Abstract
A novel algorithm to derive an approximate cellular envelope of an arbitrarily thick digital curve on a 2D grid is proposed in this paper. The concept of “cellular envelope” is newly introduced in this paper, which is defined as the smallest set of cells containing the given curve, and hence bounded by two tightest (inner and outer) isothetic polygons on the grid. Contrary to the existing algorithms that use thinning as preprocessing for a digital curve with changing thickness, in our work, an optimal cellular envelope (smallest in the number of constituent cells) that entirely contains the given curve is constructed based on a combinatorial technique. The envelope, in turn, is further analyzed to determine polygonal approximation of the curve as a sequence of cells using certain attributes of digital straightness. Since a real-world curve/curve-shaped object with varying thickness and unexpected disconnectedness is unsuitable for the existing algorithms on polygonal approximation, the curve is encapsulated by the cellular envelope to enable the polygonal approximation. Owing to the implicit Euclidean-free metrics and combinatorial properties prevailing in the cellular plane, implementation of the proposed algorithm involves primitive integer operations only, leading to fast execution of the algorithm. Experimental results including CPU time reinforce the elegance and efficacy of the proposed algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Image Analysis. Morgan Kaufmann, San Francisco (2004)
Klette, R., Rosenfeld, A.: Digital straightness: A review. Discrete Applied Mathematics 139, 197–230 (2004)
Aken, J.R.V., Novak, M.: Curve-drawing algorithms for raster display. ACM Trans. Graphics 4, 147–169 (1985)
Attneave, F.: Some informational aspects of visual perception. Psychological Review 61, 183–193 (1954)
Imai, H., Iri, M.: Computational geometric methods for polygonal approximations of a curve. CVGIP 36, 31–41 (1986)
Perez, J.C., Vidal, E.: Optimum polygonal approximation of digitized curves. PRL 15, 743–750 (1994)
Schröder, K., Laurent, P.: Efficient polygon approximations for shape signatures. In: Proc. ICIP, pp. 811–814 (1999)
Schuster, G.M., Katsaggelos, A.K.: An optimal polygonal boundary encoding scheme in the rate distortion sense. IEEE Trans. Circuits and Systems for Video Technology 7, 13–26 (1998)
Tanigawa, S., Katoh, N.: Polygonal curve approximation using grid points with application to a triangular mesh generation with small number of different edge lengths. In: Proc. AAIM 2006, pp. 161–172 (2006)
Teh, C.H., Chin, R.T.: On the detection of dominant points on digital curves. IEEE Trans. PAMI 2, 859–872 (1989)
Yin, P.Y.: Ant colony search algorithms for optimal polygonal approximation of plane curves. Pattern Recognition 36, 1783–1797 (2003)
Rosin, P.L.: Techniques for assessing polygonal approximation of curves. IEEE Trans. PAMI 19, 659–666 (1997)
Yin, P.Y.: A new method for polygonal approximation using genetic algorithms. PRL 19, 1017–1026 (1998)
Devillers, O.: Inner and outer rounding of set operations on lattice polygonal regions. In: Proc. 20th Ann. Symp. Computational Geometry, pp. 429–437 (2004)
Cohen, J., et al.: Simplification Envelopes. In: Proc. SIGGRAPH, pp. 119–128 (1996)
Bhattacharya, P., Rosenfeld, A.: Contour codes of isothetic polygons. CVGIP 50, 353–363 (1990)
Bhowmick, P., Biswas, A., Bhattacharya, B.B.: Isothetic polygons of a 2D object on generalized grid. In: Pal, S.K., Bandyopadhyay, S., Biswas, S. (eds.) PReMI 2005. LNCS, vol. 3776, pp. 407–412. Springer, Heidelberg (2005)
Biswas, A., Bhowmick, P., Bhattacharya, B.B.: TIPS: On finding a Tight Isothetic Polygonal Shape covering a 2d object. In: Kalviainen, H., Parkkinen, J., Kaarna, A. (eds.) SCIA 2005. LNCS, vol. 3540, pp. 930–939. Springer, Heidelberg (2005)
Yu, B., Lin, X., Wu, Y., Yuan, B.: Isothetic polygon representation for contours. CVGIP 56, 264–268 (1992)
Fam, A., Sklansky, J.: Cellularly straight images and the hausdorff metric. In: Proc. Conf. on Pattern Recognition and Image Processing, pp. 242–247 (1977)
Geer, P., McLaughlin, H.W.: Cellular lines: An introduction. Discrete Mathematics and Theoretical Computer Science, 167–178 (2003)
Kim, C.E.: On cellular straight line segments. Computer Graphics Image Processing 18, 369–391 (1982)
Klette, R.: Cell complexes through time. In: Proc. Vision Geometry. SPIE, vol. 4117, pp. 134–145 (2000)
Rosenfeld, A.: Digital straight line segments. IEEE Transactions on Computers 23, 1264–1268 (1974)
Freeman, H.: On the encoding of arbitrary geometric configurations. IRE Trans. Electronic Computers EC-10, 260–268 (1961)
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. Prentice Hall of India Pvt. Ltd., Englewood Cliffs (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bhowmick, P., Biswas, A., Bhattacharya, B.B. (2006). PACE: Polygonal Approximation of Thick Digital Curves Using Cellular Envelope. In: Kalra, P.K., Peleg, S. (eds) Computer Vision, Graphics and Image Processing. Lecture Notes in Computer Science, vol 4338. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11949619_27
Download citation
DOI: https://doi.org/10.1007/11949619_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68301-8
Online ISBN: 978-3-540-68302-5
eBook Packages: Computer ScienceComputer Science (R0)