative mea-
539 sure of the degree of irregularity of the object; irregularity itself becomes a
540 quantifiable aspect of morphology.
541 ConvexPerimeter
(5.8)
542 perimeter
543
544 (8) Compactness. It is defined as the ratio between the length of the object’s
contour (the perimeter) and the perimeter of a circle with the same area. It
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545
546 is always greater than 1 and approaches unity when the basin approaches a
547 circular shape
548
√
549 (4/π ) Area
(5.9)
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550 MaximumDiameter
551
552
This way of measuring shape parameters is very popular in archaeozool-
553
ogy, paleontology and physical anthropology (Mafart and Delinguette 2002;
554
Rovner 2006; Rovner and Gyulai 2007), but not so common in mainstream
archaeology, where qualitative and subjective descriptions are the rule. In
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555
556
the case of lithic tools, Rovner (1993), Russ and Rovner (1989), Rovner
557
(2006) has used such methods to analyze the shape of tools. J.A. Barceló and
558
J. Pijoan (Barceló et al. 2001; Adán et al. 2003; Barceló and Pijoan 2004;
559
Pijoan 2007) have used geometric relational indexes to describe variation in
560
use-wear textures in terms of the shape variability of determinable areas with
homogenous micro texture. Beyond the study of objects’ shape, Bardossy and
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561
562
Schmidt (2002) have used indexes of compactness, circularity and elongation,
563
for the study of macro-scale entities, like landscape features (drainage basins
564
morphology).
There are many other relational indexes specific of particular categories of
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565
566
archaeological evidences. In the case of pottery, the work by Ericson, Read and
567
Burke (1972) pioneered such approaches, introducing measures based on the
568
calculation of the centre of gravity, to study the relationship between shape
569
and equilibrium, associated with the assumed function of a pottery vase as
container. To locate the center of gravity, Bishop et al. (2005) suggest looking
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570
571
for the intersection of the length of the major and minor axis of the ellipse with
572
the same normalized second central moments as that part of the object – i.e.,
573
the vase’s mouth, its body or its base. As a result, we can measure:
574
(9) The Equivalent Diameter is the diameter of a circle with the same area as this
part of the vase, computed as:
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575
576 √ ∗
577
(4 Area/π ) (5.10)
578
579
(10) The eccentricity is the ratio of the distance between the foci of the ellipse and
580
its major axis length. The value is between 0 and 1. Orientation is the angle
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581
(in degrees) between the x-axis and the major axis of the ellipse that has the
582
same second-moments as the region. Solidity is the proportion of the pixels in
583
the convex hull that are in the region computed as:
584
585 object’s area/convex area (5.11)

106 J.A. Barceló
586 (11) Extent is the proportion of the pixels in the bounding box that are also in the
587 region, computed as:
588
589 object’s area/area of bounding box (5.12)
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590
591
M. Smith (1983) suggested measures like the relative restrictedness or the
592
absolute orifice size. Working on Smith’s approaches, K. Juhl (1995) has pro-
593
vided a detailed list with more than 40 relational indexes based on the size
594
differences between different parts of the same artifact. Among them:
(12) Restriction ratio: the surface are of a vessel mouth at its most restricted point
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595
596
divided by the area of a circle with the same maximum radius as the vessel
597
(13) Relative access factor: the surface area of the vessel’s mouth at it most
598
restricted point divided by the volume below this point.
599
(14) Relative restriction factor: the circumference of the rim divided by the total
surface area of the vessel
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600
601
(15) Leverage factor: vertical distance between the base and the centre of gravity
602
divided by the bottom radius.
603
(16) Relative position of the centre of gravity: vertical distance between the bottom
604
and the centre of gravity divided by the total vessel height.
605 Porter et al. (2005) add the following shape parameters for pottery
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606 containers:
607 (17) Relative width at base. This is the maximum width at which the object touches
608 the ground relative to the object’s maximum width.
609 (18) Relative headroom. This is the maximum clearance of a concave base relative
to the object’s height.
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610
611 (19) Clay efficiency. This is the ratio of a vessel’s capacity relative to the volume
612 of the raw material (clay) used for manufacture.
613 (20) Relative centre of gravity. This is based on the assumption of a homogeneous
614 density of the clay and set in relation to the object’s height.
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615 (21) Relative access width.. This is the width of the inner point with the biggest
616 (inner) height-to-width ratio relative to the maximum width.
617 (22) Angle of access. This is the maximum angle under which one can directly
618 reach the middle of the vessel’s bottom, calculated from the access width.
619 (23) Mean relative wall thickness. Computed along the dominant skeleton arm (by
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620 doubling the minimum distance to the next contour point), and relative to the
621 diagonal of the smallest enclosing rectangle (to incorporate both very wide as
622 well as very tall vessels). Feet and other decorations and ornaments lead to a
623 slightly overestimated value.
624 (24) Skeleton-complexity. The number of additional skeleton arms (belonging to at
625 least 5 profile points).
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626
Very similar are relational indexes for lithic tools, bronze artifacts, or even
627
in the case of macro-scale entities, like walls, buildings, settlements or even
628
landscape or territorial features. The differences come from the assumed func-
629
tional relevance of some of the indexes. For instance, in the analysis of the
630
shape of lithic tools, there are:
631 (25) Index of robustness, calculated as

632
633 width∗ thickness/length (5.13)

634
Higher values imply more robust tools.
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635
636
(26) Index of reactivation, calculated as:
637
638
639
Length/thickness (5.14)
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640
(27) An alternative measure for the reactivation index is an estimate of the quan-
641
tity of modification affecting the original shape of the tool generated by the
642
successive removing of flakes from a blank core Kuhn’s (1990) geometric
643
index is:
644
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645
height of retouch/blank thickness. (5.15)
646
647
(28) Clarkson’s “index of invasiveness” assumes the tool begins its use life as an
648
unmodified flake, and gradually as the tool undergoes use, it also undergoes
649
resharpening and retouching. The measure is based upon the relative propor-
650
tion and size off take scars to the unflaked surface of the tool blank. A tool
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651
with two completely flaked surfaces would have the highest retouch value and
652
a tool with no flakes removed from its surface would have the lowest retouch
653
value (Clarkson 2002. See also Andrefsky 2006).
654
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655
Many other relational indexes are possible, notably in the case of arrow or spear
656
points. In these cases, angles serve as a major shape descriptor.
657
Measuring the degree of symmetry is another shape parameter that has rele-
658
vance for studying the form-function relationship of an archaeological artifact. A
659
very simply approach would imply to divide width measurements into a left and
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660
right measurement, and measuring them along the major axis of the object divided
661
into ten or more segments. The idea is to consider changes in the object’s width
662
in equally spaced intervals increments down its length. When displayed as a his-
663
togram, changes of width of the point appears as bars above (expansion) and below
664
(contraction) a centerline. Families of similar points should produce repeated and
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665
recognizable patterns in the change in width, with respect to length (in the case of
666
lithic tools: Dibble and Chase 1981; Morris and Scarre 1981; Henton and Durand
667
1991; Lycett et al. 2006, in the case of pottery objects: Wilcock and Shennan 1975;
668
Richards 1987). In such a way, the degree of curvature – a shape parameter- of the
669
outline can be estimated. An alternative is based on a relational index expressing the
670
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degree of asymmetry between two bilateral measurements as a ratio of the overall

671
width. Lycett suggests computing
672
√

673 n 2
(xi − yi)
674
S= (5.16)
675 xi + yi
i=1
108 J.A. Barceló
676 where xi is the width value left of the length line taken at a particular percentage
677 point, yi is the width value right of the length line taken at the corresponding per-
678 centage point and n is the number of percentage points at which xi and yi are taken.
679 Hence, a value of zero would correspond to perfect bilateral symmetry (Lycett 2008;
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680 Lycett et al. 2006. See also Hardaker and Dunn 2005).
681
682
683
684 Edge Curvature as Shape Encoding
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685
686 To completely characterize a shape means to be able to re-create the shape using
687 only the measurements made over the shape. D. W. Read formalizes this require-
688 ment in the following definition: “An ordered n-tuple of measurements completely
689 characterizes a shape without redundancy if (a) there is a set of drawing rules that
permits reconstruction of the shape outline using only this ordered n-tuple of mea-
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690
691 sures, and (b) there is no ordered k-tuple of measures, k<n, such that the shape
692 outline can be reconstructed from the ordered k-tuple” (Read 2007: p. 157). We can
693 also refer to a set of measures that completely characterizes a shape as satisfying
694 the archival property: the shape can be reconstructed from the measures that have
695 been taken. The archival property is a weaker requirement that non-redundancy in
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696 that a set of measurements satisfying the archival property may possibly be a redun-
697 dant set of measures. Additionally, it should be a measure (or series of measures)
698 not changing under similarity transformations: translations, rotations, and changes
699 of geometric scale (enlargements or reductions).
The general characteristics of contour or silhouette of a solid is a good candidate
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700
701 for a complete shape descriptor following Read’s conditions. It is usually defined
702 as the longest elongation around – or cross-section through – the outer limit of the
AQ7 703 object defined by its rotational axis (axis of symmetry) (Mara and Sablatnig 2008).
704 It provides a relatively compact way of representing the shape of an object, with
the assumption that the region between the edges defining the contour is relatively
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705
706 homogenous.
707 There are many ways of considering the overall geometry of contour invariant to
708 size, scale and transformation. A relatively simply approach would be using a form
709 of run-length encoding (also called chord encoding). This treats the image as a series
of scan lines. For each sequential line across each region of feature, it stores the line
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710
711 number, start position, and length of the line. A simple polygonal approximation to
712 the boundary can be produced from the run-length table by using the endpoints of
713 the series of chords. A special form of this polygon can be formed from all of the
714 boundary points, consisting of a series of short vectors from one boundary point to
715 the next. On a square pixel array (a bit-map image), there are only eight possible
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716 directions. Assigning a digit from zero to seven to each direction and writing all
717 of the numbers for the closed boundary, we will produce a chain code representing
718 shape (Russ 2002; pp. 373–375). This approach was used in some of the first essays
719 of shape analysis in archaeology (Kampffmeyer et al. 1988; see also Hagstrum and
720 Hildebrand 1990).
721 The way G. Laplace described the operative edge of prehistoric lithic tools
722 can also be classified within this group of applications (Laplace 1972). He advo-
723 cated for an objective description of induced modifications along the cutting edge
724 (“retouches”) based on qualitative identifications and quantitative properties like
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725 mode (angle), amplitude, orientation, linearity. A combination of qualified retouches
726 along the edge allowed a semi quantitative description of the edge’s shape.
727 A much more efficient alternative implies converting the contour directly into a
728 mathematical representation of the digitized boundary. In some cases, the represen-
729 tation may be achieved through characterizing the boundary with a mathematical
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730 expression such as a polynomial equation. A polygon can fit a contour line through
731 the points interpolated between the raw image pixel centers for all such pairs of
732 pixels that bracket the contour value. This mathematical expression also serves as
733 the rule for drawing the curved segment from the parameter values. The main dif-
734 ficulty in implementing this method for representing a boundary lies in identifying
the appropriate mathematical expression. Smoothing will eliminate variation that
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735
736 represents the idiosyncrasies of artifact production and individual variability. For a
737 smooth curve, parameter estimation using statistical curve fitting methods will be
738 quite accurate and constructing confidence intervals for parameter estimations will
739 not be needed. The size of the confidence interval is likely to be on par with or
740 smaller, than measurement error introduced though digitization (Arlinghaus 1994;
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741 Dierckx 1995; Hermon et al. 2001; Schurman et al. 2002; Sarfraz 2007).
742 Polynomial expressions (or their equivalents: Bezier curves/B-splines) will pro-
743 vide a mathematical expression for virtually any curve that can be interpreted as
744 representing the geometry of an object’s contour. Polynomial interpolation has a
long history in archaeology, especially in the case of pottery studies. Hall and Laflin
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745
746 (1984) used B-spline methods to convert digitized pottery containers silhouettes
747 into one or more mathematical curves (see also Smith 1985; Kampel and Sablatnig
748 2003b; Mom 2005, 2006; Nautiyal et al. 2006). Since the interpolated profile curve
749 is a planar curve, we can assign a sign to the curvature: i.e. positive or negative
(Simon et al. 2002). If the curvature at each point on the object’s contour is defined
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750
751 by constructing the circle, which osculates the curve at the point of interest, then
752 the curvature will be the inverse of the circle’s radius, and it will be positive if the
753 curve is convex at that point and negative when it be concave. In this way, we can
754 obtain additional shape qualifiers. For instance, a contour can be concave, convex,
or planar. A concave edge is characterized by inflection points where the adjacent
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755
756 sections of the curve form an angle of less than 180 degrees. A convex edge is char-
757 acterized by inflection points where the adjacent sections of the curve form an angle
758 of more than 180 degrees. A planar edge is characterized by inflection points where
759 the adjacent sections of the curve are coplanar (Jang et al. 2006).
760 When constructing interpolating polynomials to represent boundary curves, there
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761 is a tradeoff between having a better fit and having a smooth, well-behaved fitting
762 function. The more data points that are calculated in the interpolation, the higher
763 the degree of the resulting polynomial, resulting in greater oscillations between data
764 points. Therefore, a high degree interpolation may be a poor predictor of a function
765 between points, even though the accuracy at the data points will be “perfect”.
110 J.A. Barceló
766 Quantitative measures of the intensity of curvature should also be calculated

767 to enhance the global understanding of the geometry allowed by the polynomial
768 expression of the contour. The rational behind this approach comes from a differ-
769 ential geometry theorem, which states that any two curves which have identical
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770 curvature and torsion are the same curve regardless of translation and rotation
771 (Lu et al. 2007).
772 One of the earliest methods is the “tangent profile” or its later development the
773 “sampled tangent-profile” technique (Leese and Main 1983). The curvature k of a
774 planar curve, at a point on the curve, is defined as the instantaneous rate of change
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775 of the slope of the tangent at that point with respect to arc length (for the general
776 procedure, see Bebis et al. 1998; Heideman 2005). The curve is described by pro-
777 viding the tangent as a function of the arc-length, since the curvature is the first
778 derivative of the tangent angle. A related method was used by Liming et al. (1989),
779 who expressed the shape by providing the distance of the points on the profile from
the axis of revolution as a function of the arc-length.
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780
781 Another relatively simple measure of “surface curvature” can be calculated by

782 taking the standard deviation of the z-coordinates along the x-axis and the y-axis,
783 and dividing this by the length of those. Hence, the coefficient of surface curvature
784 emphasizes relative variation over the length of each axis. In the same way, a “coef-
785 ficient of edge undulation” can be estimated by computing the standard deviation of
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786 the z-coordinates at the endpoints of the length, and dividing this by the geometric
787 mean of the lengths of the four axes (Lycett et al. 2006).
788 Additional measures of edge curvature are possible. Consider the coordinates of
789 each point on a curve x(s), y(s) where s denotes the arc length along the curve. As
the parameter s changes, the point moves along the line. x(s) can be chosen as the
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790
791 distance from the axis of cylindrical symmetry of the vessel. At each point on the
792 curve, the tangent vector can be drawn, which by convention, points in the direction
793 of increasing arc length along the curve. Denoting by θ s the direction of the tangent
794 vector with qrespect to a fixed axis (for simplicity the axis can be chosen as the
x-axis), then the tangent angle θ (s) will determine the curve. The curvature κ(s) will
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795
796 measures the rate of change of the tangent angle:

797
798 κ(s)) = (d θ/ds) (5.17)

799
An alternative definition can be given in terms of the radius ρ(s) of the circle that
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800
801 osculates the curve at the point s:

802
803 κ(s) = 1/ρ(s) (5.18)

804
805 The advantage of this representation is that κ(s) is large where the line changes
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806 its direction in the most rapid way, which are also the points of greatest interest
807 in many archaeological applications. Additionally, a small indentation that only
808 changes the curve locally will appear as a small perturbation in these representa-
809 tions. The tangent angle depends more strongly on local features (it is defined as
810 a ratio of derivatives of the Cartesian coordinates), and hence, local changes of the
811 line will show up. The curvature involves a higher derivative, and thus it is very
812 sensitive to local variations. The features of the line that provide information on the
813 gross properties of the curve will be hardly shown. The selection of the representa-
814 tion to be used is dictated by the particular application at hand, and on the features of
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815 the curve which are of relevance (Gilboa et al. 2004; Saragusti et al. 2005; Karasik
816 2008; Nautiyal et al. 2006; Mom 2005, 2006; Maaten et al. 2009).
817 More details about the geometry of a curvature can be measured on a digi-
818 tized profile. In the case of microtopographic features of bone surfaces, Bello and
819 Soligo (2008) suggests measuring the following parameters (Table 5.1). Although
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820 specifically oriented to micro-shape analysis, they can be easily generalizable to the
821 curvature of any kind of profile.
822 This way of shape encoding allows us a different way to calculate “shape
823 regularity”. In this case, it is meant as a measure of the variations in corners,
824 edges and faces, or, simply a measure of directional changes in the object’s con-
tour. Consequently, and following Saragusti et al. (2005) we can quantify the
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825
826 degree of roughness of a given contour based on the degree of concavity of

827 this contour. The intuition at the root of this measure of roughness stems from
828 the following observations: the smoothest closed curves are convex. Any further
829 structure of the curve is associated with the appearance of concave sections: the
830 more there are the more complex and rough the curve is. Thus, roughness can
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831 be determined by the frequency and amplitude of the transitions between con-
832 vex and concave sections along the curve. These transitions occur at inflection
833 points. The concavity can be defined as the sum of all the deflections along con-
834 cave sections. It should be borne in mind that roughness is a relative term, and it
depends upon the scale at which it is defined and measured. A given line may look
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835
836
837
838 Table 5.1 Curvature advanced descriptors (from Bello and Soligo 2008)
839
• Slope angles (σ1 and σ2): the angles between the slopes S1 (left) and S2 (right) of the
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840
cutmark and the unaffected bone surface (R).
841
• Opening angle of the cutmark (δ): the angle between the slopes S1 and S2 (δ = 180◦ –
842 [σ1 + σ2])
843 • Bisector angle (BAC): angle of the bisector of the opening angle of the cutmark relative to
844 the unaffected bone surface (expected to reflect the impact angle of the tool relative to the
bone surface; γ = σ2 + (180◦ – [σ1 + σ2])/2)
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845
• Shoulder heights (SH, left and right): the height of the shoulders formed on either side of
846
the cut (SH = sin β L, where L is the distance from the tip of the shoulder to the
847 corresponding intersection between the cutmark profile and regression line R, and where β
848 is the angle between L and R).
849 • Floor radius: the radius of a circle fitted to the floor of the cutmark profile, with the floor
defined as lying between the two points where the profiles of the left and right slopes start
850
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to converge (i.e., where the cutmark profiles start to diverge from the regression models S1
851
and S2).
852 • Depth of cut (DC): the perpendicular depth of the cut relative to the unaffected bone
853 surface (DC = sin α - H, where H is the distance from the lowest point of the cutmark
854 profile (point A) to the intersection between the left slope of the cutmark profile and the
regression line R (point B), and where a is the angle between H and R).
855
112 J.A. Barceló
856 relatively smooth at one scale, and rougher as the resolution increases. Therefore,
857 setting the scale at which the roughness is to be measured is a prerequisite in
858 any quantitative assessment of its degree. Archaeological considerations dictate the
859 choice of scale, and may change when different properties are to be addressed.
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860 It is convenient to set the scale by assessing the size of an arc along the curve,
861 within which variations of the curvature are irrelevant and can be smoothed away.
862 The fluctuations of the curve that occur on an interval of smaller length are
863 damped out.
864
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865
866 Alternative Descriptors of Global Shape

867
868 The method of archaeological curvature estimation and measuring is not without
869 its limitations. One limitation lies in the requirement that (a) the curve be placed
in a rectangular coordinate system in such a way that a line vertical to the x-axis
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870
871 will intersect the curve at most one and (b) the curves to be compared to each
872 other have a comparable way for orienting them in the coordinate system. Closed
873 curves, clearly contradict the first requirement. However, if the outline is divided
874 into segments based on the corners of the object outline, each segment can be ori-
875 ented with its own coordinate system. On the other hand, if the orientation of the
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876 object’s profile is changed, then the polynomial curve parameters will assume dif-
877 ferent values even though there is no change in the form of the curve. One needs
878 a “natural” orientation of the curves with respect to the coordinate system, and
879 whether there is, a natural orientation depends on how the archaeological artifact
was conceptualized and produced in the past, and the level of its preservation in the
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880
881 present. It is perfectly possible that the object has no predefined orientation because
882 the shape responds to local attributes. Re-sharpening a lithic tool by subsequent
883 flake removal (retouching) would be an example (Read 2007). Additional problems
884 have been remarked by van Maaten et al. in the case of uncentered objects with
excessive wear, irregular shape and/or edges, deterioration, and so on (Maaten et al.
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885
886 2006; see also Mom and Paijmans 2008). If an object does not have exact rota-
887 tional symmetry (e.g. hand shaped vessels, arrow points, flint axes, etc.), one may
888 obtain several different shapes by drawing a single object from different angles,
889 in which case the calculation will produce a measure for the internal symmetry of
the object. On those cases, edge-based statistical tend to provide insufficient shape
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890
891 description.
892 Some of these problems can be solved using alternative procedures for global
893 geometry description. For instance, it has been argued that the perimeter of complex
894 entities may have a fractal nature. A fractal analysis of shape should therefore an
895 alternative that merits to be explored.
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896 The fractal dimension is a statistical quantity that gives an indication of how
897 completely a fractal appears to define an interfacial boundary, as one zooms down
898 to finer and finer scales. Consequently, it can be understood as the rate at which
899 the profile or contour of an object increases as the measurement scale is reduced
900 (Russ 2002; Rovner 2006). There are different ways to measure it. Perhaps the most
901 widely used fractal measurement tool is the so-called Richardson plot. In analysis
902 of an irregular contour curve, the effects of varying resolution can be mimicked by
903 walking a real or virtual map divider along the curve, varying divider step size to
904 obtain different estimates of curve length. Setting a pair of dividers to a known dis-
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905 tance, the user starts at some point on the boundary and strides around the perimeter.
906 The number of steps multiplied by the stride length produces a perimeter measure-
907 ment. As the stride length is reduced, the path follows more of the local irregularities
908 of the boundary and the parameter increases. A logarithmic plot of results shows
909 degrees of curve wandering at specific scales by the plot slope values associated
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910 with particular step sizes. A plot or plot segment that is linear shows statistically
911 self-similar character over the relevant range of scales. In many cases, Richardson
912 plots from natural curves show abrupt shifts in slope with changing resolution. It
913 means the presence of contrasts in curve geometry between two ranges of scale, and
914 suggests scale thresholds in formative process. The slope of the regression line on
the plot gives the fractal dimension, and it can be used to measure the relationship
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915
916 between the measuring scale and the attribute of the image being measured (Jelinek
917 et al. 1998).
918 This method of shape description has been used in the analysis of prehistoric
919 lithic tools by Kennedy and Lin (1988) and by Brown (2001). Fracture lines charac-
920 terizing the contour of a broken fragment can also have a fractal nature (Pande et al.
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921 1987; Brown and Wiltschey 2003a; Brown et al. 2005; Leitao et al. 2005). Some
922 other relevant applications are in the domain of the shape of spatial features, like
923 territorial limits and the like. In fact, the body of literature in modern geography on
924 the fractal characteristics of the global shape of human settlement is significant and
growing. The shape of several different kinds of modern and ancient settlements has
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925
926 been shown to be fractal in form. A number of investigators have studied the bound-
927 aries of modern cities and prehistoric settlements and concluded that they are fractal
928 curves that can be modeled by a process called diffusion limited aggregation. In
929 any case, not all settlement patterns should be fractal. For example, the orthogonal
grid pattern of an archetypal Roman city tends to be Euclidean rather than fractal,
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930
931 although its fractality depends on the details of the grid squares (Puente and Castillo
932 1996; Willemin 2000; Bardossy and Schmidt 2002; Brown and Witschey 2003b).
933 Fractal dimension produces a single number that summarizes the regularity of
934 “roughness” of the contour line. However, there can be an unlimited number of visu-
ally different boundary shapes with the same fractal dimension or local roughness.
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935
936 The most usual alternative is then harmonic analysis, a shape unrolling method that
937 converts the observed boundary to a function of the angles of radii drawn from the
938 object’s centroid until the points delimiting the contour. In both cases, the contour
939 is described in terms of its polar coordinates: a point on the outline is located by the
940 angle from a reference line and the distance from the center of the polar coordinate
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941 system (the reference point) along a ray at that angle to the outline. By using a fixed
942 set of angles (even unequal angles), the measurements can be reduced to the length
943 of the rays. The combination of increasing accuracy of the representation through
944 increasing the number of rays or diameters (widths) and then treating each ray or
945
114 J.A. Barceló
946 width as a variable has the drawback that it introduces redundancy into the sys-
947 tem of measurements. Two methods of harmonic analysis have been most used in
948 archaeology: Fourier analysis and Hough Transform.
949 The power of the Fourier representation of a curve lies in the ability to represent
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950 even irregular curves simply by including a sufficient number of terms with nonzero
951 coefficients in the Fourier series. The problem is that corners are difficult to approx-
952 imate with the waveform represented by sine and cosine functions; hence a large
953 number of terms are needed to force the curve represented by the Fourier series to
954 represent a corner. Fourier decomposition has been applied to archaeological data by
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955 Gero and Mazzullo (1984), Cardillo (2005), Cardillo and Charlin (2007), to inves-
956 tigate the outlines of flake tools; by Karasik et al. (2005) and Goel et al. (2005) in
957 the analysis of asymmetrical deformations of pottery vases; by Forel et al. (2009)
958 to study the morphometry of European Bronze Age (2300–800 BC) bronze tools;
959 and by Peterson (1992) to estimate shape differences among the limits of particular
fields during Roman Times.
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960
961 Alternatively, transformation into Hough spaces have been used to find and
962 understand alignments of points along a contour and fit some kinds of shapes,
963 although it is necessary in most cases to have a pretty good idea of the type of
964 line or other arrangement that is to be fit to the data. Hough transform algorithms
965 use the polar coordinate representation of the contour consisting of radius length
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966 and angle. Each point in the real-space image produces a sinusoidal line in Hough
967 space representing all possible lines that can be drawn through it. Each point in
968 Hough space corresponds to a line in real space. The superposition of the sinusoids
969 from several points in real space causes the lines in Hough space to add together
where they cross. These crossing points identify the contours that go through the
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970
971 points in the real-space (Russ 2002). Generalized Hough transforms was one of
972 the first algorithms to be explored for the automatic documentation of pottery pro-
973 files (Lewis and Goodson 1990; Durham et al. 1990). Modern applications include
974 Kampel and Melero (2003) on pottery profiles, and Keogh et al. (2009) have used the
same approach for describing the shape of petroglyphs and pictographs in rock-art.
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975
976
977
978 Salient Points as Shape Encoding

979
In many applications of archaeological shape analysis, we need to assume that the

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980
981 shape of any object is effectively captured by a finite subset of its contour points, in
982 such a way that we do not need the entire geometry of the contour for understanding
983 it. Selected points along the contour should correspond to salient points. “Salient”
984 means that the point is in some way “special” or “distinct from its neighbors”.
985 Attempts to define what a salient point is suffer from the problem that an isolated
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986 point cannot be special by itself, but only in comparison to its neighbors along the
987 object’s contour. Hence, saliency makes sense only with respect to the surround-
988 ings. In the literature there have been various synonyms for these selected contour
989 points: inflection points, vertices, anchor points, control points, profile points, sam-
990 pling points, key points, facets, nodes, markers, fiducial markers, and so long. The
991 most usual way to refer to them is the term landmark (Adams et al. 2004; Slice
992 2007).
993 Dryden and Mardia (1998) define a landmark as a point of correspondence on
994 each object that matches between and within populations. Landmarks with the
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995 same name, homologues in the purely semantic sense, are presumed to corre-
996 spond in some sensible way over the forms of a data set. These authors consider
997 three basic types of landmarks: functional, mathematical and pseudo-landmarks (or
998 semilandmarks).
999
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1000 (1) A functional landmark (or “anatomical”, as it is used in paleontology and biol-
1001 ogy) is a point assigned by an expert that corresponds between objects in some
1002 explanatory meaningful way. Functional landmarks designate parts of an object
1003 that corresponds in terms of functional derivation and these parts are consid-
1004 ered homologous. From a production viewpoint, for instance, a corner along a
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1005 digitized contour and formed by the intersection of adjacent sides involves a
1006 discontinuity representing the intersection of two different formation processes
1007 (Leyton 1992; Read 2007; Collins 2008). For instance, in pottery studies, the
1008 transition between the neck and body of a vessel is a functional landmark (Rowe
1009 and Razman 2003). Considering the process of making a pottery vessel from
1010 the original amount of clay, Kampel and Sablatnig (2007) have suggested, in
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1011 the case of pottery a series of functional landmarks (Table 5.2).
1012 In the case of lithic tools, S. Crompton (2007) has selected a series of func-
1013 tional landmarks to describe attributes, which affect the point usefulness as
1014 throwing or thrusting spear. Based on these functional principles, Crompton has
selected 11 landmarks on specific points of the tool: on the tip, on the maximum
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1015
1016 length and width, on the hafting edge, etc.

1017 In other examples of lithic tools, it has been argued the impossibility of mark-
1018 ing functionally homologous landmarks because of the lack of such knowledge
1019
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1020
1021
Table 5.2 Functional landmarks for pottery studies (from Kampel and Sablatnig (2007))
1022
1023 • SP, starting point: in the case of vessels with a horizontal rim: innermost point, where the
1024 profile line touches the orifice plane.
• OP, orifice point: outermost point, where the profile line touches the orifice plane.
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1025
• IP, inflexion point: point, where the curvature changes its sign, i.e. where the curve changes
1026
from a left turn to a right turn or vice versa.
1027 • MI, local minimum: point of vertical tangency; point where the x-value is smaller than in
1028 the surrounding area of the curve.
1029 • MA, local maximum: point of vertical tangency; point where the x-value is bigger than in
the surrounding area of the curve; the y-value refers to the height of the object (e.g.
1030
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MA(y)).
1031
• CP, corner point: point where the curve changes its direction substantially.
1032 • BP, base point: outermost point, where the profile line touches the base plane.
1033 • RP, point of the axis of rotation: point where the profile line touches the axis of rotation.
1034 • EP, end point: point where the profile line touches the axis of rotation; applied to
incomplete profiles.
1035
116 J.A. Barceló
1036 (Cotterell and Kaminga 1992). It seems easier in the case of metallic objects,
1037 like iron brooches (Small 1996; Le and Small 1999).
1038 (2) Mathematical landmarks are points located on an object according to some
1039 mathematical or geometrical property of the figure, irrespective of its func-
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1040 tional value. It may be a matter of subjective evaluation if such points have
1041 or not functional value, given what we have suggested previously about the
1042 causal nature of corners and inflection points along a contour (Leyton 1992).
1043 Mathematical landmarks can be defined, in fact, as any set of points that are
1044 characterizable and searchable upon a surface. They can be based either on
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1045 geometric properties of profile (maxima of curvatures, umbilic points, crest
1046 lines) or, if available on the retinal properties of the underlying images – i.e.,
1047 color, luminance (Caldoni et al. 2006; Maiza and Gaildrat 2006). A suggested
1048 method is based on finding points of inflection on a curve at varying levels of
1049 detail (i.e., curvature zero-crossings). In general, points on the outline of a curve
where there is an abrupt change in slope are usually considered as control points
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1050
1051 (Rowe and Razman 2003; Goel et al. 2005). Control points are chosen on the
1052 pot’s profile by removing the irrelevant shape features and keeping the rele-
1053 vant ones. This is achieved by iteratively comparing the relevant measure of all
1054 points on the profile. For each of these iterations, the vertex that has the lowest
1055 relevance measure is removed and a new segment is established by connecting
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1056 the two adjacent points. A point with a higher relevance value signifies that it
1057 has a larger contribution to the shape of the curve. Goel et al. (2005) give the
1058 following formula to compute the relevance of a point on the curve.
1059
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1060
1061
1062
1063
1064
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1065
1066 S, S1, S2 are points on the profile of a curve

1067
K(S) is the relevance of the point S to the shape of the curve
B is the turn angle of S with points S1 and S2 is the length between S and either of the
1068
other points.
1069
Martín et al. (2009) take a somewhat different approach to the selection of

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1070
1071 salient, by simplifying a scanned point cloud.

1072 (3) Unfortunately, landmarks cannot always be easily defined and located with pre-
1073 cision. In any case, such a process could not be straightforwardly when the
1074 complete objects silhouettes do not fully correspond to their original shapes.
1075 In some case, the cutting edge may have been drastically reworked due to
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1076 repetitive sharpening operations accentuating the curvature by plastic defor-

1077 mation. Pseudo-landmarks (also called semilandmarks) are constructed points
1078 on an object, located either around the outline or in between anatomical or
1079 mathematical landmarks. For instance, when we consider equally spaced points
1080 along a contour. We can select any points (usually 100 or more), provided we
1081 are sampling the shape with roughly uniform spacing (for the general proce-
AQ8 1082 dure, see Belongie et al. 2002). M. Cardillo makes the important point that we
1083 need a minimum of two mathematical or functional landmarks (for instance
1084 both extremes of the symmetry axe) to fix a series of pseudolandmarks for
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1085 comparison purposes (Cardillo 2006. See also Otárola-Castillo et al. 2008).
1086 An example is the configuration of 51 landmarks employed for the 3D anal-
1087 ysis of nuclei surface morphology by Lycett et al. (2006). In the case of pottery,
1088 and related with pseudo-landmarks measuring a profile at specific intervals,
1089 Maaten et al. (2006) suggests a related technique called shape context. In a
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1090 shape context representation, a shape is represented by a number of points that
1091 is sampled from the boundary of the shape contour. The points are described as
1092 shape context descriptors. Shape context descriptors describe the distance and
1093 angle of a point to all other points in a discretized log-polar space (see also
1094 Maaten et al. 2009).
A good example of the object identification possibilities offered by these
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1095
1096 procedures for shape encoding has been provided by E.S Lohse and his
1097 Archaeological Auto Classification System (Lohse et al. 2004), which uses neu-
1098 ral network technology to classify stone arrow points based on the geometrical
1099 information of their contours, described in terms of pseudolandmarks.
1100
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1101
1102 Geometric Morphometrics analyzes the variability in the relative position of

1103 landmark-coordinates said to be “homologous” in terms of the phenomenon being
1104 studied. The procedure begins by determining a configuration in the set of func-
tional, mathematical or pseudo-landmarks on a particular object. The configuration
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1105
1106 matrix X is the k x m matrix of Cartesian coordinates of the k landmarks in m

1107 dimensions. We shall consider a shape space obtained directly from the landmark
1108 coordinates, which retains the geometry of a point configuration. The configuration
1109 space is the space of all possible landmark coordinates. Consequently, the selec-
tion of the appropriate coordination system is of paramount importance. A suitable
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1110
1111 choice of coordinate system for shape should be invariant under translation, scaling
1112 and rotation of the configuration. Among the most common, bookstein coordinates
1113 are the remaining coordinates of an object after translating, rotating and rescaling
1114 the baseline to preserve the original geometry (Bookstein 1991; Dryden and Mardia
1998; Adams et al. 2004; Slice 2007).
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1115
1116 Between landmark points, we can define the existence of paths connecting any
1117 two-neighbor points. In this way, parametric curves are connected sets. Two types of
1118 paths are usually considered: polygonal and continuous. A polygonal path is defined
1119 as a sequence of connected straight-line segments, i.e., straight segments sharing
1120 their extremities. In case there is a polygonal path with all its points contained in
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1121 the set, and linking any two points, then this series of landmark points is polygon
1122 connected. A polygonal path corresponds to a particular case of a continuous path.
1123 A series of landmark points is path wise connected in case any two of its points can
1124 be joined by a continuous path entirely contained in this particular set of landmark
1125 points.
118 J.A. Barceló
1126 Shape differences between the objects described by landmarks and paths between
1127 landmarks are demonstrated by superimposing the landmark configurations accord-
1128 ing to some criteria or by making them to coincide (Slice 2007). A geometrical
1129 morphometric analysis based on Generalized Procrustes Superimposition Analysis
OF
1130 (GPA) is applied to remove variation in location, orientation and scale. This proce-
1131 dure specifically addresses shape differences apart from size. After superimposition,
1132 GPA creates a “mean” or average shape from which the variability in overall
1133 landmark positions can be quantified. This is analogous to scaling and rotating
1134 photographic negatives of the samples as represented by the landmarks and super-
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1135 imposing their corresponding landmarks to obtain an overall best fit. After fitting
1136 the landmark configurations to the computed mean shape, the shape differences are
1137 recorded as residuals from this mean shape.
1138
1139
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1140
1141 Shape at the Third Dimension

1142
1143 In all precedent references, shape has been referred as a bi-dimensional geometri-
1144 cal parameter, that is, as a curve. However, archaeological evidences, as any material
1145 object are three-dimensional entities, and their bi-dimensional contour is but a crude
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1146 surrogate of their real shape. Interfacial boundaries of archaeological material evi-
1147 dences (ceramic vessels, bones, stone tools, ancient walls, occupation floors, burials
1148 or pit holes) have the appearance of irregular surfaces, more than curves, and we
1149 should take into account that due to “the loss of dimension”, 2D images of 3D
objects suffer from ambiguities.
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1150
1151 We may need then more sophisticated mechanisms for analyzing archaeologi-
1152 cal shapes in all their complexity. As objects become more complex in terms of
1153 variety of shape and changes in curvature, they become more difficult to quantify
1154 and analyze. We need then more sophisticated mathematical techniques to repre-
sent complex surface geometries of intrinsically three-dimensional objects (Hermon
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1155
1156 2008). Some of the 2D shape descriptors examined in previous sections can be
1157 adapted to be used in the 3D case. Examples of global descriptors would be the
1158 statistical moments of the volume of the model, volume-to-surface ratio, or the
1159 Fourier transform of the volume of the shape. Landmarks can also be defined three-
dimensionally (Lycett et al. 2006). Other global features for 3D shape are bounding
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1160
1161 boxes, cords-based, moments-based and wavelets-based descriptors, convex-hull

1162 based indices like hull crumpliness (the ratio of the object surface area and the sur-
1163 face area of its convex hull), hull packing (the percentage of the convex hull volume
1164 not occupied by the object), and hull compactness (the ratio of the cubed surface
1165 area of the hull and the squared volume of the convex hull).
UN
1166 Rather than working with 3D enhancements for 2D originally designed methods,
1167 al alternative approach is based on building a grid mesh joining the vertices, edges
1168 and faces defining the shape of a polyhedral object (Georgopoulos et al. 2008). The
1169 faces usually consist of triangles, quadrilaterals or other simple convex polygons.
1170 Here, a polygonal mesh acts as a 3D boundary representation (B-rep) describing the
1171 archaeological object as a set of surfaces that separate the object interior from the
1172 environment and is a geometrical approximation of a curved object surface.
1173 Polygonal meshes are usually displayed as wire-frames, built by specifying each
1174 edge of the physical object where two mathematically continuous smooth surfaces
OF
1175 meet, or by connecting an object’s constituent vertices using straight lines or curves.
1176 The object is projected onto the computer screen by drawing lines at the location of
1177 each edge. Since wireframe renderings are relatively simple and fast to calculate,
1178 they are often used in cases where a high screen frame rate is needed. When greater
1179 graphical detail is desired, surface textures can be added automatically after com-
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1180 pletion of the initial rendering of the wireframe. This allows the designer to quickly
1181 review changes or rotate the object to new desired views without long delays asso-
1182 ciated with more realistic rendering. The study of such polygon meshes is a large
1183 sub-field of computer graphics and geometric modeling (Bertolotto et al. 1997;
1184 Howard 2005; Movchan and Movchan 1998; Ghali 2008). Different representations
of polygon meshes are used for different applications and goals.
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1185
1186 The use of thin-plate splines can be considered as another surface method of
1187 3D shape encoding. A thin-plate is a thin sheet of some stiff material with infi-
1188 nite extension. When specific control points along the plate are displaced, then the
1189 plate undergoes a deformation minimizing the total bending energy E implied by
1190 the transformation (Costa and Cesar 2001).
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1191 Neural networks can be used to build connected surfaces from edge and vertices
1192 input information. In general, such programs rely on coordinate information (x, y, z
1193 or length, width, height), that is to say, the spatial location of interfacial boundaries
1194 for interpolating a geometric mesh (see Gu and Yan 1995; Piperakis and Kumazawa
2001; Barceló 2008). The advantages of using neural network for such a task are
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1195
1196 diverse (Peng and Shamsuddin 2004):

1197
1198 • Neural networks with backpropagation technique are able to estimate the depth
1199 (z) of an object with higher accuracy than other methods. It also means that neural
networks are able to reconstruct object from 2D image to 3D after training.
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1200
1201 • This type of reconstruction is able to produce more points of an object or sur-
1202 face. Therefore, a neural network is able to reconstruct more complex object with
1203 smoother surface.
1204 • Even with scattered or unorganized data of an object is provided, neural networks
are able to regenerate the object when outliers are removed and the smoothness
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1205
1206 of the surface is maintained.

1207
1208 Nowadays, laser scanned data can be easily integrated into specific soft-
1209 ware that translates clouds of points defined by x, y, z coordinates into the
1210 polygon mesh (Petersen et al. 2006; Karasik 2008). There are computer programs
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1211 like PolyWorks (www.innovmetric.com), 3DReshaper (www.3dreshaper.com),

1212 PRISM 3D (www.quantapoint.com), Geomagic (www.geomagic.com), RapidForm
1213 (www.rapidform.com), among many others. In this way, scanned points can be
1214 “automatically” connected together through some algorithm, which results in a
1215 “mesh-frame” from which a smooth surface may be extrapolated. Although there
120 J.A. Barceló
1216 is a huge quantity of algorithms and procedures for building the polygonal mesh
1217 from scanned 3D data, in most archaeological applications the particularities of
1218 the interpolation method are not usually took into account. It is impossible to give
1219 here an overview of current applications in archaeology. For examples of polygonal
OF
1220 meshes as 3D models of archaeological objects and entities, the reader is referred to
1221 the annual Computer Application in Archaeology series (www.caaconference.org)
1222 or the Virtual Reality, Archaeology and Cultural Heritage Research Network
1223 (www.epoch-net.org).
1224 Archaeological models created with polygon meshes must store different types of
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1225 elements. These include vertices, edges, faces, polygons and surfaces. Here, a vertex
1226 is a position along with other information such as color, normal vector and texture
1227 coordinates. An edge is a connection between two vertices. A face is a closed set of
1228 edges, in which a triangle face has three edges, and a quad face has four edges. A
1229 polygon is a set of faces. Surfaces are required to group smooth regions. They are
needed to group smooth parts of a mesh just as polygons group 3-sided faces.
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1230
1231 Once adequately formatted, the mesh model can be imported into analytical
1232 programs. The user can delineate regions by identifying high and low geometri-
1233 cal points on the wireframe, given a user-specified sensitivity level, which can be
1234 adjusted to obtain finer or coarser region definition. It is possible then to merge and
1235 split regions to transform them into analytically meaningful surfaces.
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1236 3D meshes allow capturing the usual linear measurements (i.e., length, width,
1237 thickness), but also alternative geometric attributes that researchers traditionally
1238 have had difficulty to measure accurately on real objects. Among them, we can
1239 mention the coordinates of the center of mass (CM) and the coordinates of the cen-
ter of the enclosing cube (CMEC), absolute object symmetry, a surface’s area, and
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1240
1241 the average angle at which surfaces intersect (Cignoni et al. 2001; Riel-Salvatore
1242 et al. 2002; Simon et al. 2002; Tsirliganis et al. 2002; Ozmen and Balcisoy 2006;
1243 Grosman et al. 2008). The use of 3D polygon meshes also makes possible new mea-
1244 sures based on topology and global or local changes in curvature that define the
shape of the original object. Volume is another parameter that can be easily calcu-
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1245
1246 lated from the polygonal mesh, although multiple lines of inference are needed to
1247 define adequately systemic function from vessel shape. Even though the attribute
1248 “capacity” for any entity working as a container (from a vessel to a building)
1249 has been relatively neglected in archaeology, it has maximum importance. Varied
methods that determine a container volume exist in the literature. Most of these
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1250
1251 techniques are relevant only when the entity analyzed is highly standardized in terms
1252 of geometry. Since the very first computer applications in archaeology, volume has
1253 been calculated using formulae derived from geometry to assess the general shape of
1254 the container. Primary examples of these methods have been discussed by Castillo
1255 et al. (1968), Ericson and Stickel (1973). Nelson (1985: pp. 312–131) estimated
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1256 pottery vessels volume through the calculus method of “stacked cylinders,” which
1257 envisions the vessel as divided horizontally into a series of regularly sized slices.
1258 Such slices represent the diameters of very thin cylinders. Stacked one on top of
1259 another, the sum of these cylinders represents the entire vessel’s volume.
1260
1261 In general, all those approaches can be classed among the very first geometri-
1262 cally based shape analysis in archaeology although these methods can generally
1263 render a useful value for a container capacity; they are usually not very accurate if
1264 the measured entity is irregular in shape and little standardized. Preliminary essays
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1265 of measuring volume capacity by computer were those by M. Smith (1983, 1985),
1266 who estimated a pot capacity by integration as the volume of the solid of revolution
1267 formed by revolving the profile curve around the x-axis (see also Senior and Birnie
1268 1995). Today we use the analytical possibilities of geometrical analysis software for
1269 taking such measurements. However, we should take into account that a polygonal
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1270 mesh is a boundary representation, defining the solid in terms of its “external” sur-
1271 face. In the case of computationally measured volume capacity, care must be taken
1272 to compensate for real world wall thickness, then, because the volume could be
1273 overestimated, even if the geometry is well suited to the container shape.
1274 Mara and Sablatnig (2005) suggest measuring the curvature of a surface based on
the geodesic distances among points in the neighborhood of some particular vertex
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1275
1276 of concave or convex region. Simon et al. (2002) show how once a 3D mesh has been
1277 calculated a 2D curvature mode can be estimated to quantify and measure various
1278 properties of the B-spline curve such as the number of inflection points, and the
1279 symmetry of the curve. Positive to negative crossings of the horizontal axis would
1280 correspond to inflection points in the profile: changes from convexity to concavity
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1281 or vise versa. A curvature plot corresponds to the complete vessel profile.
1282 The use of polygonal meshes to represent 3D solids allows the objective deter-
1283 mination of the best fitting axis of rotation from the 3D data and many profile lines
1284 can be extracted, using sections through the best symmetry axes. The large num-
ber of profiles enables the archaeologist to study the uniformity of the vessel by
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1285
1286 performing a correlation analysis based on the curvature function of each profile.
1287 The correlation analysis provides a quantitative measure for the vessel uniformity,
1288 which has direct bearing on the production technology and its development in antiq-
1289 uity or prehistory (Simon et al. 2002; Mara et al. 2004). In the same way, it is now
possible to deduce the deformations of wheel-produced pottery. A quantitative mea-
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1290
1291 sure of the deformations can be obtained by using the polar representation of the
1292 curves that are the boundaries of the horizontal sections. About this subject, in the
1293 case of pottery see archaeological applications in Saragusti et al. (2005), Karasik
1294 (2008), in the case of lithics see: Nowell et al. (2003), Bird et al. (2007); in the case
of coins see: Zaharieva et al. (2008). Doi and Sato (2005) suggest using morphing
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1295
1296 or warping techniques to study how a model can be deformed into another model.
1297 A systematic study of these deformations may reveal the technological flaws that
1298 induced them, and might possibly be used to characterize workshops methods and
1299 production patterns.
1300 Alternatively, the medial axis or skeleton representation of Blum (1973) (see also
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1301 Leymarie 2003 for archaeological applications) has been promulgated as a generic
1302 representation for describing 3D shapes from complex solids, such as those crafted
1303 by humans. The approach has shown great potential in object recognition, in solid
1304 modeling for designing and manipulating shapes, in organizing a cloud of points into
1305
122 J.A. Barceló
1306 surfaces, for volumetric mesh generation, in path planning, numerical tool machin-
1307 ing, animation, etc. On the one hand, medial or skeleton-based representations are
1308 based on a spatial layout, which is thin and tends to lie midway inside or outside
1309 the object’s contour or profile, with a branching structure related to the topology
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1310 of the object, forming a “stick figure.” The main reasons why Blum’s shape rep-
1311 resentation approach has been viewed with great promise as a universal model for
1312 shape are:
1313
1314
(i) A medial representation is an intuitive one to represent elongated objects,
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1315
such as anthropomorphic forms, which are built upon a real skeletal frame.
1316
(ii) It can also encode the “blobbyness” of a shape, that is, the varying width of
1317
forms, by its radius function.
1318
(iii) Important contour features, such as curvature extrema and ridges, are made
1319
explicit by the medial branch “tips”.
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1320
(iv) Both the interior and exterior regions of space can be described and seg-
1321
mented as a function of their relative closeness to the object’s outline.
1322
(v) A partitioning of shape is made possible by combining width and elongation
1323
properties, where, e.g., different skeleton branches relate to different object
1324
parts.
1325
(vi) A hierarchy of scales is also “built-in” via this combination of spatial and
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1326
time-like properties, i.e., smaller features can be distinguished from larger
1327
ones and ranked accordingly.
1328
(vii) The skeleton representation is complete, i.e., unless one starts pruning its
1329
branch structure, all of the object is represented, from the small bumps along
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1330
the boundary to the large engulfing of its main parts – completeness ensures
1331
that an exact reconstruction is always possible.
1332
(viii) The skeleton representation addresses the issue of dimensional reduction, i.e.,
1333
it maps the entire object and the space it occupies into a thin set and has the
1334
potential for high data compression and information concentration, a property
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1335
explored in particular in the domain of Computer-Aided Design (CAD).
1336
1337
1338
1339 From Fragment to Object

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1340
1341 Archaeological evidences are fast always incomplete: not all past material things
1342 have remained until today. Even more, most of those few items from the past that
1343 we can observe today are broken. The only possibility to “see what cannot be seen”
1344 is as a generalization of fragmented observable data, representing partially the view
1345 of a lost physical world reality. This can only be done by generating simulated data
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1346 (Barceló 2008). That is to say, archaeologists need a complete “model” of the orig-
1347 inal entity the fragment comes from in order to complete damaged input data. The
1348 general idea is to use a hypothetical shape model of the thing, and to fit it to the
1349 incomplete input data to simulate what is not preserved. We can use the following
1350 kinds of knowledge:
1351 • If all we know to simulate missing data are analogies and some other “simi-
1352 lar” cases, then we can build a qualitative model. This is the case of ancient
1353 buildings. In most cases, preserved remains do not shed light on the structure of
1354 vertical walls, which therefore remain unknown. In general, the reconstruction of
OF
1355 the shape of archaeological badly preserved ancient buildings is largely based on
1356 these types of sources:
1357 ◦ Pictorial evidence from plans and photographs of the building before it became
1358 a ruin.
1359 ◦ Evidence shown by contemporary buildings in other neighboring places or
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1360 culturally related areas, which gives clues as to likely construction methods
1361 and spatial forms.
1362
• When old drawings and photographs are not available, external data can be
1363
estimated from ethnographic records.
1364
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1365
The problem in all those cases is that analogical knowledge is not being added to
1366
the model in a systematic way. That is to say, the hypothesis of the original shape
1367
is not organized in rules and facts, but selecting additional information in a sub-
1368
jective way, using what the illustrator wants, and not what the archaeologist really
1369
needs.
1370
In ideal terms, we should follow the rule: “The most similar is taken for the
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1371
complete simulation”. The procedure is as follows: we transform perceived data as
1372
a geometric data set (shape, size, texture), and we try to interpret the visual type,
1373
assuming some dependent preference function. Once the type is decided, the closest
1374
fit is determined using different numerical techniques (Barceló 2002).
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1375
1376 IF b (x,y,z) FITS THEORY,

1377 And MODEL A IS A PROJECTION OF THEORY,
1378 THEN b (SHAPE) DERIVES FROM MODEL A.
1379 For instance:
IF the geometric model of (x) has geometric properties A,B,C,
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1380
1381 THEN (x) is an example of MODEL ABC.

1382 IF (x) is an example of MODEL ABC,
1383 And (x) has not property D,
1384 THEN JOIN property D to the geometric model of (x).
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1385
Where JOIN is an operator implemented as a command able to add some geometric
1386
unit to those already present in a preliminary model of the partial input. As a result,
1387
some new visual features (property D) are added to the geometrical model of the
1388
original data.
1389
To deal with uncertain knowledge, a rule may have associated with it a confidence
1390
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factor or a weight. For instance:

1391
1392 IF the geometric model of (x) has geometric properties A,B,C but not
1393 properties
1394 D,E,
1395 THEN (x) is an example of MODEL ABC (with probability 0.7).
124 J.A. Barceló
1396 IF the geometric model of (x) has geometric properties A,B,C, D,E,
1397 THEN (x) is an example of MODEL ABC (with probability 1.0).
1398 IF (x) APPROXIMATELY fits MODEL ABC,
1399 THEN VISUALIZE the incomplete parts of (x) using A,B,C properties.
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1400
1401 Archaeologists need to build the model first, and then use it for simulating the
1402 unseen object. They create a geometric model of the interpreted reality, and then
1403 they use information deduced from the model when available visual data fit the
1404 model. In most cases, they create “theoretical” or “simulated” shape models. Here
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1405 “theory” means general knowledge about the most probable shape of the object
1406 to be simulated or prior knowledge of the reality to be simulated. An example of
1407 this approach is provided by De Napoli et al. (2001). The reconstructed surface of
1408 each fragment is compared with the surface of a series of virtual vessels, formerly
1409 modeled, until its position is well determined on it. This operation of matching is
repeated for every piece. Since vessels are typically surfaces of revolution, every
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1410
1411 typology of vessel is stored with a table of significant values of Gaussian curvature
1412 measured on a meridian with a suitable pitch value (about ten values for each item
1413 seems to be significant). Comparing the estimated datum with the stored one, the
1414 fragment can be matched to the proper parallel.
1415 The approach by Maiza and Gaildrat (2005) is based on the exploration of a solu-
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1416 tion space constituted by all the positions that a fragment can take, relatively to a
1417 tested shape model. To achieve this exploration, it is needed a technique to evalu-
1418 ate the distance between a particular position of the fragment and the tested model.
1419 Genetic algorithms can be used in order to determine an optimum position of the
fragment. This search is based on the previous computation of the distance to evalu-
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1420
1421 ate the quality of the fragment’s matching. Melero et al. (2003) have also developed
1422 a semi-automatic system that uses genetic algorithms to carry out classification of
1423 potteries using rim-fragments, by mimicking the method of the archaeologists (ori-
1424 entation, diameter estimation, profile extraction, drawing of the fragment, additional
measurements). The use of genetic algorithms permits a flexible approach adapted
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1425
1426 to the noise produced by the digitalization of the objects. A genetic algorithm based
1427 approach allows seeking multiple positions of a fragment in order to minimize the
1428 distance criterion between this fragment and an object model.
1429 As these examples suggest in the case of regular and symmetric objects like pot-
tery vases or containers, reconstructions are apparently easier, because we assume
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1430
1431 the pottery vessel has been created on a potter’s wheel and it is hence symmetric
1432 about the Y-axis, fitting better with a single geometric model. That means that a
1433 reconstruction can only be performed if a priori knowledge on the type and class
1434 of vessel of which the fragment is a part is provided (Kampel and Sablatnig 2003b;
1435 Mara and Sablatnig 2008).
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1436 The reconstruction of a given object seems to be most of the times a direct
1437 generalization of fragmented observable data by mathematical object description.
1438 The fragmented spatial information available can be extrapolated to complete a
1439 closed surface. The procedure may be illustrated by the mathematical ovoid and
1440 the eggshell compared. The eggshell is a solid formed by a fine closed surface.
Continuity and dynamics are bound to the shape of the eggshell, in such a way
1441 that it is possible to locate the fragments of a broken eggshell as well as to define
1442 the whole by only very few spatial measurements. It would seem that to model the
1443 geometry of an eggshell, it would be sufficient to pick from the fragments of a bro-
1444 ken eggshell some spatial world data to simulate the entire eggshell. The spatial
OF
1445 continuity and dynamics of the ovoid is included in the mathematical description, to
1446 simulate the missing information.
1447 The initial hypothesis is that the ceramic pot to which the fragment belongs was
1448 inscribed in a revolution solid. The correspondent revolution axis and the generator
1449 profile should be defined through a sequence of operations based on the geomet-
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1450 ric properties of a revolution solid. The oldest and well-approved approach is the
1451 manual method used by archaeologists for several decades. This manual approach
1452 is based on the knowledge about the production process of ceramics, manufactured
1453 on rotational plates for thousands of years. Consequently, the rotational axis is esti-
1454 mated orthogonal to the container orifice plane. In the same way, any horizontal
section of the symmetrically produced pottery container is a circle with centre con-
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1455
1456 tained in the rotation axis. If the preserved rim fragment is sufficiently complete, the
1457 archaeologist can select three points on what she/he considers as defining the rim
1458 plane or any other plane parallel to the rim. This plane is used as a first approxima-
1459 tion for the determination of the axis of symmetry of the fragment (Yao and Shao
1460 2003). In this way, by placing the fragment with some rim vestiges on a planar plate
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1461 such that the contact between the rim and the plate is maximal, the plate will then
1462 define the tangent plane, which is parallel to the plane of the original wheel. The
1463 vessel axis of symmetry is perpendicular to the tangent plane, and goes through the
1464 center of the arc generated by the contact points of the rim with the table. If we
intersect the vessel with any other plane which is parallel to the tangent plane, we
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1465
1466 would find two concentric circles (or arcs when we deal with fragments), and their
1467 common center lies on the axis of rotation. Thus, by cutting the 3D representation
1468 of the vessel by several parallel planes, we can identify the axis of rotation as the
1469 line going through the centers of the concentric circles. The tangent to the rim is just
a special case of this family of planes where the two circles, corresponding to the
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1470
1471 inner and outer surfaces, coalesce to a single circle.

1472 A broken fragment reconstruction algorithm based on the above understanding,
1473 will consist of two independent steps: the first, endeavors to find the axis of rotation
1474 as the line of centers of the circular arcs, which result from the planar intersections
(the horizontal sections method). The second makes use of the points of the rim, and
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1475
1476 attempts to find the best fitting plane tangent to the rim (the rim-tangent method).
1477 The horizontal sections method takes advantage of the entire information on the
1478 surface and therefore it is usually more stable and reliable. The rim-tangent method
1479 is used for fine-tuning, and it improves the quality factor in some cases.
1480 Nevertheless, it is in no way obvious how to determine from an isolated fragment
UN
1481 the rotational axis and the profile of the pot, even though we assume the original
1482 object was symmetric. We would need to fix the position and the orientation of each
1483 fragment so as to put the base of the object it comes from on the (x, y) plane and
1484 to align the original axis of rotation with some relevant z axis. For the fragments,
1485 the major problem is to specify their orientation respective to the z-axis. This is
a non trivial problem because a great number of fragments have no determining
126 J.A. Barceló
1486 characteristics (a piece of rim or base, presence of external or interior grooves due to
1487 handling during the molding of the object before cooking, a more important quantity
1488 of material in one of the ends of the object).
1489 In the two-dimensional (2D) case, where the complementary matching is reduced
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1490 to a “jigsaw puzzle”, many solutions have been proposed in terms of matching
1491 planar curve segments (Zhou et al. 2009). Current approaches of matching can be
1492 divided into two types:
1493
1494 1. Matching the fragments based on region: two sherds have common surface, and
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1495 the adjacency relation between two fragments is decided my matching the shape
1496 of surfaces. The ideal situation is that two surfaces are able to match.
1497 2. Re-assembling the fragments based on contour lines: two adjacency sherds have
1498 the same or similar edges; the object is restored by matching and aligning
1499 between each other.
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1500
AQ9 1501 Papaioannou et al. (2001) presented a method designed for 3D complementary
1502 matching of archaeological fragments, in the form of arbitrary polygonal surfaces.
1503 The method introduces a matching error between complementary surfaces that
1504 exploits the z-buffer algorithm. For each pair of surfaces the algorithm employs a
1505 stochastic search technique to minimize this matching error and derive the transfor-
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1506 mation that aligns the two fragments. The pair wise matching errors are used in an
1507 optimization scheme to cluster the fragments into reconstructed objects. The method
1508 uses only surface information and does not take advantage of possible boundary
1509 curve similarities. If a large percentage of the fragments are expected to participate
in valid combinations, an approach that minimizes the sum of the matching errors
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1510
1511 of the individual combinations is adopted. This approach generates more fair solu-
1512 tions but diminishes the importance of perfect matches. More specifically, the set of
1513 fragment combinations that yields the smallest cumulative error is determined using
1514 exhaustive search.
Nevertheless, not always we can estimate the complete profile to which a frag-
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1515
1516 ment belongs using only information contained in the shape of the fragment. The
1517 problem is that usually the fragments usually cover a rather small part of the full
1518 perimeter of the original vessel. The smaller the fragment, the harder it is to establish
1519 its correct positioning. Furthermore, common geometric features of the fragmented
objects, assuming a moderate surface deterioration, are the irregularity of the bro-
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1520
1521 ken surfaces and the sharp curvature transition from an intact surface to a broken
1522 one. (Papaioannou and Karabassi 2003). When a human tries to determine a pos-
1523 sible match between two solids, the correspondence between the boundary lines of
1524 the surfaces is difficult to establish, because it is not always clear what faces of the
1525 polygonal mesh belong to the original surface and which ones to the fracture.
UN
1526 Although there are photometric pairwise approaches (Sagiroglu and Erçil 2005;
1527 Boon et al. 2008; Zhou et al. 2009) based on the estimation of the photographic
1528 affinity between the visually perceived texture of neighboring fragments, existing
1529 techniques for solving the fragment reconstruction problem mainly focus on the
1530 analysis of the break curve. Kong and Kimia (2001) were among the first to propose
1531 an algorithmic solution for reassembling broken two-dimensional fragments com-

1532 paring the curvature-encoded fragment outlines. The method was based on local
1533 shape matching followed by global search and reconstruction in the case of flat
1534 objects. Leitao et al. (2001, 2002) have proposed an alternative approach based on
OF
1535 multiscale matching and constrained dynamic programming. Geometric pair wise
1536 matching is based on the estimating the original curvature of the complete profile
AQ10 1537 and has been proposed by Papaioannou et al. (2002, 2003). They used a global
1538 optimization method to minimize an error measurement of the complementary
1539 matching between two object parts at a given relative pose, based on a point-
RO
1540 by-point distance between the mutually visible faces of the objects. Huang et al.
1541 (2006) followed a feature-based approach in combination with a non-penetrating
1542 iterative closest point algorithm. Among other pair wise matching approaches for
1543 fragments of pottery surfaces based on the previous estimation of the axis/profile
1544 curves there is the one proposed by Cooper et al. (2002), Willis et al. (2003), Willis
and Cooper (2004). They have developed a method for fragment matching based
DP
1545
1546 on a Bayesian approach using break curves, estimated axes and profile curves. The
1547 procedure estimates the axis/profile curve for a sherd by finding the axially sym-
1548 metric algebraic surface which best fits the measured set of dense 3D points and
1549 associated normals, not requiring any local surface computations such as differen-
1550 tiation. Kampel and Sablatnig (2002, 2003a) proposed a matching algorithm based
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1551 on the point-by-point distance between facing outlines. As the orientation of the
1552 candidate fragments was already known, the alignment of two fragments could be
1553 achieved in a two-degrees-of-freedom search space. Zhu et al. (2006) follow a par-
1554 tial curve matching method to find candidate matching fragment pairs. The fragment
contours are represented by their turning functions and the matching segments are
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1555
1556 found by analyzing the difference curve between two turning functions directly. The
1557 curve similarity is evaluated as the residual distance of corresponding points after
1558 optimal transformation between two matching segments (see also Winkelbach et al.
1559 2008).
When the original orientation of the pottery fragment within the complete object
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1560
1561 is not known beforehand, we may use of the fact that in wheel-produced ceramic,
1562 any horizontal section of a broken fragment is a circular arc. Thus, it will be possible
1563 an approach based on the examination of the vessel horizontal sections defined by
1564 circular arcs whose centers provide an improved estimate of the axis. This algorithm
is very efficient for surfaces defined in terms of parts of cylinders or cones, that is,
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1565
1566 for surfaces whose profiles consist of straight lines. However, for surfaces formed by
1567 the rotation of a more complex profile (e.g., archaeological potsherds) this method
1568 fails (Halir 1999).
AQ11 1569 Kampel et al. (2005, 2006) suggest finding it from the identifications of circular
1570 rills on the surface of the fragments. These rills are artifacts created from tools or
UN
1571 fingers during the manufacturing process on the rotational plate. The first step of
1572 this method is to identify the inner side of the sherd where the rills are located. This
1573 can be done by measuring the curvature of the surface. This algorithm is based on
1574 the manual approach, where the sherd is tilted and rotated, so that the concentric
1575 rills can be seen as parallel lines, which are orientated horizontally. The authors
128 J.A. Barceló
1576 estimate the center of gravity and the balancing plane of the remaining vertices of
1577 the reduced inner surface. The balancing plane is described by the two longest eigen-
1578 vectors of the mean-normalized vertices, which are estimated by using the singular
1579 value decomposition. In a third step a line is fitted by minimizing the least square
OF
1580 error to the centers of the concentric circles with the minimum variance. The fitted
1581 line is used as estimated rotational axis. It is tilted orthogonal towards the balance
1582 plane to find the best line fit. After each tilt, the centers of the concentric circles are
1583 estimated. For these centers, different lines are fitted. The line with the best fit is
1584 chosen as the rotational axis (see also Pires et al. 2007).
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1585 The vectors of unit length perpendicular to a surface are referred to as the nor-
1586 mal vectors. Vectors normal to the fracture surfaces or to other features violating the
1587 assumed axial symmetry are not perpendicular to a surface of revolution, nor do they
1588 intersect at the axis of symmetry. Thus, the line minimizing the squared distances to
1589 the normal vectors defines the axis of symmetry. This fact can be used effectively to
eliminate the irrelevant parts of the scanned surface. Grosman et al. (2008) (see also
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1590
1591 Karasik and Smilansky 2008) have suggested studying the distribution of perpen-
1592 dicular vector normals on the surface of the object to determine symmetry planes
1593 which estimate the object position in 3D space in terms of surface tensor calcula-
1594 tions. Although very effective in many cases, Mara and Sablatnig (2005) Kampel
1595 et al. (2006) consider that the method fails for S-shaped objects and for irregularly
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1596 shaped objects.
1597 Recently, some authors have critized the idea of a completely algorithmic
1598 approach to the reconstruction problem (Goel et al. 2005; Reuter et al. 2007; Lu
1599 et al. 2007; Karasik and Smilansky 2008). Even though automatic methods assist
the assembly task by classifying and matching the fragments, they cannot fully
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1600
1601 replace a manual user interaction. Archeologists reason not only bottom up by
1602 pairwise matching, but also top-down, by considering the assembly problem as a
1603 whole, and by taking into account the archeological context. Algorithmic solutions
1604 should always be integrated, either before the manual assembly for classification
and matching, or after the assembly for precise alignment.
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1605
1606 In this context, Goel et al. (2005) propose specific software enabling the alter-
1607 ation of pot profiles via the addition of user-defined splines along their length. A
1608 spline can be created by altering the contour control points in number and in loca-
1609 tion. A previously saved spline can also be loaded from the database. Splines can be
appended to any point along the boundary of an existing pot profile and correspond-
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1610
1611 ing changes to the profile and the 3D model of the pot can be viewed simultaneously.
1612 Archaeologists can utilize this feature for altering profiles of complete pots and for
1613 extending profiles of fragments. The classification of partial and complete pot pro-
1614 files is done with respect to a user-defined database of profiles. To add a new profile
1615 to the database, an image of the profile is provided to the tool for computations. The
UN
1616 profile is then added to the database, becoming available for comparisons. There
1617 also exists a separate database of user-defined splines. After creation of a spline, in
1618 the pot profile editor, it can be added into this database. This enables the archaeolo-
1619 gist to apply the same base to several different pot profiles without having to redraw
1620 the spline.
1621 Lu et al. (2007) use boundary curves transformed into curvature and torsion form
1622 to allow the user to view the geometrical representation of the broken fragment. The
1623 archaeologist selects one boundary curve of interest from other fragment and finds
1624 out which other fragments have a high probability of originating from the same
OF
1625 artifact based on both the automatic ranking and her own expertise.
1626
1627
1628 Too Many Different Shape Encoding. Which is the best?

1629
RO
1630 It is easy to understand that there is not a single way to encode the shape of archae-
1631 ological evidences, in the same way that there is not a single way of encoding the
1632 shape of any solid. As we suggested at the beginning of the paper, “shapes” do not
1633 exist in the real world as particular entities. They are instead, a particular way of
1634 seeing the world. There are not round, curve, linear things in the archaeological
record, but some geometric properties of the observed entity can be described using
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1635
1636 a particular list of indexes values, a non-linear polynomial (a curve), a particular set
1637 of equations (a surface), or a topologically complex mesh (a 3D grid). There are
1638 different ways of encoding different levels of geometric information, depending on
1639 the nature of the archaeological problem we want to solve.
1640 Conventional approaches assume that a comparatively small number of discrete
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1641 features based on lengths and angles may describe the shape of any object. Thus,
1642 any archaeological material can be represented as a vector in a high-dimensional
1643 space where most dimensions are constituted of spatial dimensions with a Euclidean
1644 metric, while some other dimensions represent the angles between geometric
primitives (in polar coordinates). Nevertheless, for the most part such features
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1645
1646 only capture single dimensions of variability throughout the analyzed specimens.
1647 Morphological variability is not necessarily expressed as single lines, nor do arbi-
1648 trarily defined vectors constitute the only places where morphological variability is
1649 expressed.
The problem with this feature-based approach of shape is that its measurement
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1650
1651 schemes have been devised, with few exceptions in a more or less ad hoc manner
1652 based on what the researcher views as important dimensions of the artifact. Given
1653 that dozens of size parameters are possible, they can be combined in hundreds of
1654 ways into a formally dimensionless expression that might be used as a shape descrip-
tor. In fact, only a few relatively common combinations are possible, but even these
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1655
1656 are plagued by inconsistency in naming and calculation conventions. The burden
1657 placed on the user is to be sure that the meaning of any particular shape descriptor
1658 is clearly understood and that it is selected because it bears some relationship to
1659 the observed changes in the shape of features, because it is presumably being mea-
1660 sured in order to facilitate or quantify some comparison. It is important to take into
UN
1661 account that an unlimited number of visually quite different shapes can be created
1662 with identical values for any of these dimensionless shape parameters.
1663 Global shape encodings using all 4D information (length, width, height, and tex-
1664 ture) can be useful in some cases, but in some other cases, they can be imprecise.
1665 Although they deliver a polygonal mesh based on a high density of scanned points,
130 J.A. Barceló
1666 the accuracy of individual points barely reaches 1 cm (cf. Koch 2009). On the other
1667 hand, such models usually contain too much redundant information. Of course, it
1668 can be very useful working with the full geometry of the archaeological evidences,
1669 but we do not necessary need all that information, because in most cases, we do
OF
1670 not need all the sensorial possible visual information to solve the archaeological
1671 problem. For instance, the profile of a symmetric object can be enough for under-
1672 standing the way the object was manufactured and used in the past. Even in the case
1673 of asymmetric archaeological solid materials, an analysis of the geometric irregular-
1674 ities along the profile or contour can allow the analysis. An excess of 4D information
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1675 can exaggerate shape variability in our data. It is important to take into account that
1676 archaeological samples have been formed along periods of time comparatively long.
1677 We do not have access to all instruments made by a single artisan during a year, but
1678 a series of instruments, supposedly equivalents, made by many different artisans
1679 along three or four generations (100 years). If we are looking for regularities in
shape, and interpreting measured regularity in terms of social activities performed
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1680
1681 in the past, we need to go beyond individual variability. An excess of redundancy

1682 can prevent such an analysis.
1683 Although impressive results can be achieved for these shape encoding mech-
1684 anisms, there is still the lingering problem that current automatic simplification
1685 algorithms perform poorly on a large class of manufactured objects. In contrast
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1686 to current 4D visualization formats, which are dominated by smooth differen-
1687 tial surfaces, human made objects are usually dominated by discrete features.
1688 Archaeological evidences are for the most part artificial solids containing many
1689 sharp edges and for large parts of them, a polygonal mesh is not an approximation
of a smooth differential surface. Instead, the mesh represents the actual piecewise
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1690
1691 linear surface (Jang et al. 2006). If we apply current smooth simplification methods
1692 to this kind of objects, resulting simplifications may deviate from the ideal.
1693
1694
– Small features are merged into new larger ones. The larger features have charac-
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1695
teristics not present in the smaller features. In this case, new face orientations are
1696
introduced.
1697
– Many intermediate steps of the calculated simplifications are not correct
1698
– It is not clear which intermediate simplification steps are meaningful.
1699
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1700
1701 Another important criticism to approaches based on interpolated global shape

1702 models is the fact they are for the most dependent on the orientation of the
1703 object, especially in the case of uneven and asymmetric objects (lithic tools, ani-
1704 mal carcasses, material accumulations on the ground, complex buildings, etc.). Each
1705 orientation may produce a different representation, which clearly indicates that the
UN
1706 shape of the object cannot be uniquely derived from its surface or curvature alone.
1707 Current shape modeling systems traditionally based on polygonal meshes and
1708 other boundary representation formats presently lack the construction history and
1709 constructive object structure. From an archaeological and culture history point of
1710 view, internal structures (revealing the logic of construction and the properties of
1711 solid material) of objects, as well as their time, and other parametric dependencies
1712 also should also be added to the consideration (Boss et al. 2009).
1713 Therefore, shape-encoding procedures that make emphasis in the main aspects of
1714 the social processes that generated perceived shape are more convenient, than visu-
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1715 ally rich but excessively complex encodings. Of course, there is a balance between
1716 complexity of the encoding format and quantity of necessary geometric informa-
1717 tion. Sometimes harmonic analysis (Fourier decomposition or Hough transforms)
1718 can simplify excessively available information. Geometric Morphometrics based on
1719 functional landmarks (especially in the 3D case) can produce then better results than
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1720 most alternative approaches.
1721 Nevertheless, nothing in shape analysis is simple. As we will see in the next sec-
1722 tions of this chapter, the analysis of visual data in archaeology should be based on a
1723 more advanced assumptions that the mere encoding of perceived visual regularities.
1724
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1725
1726 Intermediate–Level Visual Analysis

1727
1728 Archaeologists do not define what they “perceive” as an unorganized set of discrete,
1729 irregular, discontinuous geometrical boundaries or edges, but in terms of complex
1730 objects. That is to say, archaeological evidences are a complex aggregate of individ-
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1731 ual elements interacting with different formative/modificative agents in a statistical
1732 way. We need to distinguish between the shape of an artifact (by virtue of it being a
1733 material object) and those aspects of the material shape that arose through deliberate
1734 modification of the artifact as a whole – or in part- during its production or use.
It is then necessary to merge geometric shape encoding into larger, more complex
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1735
1736 explanatory entities. If archaeologists would simply have encoded the geometry or
1737 size of each observed entity completely separately in a spatially invariant fashion,
1738 and then tried to analyze archaeological observable evidences on the basis of the
1739 resulting collection of features, they would have lose track of the causal arrangement
of these features relative to each other. We would have confused archaeological
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1740
1741 evidences that had the same features but in different arrangements. In other words,
1742 an object’s shape is a function of the visible arrangements of its component parts
1743 relative to each other. Therefore, in identifying an archaeological artifact or object,
1744 or when trying to determine its functional relevance one must encode not only single
unitary geometries but also the relationships between all shape components.
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1745
1746 The solution is not to give a mere enumerative list of individual shape features
1747 or global measurements but a composite model supervenient on its individual parts.
1748 The hypothesis that parts have their own meaning and function to understand the
1749 complexities of a distinct shape has led many researchers to assume that shape can
1750 be specified in terms of sets of parts; the idea of an alphabet of shape minimal units
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1751 is then a prevalent one. This assumption implies both a decomposition approach and
1752 a constructive framework, which describe the shape of an object as a combination
1753 of primitive elements. This different approach is based on the idea that the shape of
1754 an object has an important aspect that cannot be captured by the description of its
1755 edges and interfacial boundaries: the fact that most complex objects are perceived
132 J.A. Barceló
1756 as being composed of distinct parts. As we will use the term, a part is a restricted
1757 portion of an object that has semiautonomous status in visual perception. In other
1758 words, it is any portion detectable by a set of rules or procedures (Jang et al. 2006).
1759 Thus, shape features can be arbitrarily complex. For example, parts of a figurine’s
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1760 face, such as the eyes, nose, and mouth, can be considered as parts and modeled
1761 separately. In another case, a chair has four legs, a set and a back. In addition to
1762 such decomposition in parts, object perceptions should include the spatial relations
1763 among them (Palmer 1999, p. 348). The part decomposition approach assumes that
1764 each object can be decomposed into a small set of generic components that combine
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1765 to form units depending on the relationships between the components. However, it
1766 should be emphasized that parts need not have a natural meaning to us (such as the
1767 nose or the eye of a prehistoric cultic figurine). Shape can be defined as a group
1768 of geometric units satisfying certain mathematical properties and coinciding with
1769 what we know of the fabrication process that produced the figurine. In addition,
these parts do not have to be composed from disjoint groups of variables; a vari-
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1770
1771 able can be re-used in a multiple parts composite ensemble. A parts-based approach
1772 selects each part to represent a small group of variables that are known to be statis-
1773 tically dependent. Such an approach avoids devoting representational resources to
1774 weak relationships and instead allocates richer models to the stronger relationships
1775 (Schneidermann and Kanade 2004).
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1776 Consider the decomposition schema for a prehistoric cultic figurine; it has several
1777 parts as a head, a neck, a body, two arms, 2 legs. Each part has a characteristic shape
1778 and size: the head may be geometrically represented as a small triangular block, the
1779 neck a short cylinder, the body a large rectangular block, and the arms and legs long
and slim cylinders. The parts are arranged in more or less specific locations. The
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1780
1781 head is attached to one end of the neck, and its other end is attached to the body. The
1782 legs are attached to the bottom of the body to support it. All such information must
1783 be represented in order to recognize some visual input as an instance of a figurine.
1784 In such a decomposition-based approach, the input variables are grouped into sets.
Furthermore, the relationships within each set are more accurately modeled than
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1785
1786 those across.

1787 Shape analysis of archaeological evidences should be based on the decomposi-
1788 tion of solid object shapes into discrete parts, followed by the identification of those
1789 parts and their spatial and temporal relationships. We have not to forget, however,
that the relationship between parts (their configuration) is equally important. It is not
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1790
1791 just that the spatial arrangements of visual features are important. The intrinsic or
1792 extrinsic features of other parts may influence the internal descriptions of the parts
1793 themselves. The perception of shape will then depend critically on the part structure
1794 of distinct features. It also depends on how these various parts are related to one
1795 another in terms of their relative positions, relative orientations, relative sizes, and
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1796 so forth.
1797 The segmentation process is one of the most difficult tasks for shape decompo-
1798 sition. A robust segmentation is essential for shape problems that require objects
1799 to be classified or identified individually. A weak segmentation algorithm causes
1800 the eventual failure of the whole recognition or classification process. In general,
1801 shape segmentation algorithms follow three approaches. The first group partitions
1802 an image based on abrupt changes in detected corners along a contour; for instance,
1803 the shape of some artifacts seems to have been produced as a series of connected,
1804 curved segments, formed by the intersection of different formation processes. The
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1805 ancient manufacture of the object should be reconstructed from a representation of
1806 the edges and the angles through which the edges are joined together to make a
1807 corner, the complete shape can be decomposed into a sequence of nonintersecting,
1808 connected C- or S-Shaped curved line segments. Additionally, an S-curve can be
1809 represented as two C-curves joined (one in a convex direction and the other in a
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1810 concave direction) that are joined smoothly with the connection point forming an
1811 inflection point rather that a corner. In this way, not only the final shape is correctly
1812 described, but also, the formation process – human labor performed in the past- can
1813 also be reconstructed.
1814 The second category identifies the individual shape features that are similar
to a set of predefined criteria. In the case of pottery, we can mention rim, wall,
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1815
1816 and base; in the case of spears and arrows, there are distinctions among base,
1817 blade, edge, point, etc. Decomposing the shape of archaeological objects has been
1818 usually reduced to a subjective procedure of identification based on personal expe-
1819 rience assumptions. Many authors (Henton and Durand 1991; Lycett et al. 2006;
1820 Nautiyal et al. 2006; Read 2007; Kampel and Sablatnig 2007; Hörr et al. 2007;
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1821 Crompton 2007) have presented objective treatments for a rule-based or criteria
1822 based morphological segmentation.
1823 Another group of segmentation techniques is based on finding the parts of a com-
1824 plex shape directly (e.g. region splitting and merging). To differentiate among parts,
the perceptually salient shape components are identified. In general, this process
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1825
1826 is based on predefined criteria ranging from simple measurements such as area
1827 dimensions or circularity to complex shape descriptors. In some ways, it implies
1828 the detection of landmarks (see previous section).
1829 The most common 3D shape decomposition is the Generalized Cylinder (GC)
approach (Binford and Levitt 2003). The key insight is that many curved shapes
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1830
1831 can be expressed as a sweep of a variable cross section along a curved axis. Issues
1832 such as self-intersection and surface singularities do arise but shapes like a coffee
1833 pot or cup are easily handled. Compound object models, called “parts” or “assem-
1834 blies,” are graphs of GC primitives with affixment arcs labelled by rotation in Euler
angles and a translation represented in the object-centred reference frame of the part.
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1835
1836 Transformations between objects in an assembly can be parameterized, symbolic

1837 expressions, necessary to model articulation.
1838 Biederman (1987, 1995) calls the primitive 3-D components geons, which is a
1839 shortened form of geometric units. Each geon corresponds to an elementary shape
1840 (e.g., a brick, a cylinder, a curved cylinder), and all shapes are represented by com-
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1841 binations of geons. Biederman defined a set of 36 qualitatively different geons by

1842 making distinctions in some variable dimensions: cross-sectional curvature, sym-
1843 metry, axis curvature, and size variation. This produces a relatively small set of
1844 distinct primitive volumes from which a huge number of object representations
1845 can be constructed by putting two or more together. Because complex objects are
134 J.A. Barceló
1846 conceived in Biederman theory as configurations of two or more geons in particular

1847 spatial arrangements, they are encoded as structural descriptions that specify both
1848 the geons present and their spatial relationships. If geons are the alphabet of complex
1849 3-D objects, then spatial relations among geons are analogous to the order of letters
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1850 in words. Biederman uses structural descriptions in which 108 qualitatively different
1851 relations can be represented between two geons. Some of this connections con-
1852 cern how they are attached (e.g. SIDE-CONNECTED and TOP-CONNECTED);
1853 others concern their relational properties, such as relative size (e.g., LARGER-
1854 THAN, SMALLER-THAN). With these geon relations, it is logically possible to
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1855 construct more than a million different two-geon objects. Adding a third geon and
1856 its relations to the other two geons pushes the number of combinations into the
1857 trillions.
1858 Although geons are themselves volumetric entities, Biederman theory proposes
1859 they should be identified directly from image-based features such as edges and
vertices. Hummel and Biederman (1992) have built a neural network system to
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1860
1861 represent how shape analysis can be performed automatically.

AQ12 1862 Edelman (Edelman 1994; Edelman and Intrator 2000, 2002) has suggested giving
1863 up this compositional representation of shape by a fixed alphabet of crisp “all-
1864 or-none” explicitly tokened primitives (such as geons) in favor of a fuzzy, super
1865 positional coarse-coding by an open-ended set of image fragments. This alternative
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1866 approach has met with considerable success in computer vision. For example, the
1867 system described by Nelson and Selinger (1998) starts by detecting contour seg-
1868 ments, and determines whether their relative arrangement approximates that of a
1869 model object. Because none of the individual segment shapes or locations is critical
to the successful description of the entire shape, this method does not suffer from the
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1871 brittleness associated with the classical structural description models of recognition.
1872 Moreover, the tolerance to moderate variation in the segment shape and location data
1873 allows it to categorize novel members of familiar object classes.
1874 One of the relatively new constructive/deconstructive shape model approaches
of higher abstraction level is the function representation approach (FRep). It is a
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1875
1876 generalization of traditional implicit surfaces, constructive solid geometry (CSG).

1877 This representation supports a wide class of primitive objects and operations on
1878 them. In FRep, a 3D object is represented by a continuous function of point coor-
1879 dinates as F (x, y, z) >= 0. A point belongs to the object if the function is
non-negative at that point. The function is zero on the entire surface (called usu-
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1881 ally an implicit surface) of the object and is negative at any point outside the object.
1882 The function can be easily parameterized to support modeling of a parametric fam-
1883 ily of objects. In a FRep system, an object is represented by a tree data structure
1884 reflecting the logical structure of the object construction, where leaves are arbi-
1885 trary “black boxes” primitives and nodes are arbitrary operations. The following
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1886 types of geometric objects can be used as primitives (leaves of the construction
1887 tree): algebraic surfaces and skeleton-based implicit surfaces, convolution surfaces,
1888 objects reconstructed from surface points and contours, polygonal shapes converted
1889 to real functions, procedural objects (such as noise), volumetric (voxel) and other
1890 objects. Many modeling operations have been formulated which are closed on the
1891 representation. These modeling operations include set-theoretic operations, non-

1892 linear deformations and metamorphosis, and others. FRep also naturally supports
1893 4D (space-time) and multidimensional modeling using functions of several vari-
1894 ables. The main idea of visualization is to provide a mapping of such objects to a
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1895 multimedia space with such coordinates as 2D/3D world space coordinates, time,
1896 color, textures and other photometric coordinates. Time-dependent shape transfor-
1897 mations between given objects (metamorphosis) are among the typical space-time
1898 modeling operations (Pasko et al. 1995; Pasko and Pasko 2006; Vilbrandt et al.
1899 2004; see also Maiza and Gaildrat 2005).
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1900 Artificial Intelligence research in function from shape uses the idea that the
1901 sequence of parts provides some indication of the object function. For example,
1902 a hammer can be defined as a T-shaped object with geometric constraints like the
1903 head is nearly perpendicular to the handle, and the handle is positioned near the cen-
1904 ter of the head. These cannot be learned without spatial relations between parts and
subparts, implying that they directly relate to the affordances of an object. The most
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1905
1906 recent simulations following this approach call for constructing a generic multi-level
1907 hierarchical description of object classes in terms of meaningful shape components.
1908 Shape meaning is derived from a large set of geometric attributes and relationships
1909 between object parts. Initially, the input range data describing each object instance is
1910 segmented, each object part is labeled as one of a few possible primitives, and each
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1911 group of primitive parts is tagged by a functional symbol. Connections between
1912 primitive parts and functional parts at the same level in the hierarchy are labeled
1913 as well. Then, the generic multi-level hierarchical description of object classes is
1914 built using the functionalities of a number of object instances. During classification,
a search through a finite graph using a probabilistic fitness measure is performed
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1915
1916 to find the best assignment of object parts to the functional structures of each class.
1917 An object is assigned to a class providing the highest fitness value. The scheme does
1918 not require a-priori knowledge about any class (Froimovich et al. 2002; Pechuk et al.
1919 2005).
3D shape decomposition is much more usual in the representation of ancient
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1920
1921 buildings shape. Manferdini et al. (2008) follows an approach consisting of (i)
1922 identification of single elements, (ii) naming of the elements, (iii) identification of
1923 relations between them, and (iv) definition of the volumes they subtend. All this
1924 information is stored in a database together with find’s number, geo-location and
other useful archaeological details. This operation requires the help and support of
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1925
1926 archaeologists and architects to recognize transitions between different elements

1927 that constitute the find and semi automatically segment it. The semantic classi-
1928 fication of the finds is used in the archaeological database to decide whether the
1929 object is constituted by original pieces or some of them belong to other finds and
1930 should be re-located. Furthermore, the semantic classification of the finds leads to
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1931 the identification of classical orders, building functions and materials as well as
1932 extra information. The semantic segmentation is done directly on the 3D geome-
1933 try using a supervised classification. Additional information such as geo-location
1934 and numbering are also added to indicate a single element within the entire set
1935 of finds. Each part is connected to an instance in a knowledge base to ease the
136 J.A. Barceló
1936 retrieval process in a semantics-based context. The naming of each single element
1937 and of the classes in which they can be grouped is an important process that strictly
1938 depends on archaeological and architectural considerations. While the mesh is sub-
1939 divided and the model segmented, the semantic structure of the find suggests the
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1940 organization of single nodes and their naming. After the segmentation phase, it is
1941 possible to re-build inner subdivision surfaces, in order to define the entire volume
1942 of each single element and node. This phase is strictly dependent on the ability
1943 of archaeologists to recognize morphological elements and constructive techniques
1944 and give volumetric interpretations. The connection between the database and the
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1945 single parts of an archaeological find upgrades the traditional 2D GIS to a 3D
1946 system. Digital models are georeferenced (by point, line or area belonging to neces-
1947 sities), so that they also can be linked to 2D systems that are generally already
1948 available.
1949 It should be by now obvious that shape analysis of archaeological objects must
be based on the decomposition of object shapes into discrete parts, followed by the
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1950
1951 identification of those parts and their spatial and temporal relationships. In any case,
1952 we have not to forget, however, that the relationship between parts (their config-
1953 uration) is equally important. It is not just that the spatial arrangements of visual
1954 features are necessary. The intrinsic or extrinsic features of other parts may influ-
1955 ence the internal descriptions of the parts themselves. The perception of shape will
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1956 then depend critically not only on the part structure of objects and how its various
1957 parts are related to one another in terms of their relative positions, relative orienta-
1958 tions, relative sizes, and so forth. An approach may be the organization of the parts
1959 and their shape relationships in an explicit graph structure, i.e., in terms of nodes
and their connectivity via a network of links (Leymarie 2003).
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1960
1961
1962
1963 High Level Analysis

1964
We have already commented that current archaeological research, like most social
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1965
1966 science research seems addressed to the mere description of archaeological evi-
1967 dence. In this way, shape analysis appears as a passive presentation of the
1968 visual appearance of the archaeological record. Instead, we should analyze what
1969 really happened in the past, why or how archaeological visual data acquired
their actual properties of size, shape, texture, composition and spatiotemporal
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1970
1971 location.
1972 Why an archaeological artifact has a particular shape? A possible answer will
1973 be “because it has a distinct appearance on the sake of its proper functioning”. We
1974 have already suggested archaeologists should investigate how perceptual properties
1975 (shape) allow finding the social cause (production, use, distribution) of what we
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1976 have seen at the archaeological site. In other words, material elements found in
1977 archaeological contexts are assumed to be like they physically appear to be because
1978 they performed some particular function in the past (Beck 1980, see also Cotterell
1979 and Kamminga 1992; McGrew 1993; St. Amant 2002; Bicici and St. Amant 2003;
1980 De Ridder 2003; Hughes 2009).
1981 The same argument that says the geometry of the external contour or the surface
1982 is functional by virtue of its role in the performance of an activity equally says that
1983 a stylistic feature (i.e. decoration, color, etc.) used in a ritual or ceremony is also
1984 functional. We can distinguish between utility in these two domains by considering
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1985 whether it is possible to perform the same activity with the artifact and visual feature
1986 in question in an equally efficacious manner when characteristics of the feature are
1987 changed. That is, changes in the phenomenological domain (namely the shape of
1988 the tool) may affect the efficacy with which a task can be performed. Change in
1989 a stylistic feature (decoration) may also affect its use in a ritual but for different
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1990 reasons. The meaning associated with the artifact with the feature in the context of
1991 the ritual may have been altered by the trait change and thereby made the modified
1992 object unsuitable for use in the ritual (Read 2007).
1993 In all cases, the meaning of functioning is related to the term using. Wright’s
1994 suggestion (Wright 1973) was that functional analysis must depend on a notion of
design (intended shape). That is to say, an object has been made to do something in a
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1995
1996 particular way, and the goal it has to fulfill can only be attained when the artifact has
1997 some particular shape among many other possible shapes. According to this inter-
1998 pretation, the visual appearance of any archeological evidence could be explained
1999 because it performs some particular function. The function of each element would
2000 be distinguished from other non-functional (or “accidental”) uses by the fact that
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2001 the features that define the visual appearance of the object owe its existence to this
2002 particular use (Millikan 1999).
2003 Wright’s definition would be correct, however, only in the case of objects like
2004 huts or hats, or any other instrument-like things, which have been made according
to a clearly defined purpose (Millikan 1999; Neander 1991; De Ridder 2003; Hughes
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2005
2006 2009). A naïve transposition of Wright’s etiological account to the archaeological

2007 domain would presuppose that artifacts have causal histories of reproduction and
2008 selection; then one could identify the proper function of an artifact with the disposi-
2009 tions for which the artifact was selected or the dispositions that causally contributed
to its existence. According to Nagel (1961) the task of a functional explanation
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2010
2011 would be then to explain the presence of an item in a system, in our case, the spe-
2012 cific geometry of the object’s surfaces. The proper way to accomplish that task
2013 is by showing the geometry to be indispensable. In other words: by showing the
2014 object’s actual shape to be a necessary condition for the proper functioning of the
thing.
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2015
2016 There is however something odd about the intuition underlying Nagel’s account
2017 (see criticism in De Ridder 2003). It supposes that a functional explanation amounts
2018 to prove the indispensability of some feature. But that is certainly not an incon-
2019 testable claim. Why not suppose that a functional explanation should explain how
2020 an item helps to realize a function? This leads to two different conceptions of what
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2021 a functional explanation is: on the one hand it can be defined as an explanation that
2022 contains a function ascription (this is what Nagel does), and on the other as an expla-
2023 nation of the physical mechanisms that underlie a function ascription. On the former
2024 definition the appropriate explanatory question is: “Why object O has shape S?” but
2025 on the latter it is: “Why (or how) S contributes to perform function f ?”
138 J.A. Barceló
2026 A function ascription asserts that the archaeological artefact’s shape is good
2027 for something, specifically, for bringing about some particular state of affairs.
2028 Therefore, every instrumental function includes a functional goal: a condition which
2029 can be realized by proper use of the item. Functional goals should be considered as
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2030 propositional functions and not simple sentences (Hughes 2009). It is also impor-
2031 tant that some of them (such as, “cutting meat in a timely manner” or “containing
2032 water effectively”) can be satisfied to greater or lesser extent. Consequently, instru-
2033 mental functions apply to specific types of objects (typically, artefact types). This
2034 may be a matter of some controversy, since some accidental functions appear to
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2035 apply only to particular tokens rather than types. In fact, it seems plausible that
2036 instrumental functions do primarily refer to artefact types and only derivatively to
2037 tokens. One may argue that, even for novel artefacts, the proper subject of a func-
2038 tion ascription is an artefact type, albeit a type instantiated by a single token. We
2039 can argue that archaeological evidences functional statements should provide an
answer to the question “how does S work?” where S is a goal-directed system in
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2040
2041 which the material entity whose function we are interested in appears (Nagel 1961;
2042 Boorse 1976, 2002; Adams 1979; Cummins 1975, 2002; De Ridder 2003; Hughes
2043 2009).
2044 Defining the shape of archaeological materials in terms of functional information
2045 is not as straightforward as it might seem. The shape of archaeological objects can
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2046 be the unpremeditated material consequences of accidental actions. Tools can be
2047 used for purposes not intended by their designers and conversely, an object can be
2048 used as a tool even if it was not shaped as a tool initially (St. Amant 2002; Bicici
2049 and St. Amant 2003). The problem is that, although use actions seem to be goal-
directed activities sometimes desirable ends are achieved through the incidental or
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2050
2051 even accidental use of an object, and consequently the shape of archaeological arti-
2052 facts can also be opportunistic or even accidental, without any relationship with the
2053 hypothetical use.
2054 A central part of the idea of a function is that not all effects of a structure count
as its functions; some are simply accidental by-products. What is needed is a con-
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2055
2056 straint on which of the visual features of a mechanical system are such that may
2057 count as performing a function. We need to know which “selected effects” of a
2058 geometry count as its function or functions, presumably in virtue of contributing to
2059 the particular class of activities or conditions of the containing system which count
as that system’s working right. In short, we need a way to separate the function(s)
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2060
AQ13 2061 of a structure from its accidental effects (McClamrock 1993).

2062 To be an archaeological artifact and to be designed and manufactured in the past
2063 by a craftsperson who had some proper use - economical or ideological, subsis-
2064 tential or ritual- in mind ought not to be the same. When we specify the function
2065 of an artifact, we are typically specifying the intention with which it was made.
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2066 If the artifact was made in prehistory or ancient times for performing some func-
2067 tion, then that function is (through the intentions of the artifact’s creator) relevant
2068 to a causal explanation of its presence in the archaeological record. Nevertheless,
2069 without the mechanism of explicit design intentions at work, function does not
2070 explain archaeological presence. It is obvious that living beings can make tools
2071 without being able to design them or think consciously about their proper function-
2072 ing. Rather than there being a single sense of function, a large family of meanings
2073 exists. Of course, we cannot reject the idea of a tight relationship between form
2074 and function independently of any other observable or non-observable proper-
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2075 ties of the object, such as its physical structure, composition, contents, texture or
2076 contexts of use. Nevertheless, it should be restricted to items which are the conse-
2077 quence of purposeful activity (tools), and only in some restrictive circumstances in
2078 which the object is not being used in improper ways because there are not other
2079 alternatives.
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2080 Only certain aspects of the shape of an artifact seem to be relevant to the arti-
2081 fact being included as a member of a functionally salient category. Therefore, it is
2082 not very clever to try to recognize archaeological functionalities by just looking at
2083 the artifact’s shape; since none uses anything by looking (except in the case of a
2084 rock-art symbol or a decorative motif). An agent that is interested in learning how
a tool was used in the past either needs to look for the changes it can achieve in
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2085
2086 the physical world by using the tool or be aware of the rules governing the cre-
2087 ation of those tools. This way, tools are no longer named specifically as vases,
2088 for instance, but as a-tool-that-can-increase-my-abilities-of-containing-different-
2089 kinds-of-substances-by-using-the-governing-rule-number-X. Archaeologists need
2090 to search for where these tools come from and what is the underlying functionality
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2091 that we achieve while using them.
2092 So objects have shapes in virtue of their contributing to the general disposition
2093 of the object. Only if the object in fact has a specific shape, it will have the dispo-
2094 sition of doing something in some way. This is required by the earlier claim that
function ascriptions imply disposition ascriptions, which in turn reduce to condi-
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2095
2096 tional statements about the behavior of an object. Such statements are only true if
2097 the component in question actually possesses the required disposition. To explain a
2098 disposition manifestation we can show them to be instances of more general laws,
2099 i.e. “laws governing the behavior of things generally, not just things having d ”
(Cummins 1975). The function should then be defined because of interpretation
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2100
2101 of a behavior of the item under an intended goal. The main result will be that we
2102 need to distinguish two types of functional explanation, one showing that a thing
2103 has a function and the other demonstrating how it performs this function (De Ridder
2104 2003).
The proper function of an item can be defined therefore in terms of what the item
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2105
2106 has done in the past, or what it normally does or is disposed to do in a specific context
2107 of use (Hughes 2009), and according some well defined user-plans (Vermaas and
2108 Houkes 2003). That means that we should define explanatory functional analysis as
2109 implying that the shape of an archaeological object must be explained in terms of
2110 the role it plays in bringing something about, and in terms of user actions and user
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2111 circumstances. For example, consider the shape of a pottery vase. Each has a crucial
2112 function assigned to it: the flat bottom is for standing the vase on a surface; the
2113 handle is for grasping the vase when lifting; the inside is for containing the liquid;
2114 the rim is for supporting the object against the lips when drinking. The assignment
2115 of causal interactions to the different shape features defines the object as a vase
140 J.A. Barceló
2116 (Leyton 1992, p. 163). We may argue, then, that the use of this object is specified
2117 in terms of the actions applied to it, e.g., standing up, lifting, etc., and in terms of
2118 the resulting actions that the cup applies back to the environment, e.g., conveying
2119 the liquid upward. All that means that we are describing the artifact in terms of five
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2120 components:
2121
2122 (1) INPUTS: e.g., standing up, lifting, etc.

2123 (2) OUTPUTS: e.g., conveying liquid
2124 (3) STATES: physical characteristics of the cup, e.g., its shape
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2125 (4) FIRST CAUSAL RELATIONSHIP:
2126 e.g., lifting (input) acts on shape (state) → conveying liquid (output)
2127 (5) SECOND CAUSAL RELATIONSHIP:
2128 e.g., lifting (input) acts on shape (state) → shape does not change (dynamics:
2129 next state).
DP
2130
2131 The functional analysis of the object is understood as converting the input actions
2132 into the output actions. This approach equates function with causal links or goal-
2133 directedness rather than logical purpose. In other words, it asserts that anything can
2134 have a function regardless of how it came into existence (Henning 2005). The proper
2135 function of an item is its normal function, understood as a causal or dispositional
TE
2136 act. According to Cummins, function ascriptions can be translated into conditional
2137 statements describing the behavioral regularities of an object (Cummins 1975). We
2138 may need to extend this analysis and allow that such conditionals be explained not
2139 only by the possession of dispositional properties but also by the possession of non-
dispositional (categorical) properties.
EC
2140
2141 DiManzo et al. (1989) regarded functional reasoning as the ability to integrate
2142 shape and function with the help of planning. They described the difficulty of sep-
2143 arating the function of a tool from the plan it takes part in, since plans and tools
2144 evolve together and differentiate with time. Their simulated model is based on a
hierarchy of levels that interact with each other. At the top level, they have a task and
RR
2145
2146 plan representation that uses semantic functional descriptors and functional experts
2147 for planning based on functionality of objects. The object representation level uses
2148 functional experts and geometric primitives to describe objects. The next level car-
2149 ries out function modeling by describing some basic functions in terms of geometric
primitives, and the last level performs geometric reasoning by defining geometric
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2150
AQ14 2151 constraints (see also Zhang et al. 2002).

2152 Archaeological entities have shapes, but they should also have relationships
2153 between their physical/dynamic properties (shape) and the properties/abilities of
2154 their intended users. The affordances of any archaeological evidence become obvi-
2155 ous in its use and/or formation process. Both involve establishing and exploiting
UN
2156 constraints (between the user/producer and the artifact, the user/producer and the
2157 environment, and the artifact and the environment).
2158 In the same way we should enhance shape information with causal and planning
2159 hypotheses, we can add to our simulation information about dynamics. These mod-
2160 els will use the shape, kinematic and dynamic properties of an object (e.g. motion)
2161 to recognize its functionality while the system observes the action that is performed
2162 with the object. The motion analysis results in several motion primitives and these
2163 are compared with previously known motion-to-function mappings. Optical flow
2164 measurements are used to derive motion information for different objects. The
OF
2165 relevant motion is in object’s coordinate system and its relation to the object on
2166 which it acts. This relation is important for establishing the mapping and creating
2167 a frame of reference. Thus, the motion is derived independently of the place of
2168 action.
2169 What underlies this sense of the concept of form-function is that the concept
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2170 is essentially historical in character, what should be obvious in an archaeological
2171 investigation! The proper function of an archaeological artifact is determined not by
2172 its shape but by its history! This is what O’Brien and Lyman have recently suggested
2173 in archaeology (O’Brien and Lyman 2002; Lyman et al. 2008). However, even if the
2174 concepts of reproduction and selection are applicable in the context of archaeol-
ogy, they have a different meaning that found in a biological context, where the
DP
2175
2176 etiological conception of functions derives (Vermaas and Houkes 2003; De Ridder
2177 2003).
2178 The position I take here is that the focus of shape analysis in archaeology should
2179 be to study historical causation and not a mere form-function decision based on
2180 some classifier. If evolution is to be grounded in function, and function is to be
TE
2181 grounded on the characteristics of the entity, then perceived shape must have a par-
2182 ticular function not because we explain it, but for objective reasons that can be
2183 derived from the historical (evolutionary) role it played. I sustain the view that
2184 archaeological functional explanation is as a complex relational system that links
physical structure, intention, settings, action, and use history.
EC
2185
2186 That means that it is not the artifact’s shape what will explain its appearance in
2187 some archaeological record, but the history of social actions having used that tool
2188 for different purposes at different circumstances. An item’s function is to explain not
2189 only why the item has a distinctive shape, but also its causal disposition (Cummins
1975; 2002). Thus, perceived shape of archaeological evidences should be explained
RR
2190
2191 by the particular causal structure in which it is supposed to participate. The knowl-
2192 edge of the function of some perceived material element should reflect the causal
2193 interactions that someone has or can potentially have with needs, goals and prod-
2194 ucts in the course of using such elements. On this view, an object’s function reflects
the actions that can be performed on it, given both its physical structure and the
CO
2195
2196 physical structure of the agent interacting with it. An object’s physical structure and
2197 an agent’s action specify an affordance jointly, constituting the immediate causes
2198 of a perceived function (Kitamura and Mizogouchi 1999; Chaigneau et al. 2004).
2199 It should be possible to build a model of function based on a description of the
2200 physical structure (shape) of its ancestors, namely certain reproduced physical dis-
UN
2201 positions. In that sense, both the artifact and its ancestors are part of a genetic
2202 reproduction history and are thus products of processes (Lyman et al. 2008). In some
2203 cases, it can be proved that the physical structure of the element is approximately
2204 similar to the physical structure of those ancestors, including the dispositions that
2205 correspond to the proper functions ascribed to the artifact. Only malformed, and
142 J.A. Barceló
2206 consequently malfunctioning is an exception of the principle that the genetic struc-
2207 ture of the causal history provides partial justification for the belief that artifact
2208 A has the physical disposition (shape) that corresponds to the ascribed function.
2209 Obviously this approach cannot be applied in all circumstances, because it is wrong
OF
2210 in the case of new objects and the introduction of novelty and revolutionary changes,
2211 but it can be useful for understanding the causal history (or “genetic” reproduc-
2212 tion) of a historically connected series of objects (Chaigneau et al. 2004; Rovner
2213 2006).
2214 This approach is at odds with current practice in morphological analysis in
RO
2215 archaeology, still based on the assumption that objects most similar to a given func-
2216 tion will be found in the core of the definition, while objects that are less similar
2217 will be in more marginal parts (see discussion in Barceló 2008). What we need is a
2218 study of covariances between shape representations and other associated or causal
2219 variables (human labor). In other words, whereas shape cannot be reduced to the
identification of objects per se, it should be concerned with the degree to which
DP
2220
2221 other variables (e.g., time, space, composition, ecology, phylogeny, and use con-
2222 straints) are related to shape variation within a sample or population and the nature
2223 of that relationship. As such, morphometric data analysis (Lele and Richtsmeier
2224 2001) can be used to address a far wider range of shape-related problems than either
2225 geometry or pattern recognition. The hypothesis under examination it is whether the
TE
2226 social or historical events (in the physical sense of the word) cause or covary with
2227 the aspects of the objects’ morphological variation (= the aspect(s) of their geome-
2228 try) actually measured or extractable from the data under analysis. Confusion over
2229 this issue inevitably leads to erroneous interpretations of morphometric results (see
MacLeod 2002).
EC
2230
2231 If this approach were right, then to be able to ascribe functions to observed
2232 archaeological evidences, we would need to combine three kinds of information:
2233
2234
RR
2235
2236
• Knowledge about how the designers intended to design the artifact to have the
2237
function
2238
• Knowledge about how the designers determined the physical structure of that
2239
artifact on the basis of their technological abilities
• Knowledge about how the artifact was determined by its physical structure to
CO
2240
2241
perform that function
2242
2243
2244 Acknowledgments Special thanks to Marcelo Cardillo for comments on a previous version of
2245 this paper. Erik Otarola-Castillo and R. Sant-Amant sent me details of their interesting research.
UN
2246 Xavier Clop, Mercedes Farjas and F.J. Melero have also contributed to my understanding of what
2247
should be a comprehensive visual analysis in archaeology. Thanks also to my students at the
Dept. of prehistory (Universitat Autònoma de Barcelona, Spain) who are beginning to explore
2248
the many possibilities of shape analysis in archaeology. Parts of this research have been funded by
2249 the Spanish Ministry of Science and Innovation, The Generalitat de Catalunya and the Universitat
2250 Autònoma de Barcelona.
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AQ1 “Carbonetto et al. 2005” is changed as “Carbonetto et al. 2004” as per the
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reference list. Please confirm.
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2888 AQ2 “Flenniken et al. 1986” is changed as “Flenniken and Raymond 1986” as per
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the reference list. Please confirm.
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2890 AQ3 “Marr and Hildreth, 1980” is not listed in the reference list. Please provide.
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AQ4 Please specify whether “Mara and Sablatnig 2005” is “2005a” or “2005b” in
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2894 AQ5 “Lysett et al. 2005” is not listed in the reference list. Please provide.
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AQ6 “Russ (2002)” is not listed in the reference list. Please provide.
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AQ7 “Mara and Sablatnig 2008” is not listed in the reference list. Please provide.
2898 AQ8 “Belongie et al. 2002” is not listed in the reference list. Please provide.
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AQ9 “Papaioannou et al. (2001)” is not listed in the reference list.
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2901 AQ10 “Papaioannou et al. (2003)” is not listed in the reference list.
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AQ11 Please specify Whether “Kampel et al. 2006” is “2006a” or “2006b” in all
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occurrences.
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AQ12 “Edelman and Intrator 2000” is not listed in the reference list.
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2906 AQ13 “McClamrock 1993” is not listed in the reference list.

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AQ14 “Zhang et al. 2002” is not listed in the reference list.
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2909 AQ15 The references “Dibble, H.L., 1997”, “Leyton, M., 2005”, “Russ 1990”, “Mara
and Sablatnig 2006”, “Cummins 2000”, “Kitamura and Mizogouchi 2004”,
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2911 “Razdan et al. 2001”, “Razdan et al. 2004” are not cited in the text part. Please
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provide citation.
2913 AQ16 Please update the reference.
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Visual Analysis in Archaeology. An Artiﬁcial Intelligence Approach

Chapter in Lecture Notes in Earth Sciences · June 2010

Applications of cost-based spatial statistics in Archaeology View project

The user has requested enhancement of the downloaded file.

13 Idea and Aims

42 J.A. Barceló (B)

A.M.T. Elewa, Morphometrics for Nonmorphometricians, Lecture Notes in Earth 93

46 identifying activity areas, material accumulations or buildings at the site, interpret-

81 Following modern studies of computer object-recognition (Grimson 1991;

91 Fig. 5.1 A schema showing Raw visual data

124 What is Shape?

196 Collins 2008).

206 of any archaeological deposition should be analyzed to understand the formation

234 Low Level Visual Analysis

256 rough or irregular.

267 Visual Data Acquisition and Encoding

269 In order to be able to increase the objectivity of visual information, archaeologists

296 popular are:

298 1. Laser scanning. The measuring of the 3D points coordinates is implemented

311 digitized data generated by the scanner is composed of thousands of x, y, z coordi-

316 be recorded in order to obtain an estimation of visual marks variability to be trans-

331 of the possible internal components in a 10 mm slice. Digital geometry processing

351 of perceived boundaries (shape), and retinal properties (texture).

354 Edge Detection

424 Preliminary Approaches to Shape Encoding

511 irregularities, but not with overall elongation.

585 object’s area/convex area (5.11)

589 object’s area/area of bounding box (5.12)

631 (25) Index of robustness, calculated as

633 width∗ thickness/length (5.13)

Higher values imply more robust tools.

degree of asymmetry between two bilateral measurements as a ratio of the overall

684 Edge Curvature as Shape Encoding

766 Quantitative measures of the intensity of curvature should also be calculated

781 Another relatively simple measure of “surface curvature” can be calculated by

796 measures the rate of change of the tangent angle:

798 κ(s)) = (d θ/ds) (5.17)

801 osculates the curve at the point s:

803 κ(s) = 1/ρ(s) (5.18)

826 degree of roughness of a given contour based on the degree of concavity of

866 Alternative Descriptors of Global Shape

978 Salient Points as Shape Encoding

In many applications of archaeological shape analysis, we need to assume that the

1016 length and width, on the hafting edge, etc.

1066 S, S1, S2 are points on the profile of a curve

Martín et al. (2009) take a somewhat different approach to the selection of

1071 salient, by simplifying a scanned point cloud.

1076 repetitive sharpening operations accentuating the curvature by plastic defor-

1102 Geometric Morphometrics analyzes the variability in the relative position of

1106 matrix X is the k x m matrix of Cartesian coordinates of the k landmarks in m

1141 Shape at the Third Dimension

1161 boxes, cords-based, moments-based and wavelets-based descriptors, convex-hull

1196 diverse (Peng and Shamsuddin 2004):

1206 of the surface is maintained.

1211 like PolyWorks (www.innovmetric.com), 3DReshaper (www.3dreshaper.com),

1339 From Fragment to Object

1376 IF b (x,y,z) FITS THEORY,