2011 International Conference on Digital Image Computing: Techniques and Applications
Statistical Shape and Probability Prior Model for
Automatic Prostate Segmentation
Soumya Ghose∗† , Arnau Oliver† , Robert Martı́† , Xavier Lladó† , Jordi Freixenet† , Jhimli Mitra∗† ,
Joan C. Vilanova‡ , Josep Comet§ and Fabrice Meriaudeau∗
∗ Le2i-UMR
CNRS 5158, Université de Bourgogne, Le Creusot, France
Email: {soumya.ghose, jhimli.mitra, fabrice.meriaudeau}@u-bourgogne.fr
† Computer Vision and Robotics Group, Universitat de Girona, Girona, Spain
Email: {sghose, aoliver, marly, llado, jordif, jmitra}@eia.udg.edu
‡ Girona Magnetic Resonance Center, Girona, Spain
§ University Hospital Dr. Josep Trueta, Girona, Spain.
Gabor filters to achieve accurate prostate segmentation. However, the method is computationally expensive and therefore
probably unsuitable for TRUS guided prostate intervention
[2]. In recent years, Cosio et al. [7] reported an automatic
method for prostate segmentation with active shape models [8].
However, the optimization framework of the genetic algorithm
used is computationally intensive and unsuitable for TRUS
guided intervention.
Cootes et al. [9] proposed an efficient framework to build
a statistical model incorporating prior shape and texture information in their work of active appearance model (AAM).
To address the challenges involved with prostate segmentation
in TRUS images we propose a novel approach using multiple
mean parametric models derived from principal component
analysis (PCA) of prostate shape and posterior probabilistic
values of the prostate region to segment the prostate in a
multi-resolution framework. The performance of our method
is compared with the traditional AAM [9] and also with our
previous work [10]. In contrast, to the use of intensity and
one mean model as in [9] and to the use of texture from Haar
wavelet features of [10], posterior probabilistic information of
the prostate region obtained in a Bayesian framework is used
to train, initialize and propagate multiple statistical models
of shape and texture. Statistically significant improvement is
achieved with the use of multiple mean models when validated
with 23 datasets, that have significant shape, size, and contrast
variations of the prostate, in leave-one-patient-out validation
framework. The key contributions of this work are:
• The use of iterative graph cut in identifying the prostate
and use of the probability information of the prostate
region obtained in a Bayesian framework in building the
statistical model of texture.
• The use of the texture model in training, automatic
initialization, propagation and selection of the optimum
mean model.
Abstract—Accurate prostate segmentation in Trans Rectal
Ultra Sound (TRUS) images is an important step in different
clinical applications. However, the development of computer
aided automatic prostate segmentation in TRUS images is a
challenging task due to low contrast, heterogeneous intensity
distribution inside the prostate region, imaging artifacts like
shadow, and speckle. Significant variations in prostate shape,
size and contrast between the datasets pose further challenges to
achieve an accurate segmentation. In this paper we propose to use
graph cuts in a Bayesian framework for automatic initialization
and propagate multiple mean parametric models derived from
principal component analysis of shape and posterior probability
information of the prostate region to segment the prostate. The
proposed framework achieves a mean Dice similarity coefficient
value of 0.974±0.006, mean mean absolute distance value of
0.49±0.20 mm and mean Hausdorff distance of 1.24±0.56 mm
when validated with 23 datasets in a leave-one-patient-out validation framework.
I. I NTRODUCTION
Prostate cancer is the second most leading cause of death
from cancer in American men [1]. Accurate prostate segmentation in TRUS may aid in biopsy therapy planning, motion
monitoring, needle placement and multimodal image fusion
between TRUS and magnetic resonance imaging (MRI) to
improve malignant tissue collection during biopsy [2]. However, accurate computer aided prostate segmentation in TRUS
images encounters considerable challenges due to low contrast
of TRUS, heterogeneous intensity distribution and presence
of micro-calcifications inside the prostate gland, speckle, and
shadow artifacts. Moreover, inter patient prostate shape, size
and deformation may vary significantly.
In the last decade, a number of semi-automatic prostate
segmentation methods have been reported. These methods
require expert intervention during initialization of the model
or during refinement of the segmented contour. Often deformable models and statistical shape models are used to
achieve segmentation in semi automatic methods like [3],
[4]. However, it is necessary to use an automatic prostate
segmentation method for TRUS guided procedures. Shen et
al. [5] and Zhan et al. [6] presented an automatic method
that incorporated a priori shape and texture information from
978-0-7695-4588-2/11 $26.00 © 2011 IEEE
DOI 10.1109/DICTA.2011.64
II. P ROPOSED S EGMENTATION F RAMEWORK
The proposed method is developed on the following major
components: 1) the use of expectation maximization (EM) and
Bayesian framework to determine the posterior probability of
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Fig. 1.
Schematic representation of our approach. Abbreviations used EM = Expectation Maximization, MRF = Markov Random Field.
a pixel being prostate, 2) use of Markov random field (MRF)
model to impose spatial constraints and improve the probabilistic representation of the prostate region, 3) estimation
of the optimum label of the prostate region from graph cut
segmentation on the MRF, 4) adapting AAM to incorporate the
probabilities of the prostate region for training, initialization
and propagation of the parametric model and 5) selection of
one of the mean models depending on the error of fitting of
the posterior probabilities to segment the prostate. The global
schema of the method is given in Fig. 1.
images. In our model, the class prior probability is estimated
from the frequency of the pixels (x) belonging to a class as
pnm
xi
(2)
P (Cprs ) = i=1
m
x
j=1 j
where, P (Cprs ) gives the class prior probability of being
prostate, xi represents the pixels belonging to prostate region
(total given by pnm) and xj represents the pixels in all training
images (given by m). The probabilities of intensity (being
prostate) obtained in the EM framework, location (being
prostate) and class prior probability (prostate class) are used in
a Bayesian framework to determine the posterior probability
of a pixel being prostate. According to the Bayes rule,
A. Bayesian Framework
In traditional AAM [9], the point distribution model (PDM)
[8] of the contour is aligned to a common reference frame
by generalized Procrustes analysis [11]. Intensities are warped
into correspondence using a piece wise affine warp and sampled from a shape-free reference to build the texture model.
However, intensity distribution inside the prostate region may
vary significantly from one dataset to another depending on the
parameters of acquisition and nature of the prostate tissue of a
patient introducing larger variabilities and hence producing an
inaccurate texture model. Therefore, to reduce inter dataset
intensity variabilities we propose to use the PCA of the
posterior probabilities of the image pixels being prostate to
build our texture model.
Firstly, K-means clustering is used to roughly cluster the
pixels into two classes (prostate and non-prostate) from the
intensities. The class means and standard deviations obtained
from this rough clustering are then used as the initial estimates
in an EM [12] based algorithm on Gaussian mixture model
to determine the probability of a pixel being prostate from
intensities. The E-step assigns the probabilities to the pixels
depending on the current mean and standard deviation values
of the classes, while in M-step the means and standard deviation values are re-estimated. Maximum a posteriori estimates
of the class means and standard deviations are used to soft
cluster the pixels. The likelihood of a pixel location in an
image being prostate is obtained by normalizing the ground
truth values of all the pixels for all the training images as
N
1
GTi
P (xps |Cprs ) =
N i=1
P (Ci |X) =
P (X|Ci ) P (Ci )
P (X)
(3)
the posterior probability distribution P (Ci | X) of a class is
given by the prior P (Ci ) (i.e. P (Cprs )) and the likelihood
P (X|Ci ). P (X) being equal for all classes can be removed
from the formulation. Considering class conditional independence, the likelihood could be formalized as,
P (X|Ci ) = P (xps |Cprs ) .P (xin |Cprs )
(4)
In equation (4) the likelihood P (X|Ci ) is obtained from the
product of the probability of a pixel intensity being prostate
(P (xin |Cprs )) obtained from EM framework (xin being pixel
intensity) and the probability of a pixel location being prostate
(P (xps |Cprs )) obtained from (1).
Further to impose spatial constraints over the pixel position
we use MRF modeling over the posterior probabilities obtained
from equation (3). MRF is a random process defined on a discrete lattice. For our case, the lattice is a 2D grid on the image
plane. In 2D, we assume that S = {1, 2, ....N } × {1, 2, ....N }
is the set on N 2 points called sites. For a fixed site s, a
neighborhood N(s) is defined. Clique c is defined as a set
of sites, such that if si , sj ∈ c then sj is in the neighborhood
of si , sj ∈ N(i) and let C be the set of cliques. Assuming
Xn is a Markov chain we have,
P (Xn = xn |Xk = xk , k = n) =
P (Xn = xn |Xn−1 = xn−1 , Xn+1 = xn+1 )
(1)
(5)
i.e. the conditional distribution of Xn depends only on its
neighbors Xn−1 and Xn+1 in 2D. According to HammersleyClifford theorem [13], Xn is a Markov field if and only if it
follows a Gibb’s distribution given by,
where P (xps |Cprs ) gives the probability of a pixel position
being prostate with xps being the pixel location (ps standing
for position) and Cprs denoting pixels being prostate (prs
stands for prostate). GTi represents the ground truth of the
training image, N being the total number of ground truth
P (Xn ) =
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1
exp [U (Xn )/T ]
Z
(6)
where Z =
exp[U (Xn )] is a normalizing constant called
partition function, T is aconstant called the temperature and
Vc (Xn )c∈C is the sum of clique
the energy U (Xn ) =
potentials Vc over all possible cliques c ∈ C that depends
on the local configuration on the clique c. The local potentials
are determined using the Ising model [14].
According to the Ising model each particle can have one
of two magnetic spin orientations +1 and -1. Each particle
interacts only with its neighbor and the contribution of each
particle to the total energy of the system depends upon the
orientation of its spin compared to its neighbor. Adjacent
particles with same spin are in lower energy than those
with dissimilar spin. To model our problem using the Ising
model we assign initial labels +1 to the pixels with posterior
probability greater than zero and assign -1 to other pixels.
According to Ising model,
1
xi xj ]
(7)
P (x) = exp[J
Z
in an iterative manner in three steps. Step 1 is done by simple
enumeration of the kn values for each pixel. In step 2 Gaussian
parameters the mean and covariance are estimated with expectation maximization algorithm in standard manner. Finally,
in step 3 a global optimization is achieved using minimum
cuts as done by Boykov and Jolly [17]. Iterative minimization
from steps 1 to 3 of the total energy E with respect to the
three parameters α, θ, k ensures convergence at least to a local
minimum of E [16]. The energy minimization assigns α value
to each pixel where α = 1 corresponds to the foreground (the
prostate) and α = 0 corresponds to background. However,
the segmentation achieved does not incorporate any shape
information of the prostate and hence produce some misclassified regions. To further improve the segmentation result
we adapt AAM of Cootes et al. [9] to introduce probability
values (obtained in the Bayesian framework) of the prostate
region (obtained with iterative graph cut) to build our texture
model and to impose shape restriction. AAM or statistical
shape and texture model is discussed next with the adaptation
of the model to introduce probability values of the prostate
region in building the texture model.
i=j
where x is a particular N × N configuration of ±1, xi = xj
are pair of neighboring pixels, J represents the strength of the
interaction between xi and xj that decrease the energy when
both have same values and Z is the partition function. Use
of Ising prior in our models favors regions of coherence and
imposes spatial restriction. We use the Gibb’s sampling [14]
to produce the Ising model.
The image now consists of pixels z = (z1 , z2 , ....zn , ...zN )
with probabilistic values obtained with the Markov modeling.
The segmentation of the image is expressed as an array of
opacity values α = α1 , ....αN at each pixel. In our case we
use a hard segmentation to identify the prostate and hence
αn ∈ 0, 1 with 0 for background and 1 for the foreground
i.e. the prostate. The parameters mean and covariance (θ)
describe image foreground and background, represented with
two Gaussian mixtures model (GMM). An energy function E
is defined so that its minimum produces good segmentation
and is guided by foreground and background GMMs and that
the opacity is coherent imposing spatial constraints. This is
captured by a Gibbs energy given by,
E(α, k, θ, z) = U (α, k, θ, z) + V (α, z)
B. Statistical Shape and Texture Model (AAM)
The process of building the parametric statistical model
of shape and texture variations involves the task of building
a shape model, a texture model, and a combined model of
texture and shape and prior learning of the optimization space
from the combined model perturbation. To build the shape
model, a PDM is built by equal angle sampling of the prostate
contours to determine the landmarks automatically. The PDM
of the contours are aligned to a common reference frame by
generalized Procrustes analysis. PCA of the aligned PDMs
identifies the principal modes of shape variations. Posteriori
probabilistic information (of pixels being prostate) of the segmented region are warped into correspondence using a piece
wise affine warp and are sampled from a shape free reference
similar to the AAM. PCA of the posterior probabilities of
the segmented region obtained with Markov random field
modeling and graph cuts algorithm is used to identify their
principal modes of variation. The model may be formalized
in the following manner. Let s and t represent the shape and
posterior probability models, then
(8)
depending on the GMM component variables k. The data term
U that represents the GMM is defined as,
G(αn , kn , θ, zn )
(9)
U (α, k, θ, z) =
s = s + Φs θs ,
t = t + Φt θt
(11)
where s and t denote the mean shape and posterior probability
information respectively, then Φs and Φt contain the first
p eigenvectors (obtained from 98% of total variations) of
the estimated joint dispersion matrix of shape and posterior
probability information and θ represent the corresponding
eigenvalues. The model of shape and posterior probability
variations are combined in a linear framework as,
W θs
W ΦTs (s − s)
b=
=
(12)
θt
ΦTt (t − t)
n
where G is a Gaussian probability distribution. V , the smoothness term in equation (8) that maintains coherence in regions
of similar probabilities and is given as,
2
(10)
V (α, z) = γ
[αn = αm ]exp −β zm − zn |
(m,n)∈C
where [φ] denotes the function taking values 0, 1 for a predicate φ, C is the set of pairs of neighboring pixels and β
a constant that encourages smoothness in region of similar
contrast. The energy E defined in equation (8) is minimized
where W denotes a weight factor (determined as in AAM [9])
coupling the shape and the probability space. A third PCA of
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the combined model ensures the reduction in redundancy of
the combined model, and is given as,
b=Vc
(13)
where V is the matrix of eigenvectors and c the appearance
parameters.
C. Optimization and Segmentation of a New Instance
In our model, we incorporate the traditional AAM optimization. The objective function of our model is similar to
AAM. However, instead of minimizing the sum of squared
difference of intensity between the mean model and target
image, we minimize the sum of squared difference of the
posterior probability of the mean model and the target image.
The prior knowledge of the optimization space is acquired by
perturbing the combined model with known model parameters
and perturbing the pose parameters (translation, scale and
rotation). A linear relationship between the perturbation of the
combined model (δc) and the residual posterior probability
values (δt) (obtained from the sum of squared difference
between the posterior probability of the mean model and
the target image), and between the perturbation of the pose
parameters (δp) and the residual posterior probability values
are acquired in a multivariate regression framework as,
δc = Rc δt,
δp = Rp δt
(a)
(b)
(c)
(d)
(e)
(14)
where Rc and Rp refer to the correlation coefficients. Given
a new instance, equation (14) is used as update parameters.
Given a test image, the posterior probability values of the
pixels being prostate is determined in the Bayesian framework
section (II-A). The sum of the squared difference of the
posterior probability values with the mean model is used to
determine the residual value δt. The combined model (δc) and
the pose parameters (δp) are then updated using equation (14)
to generate a new shape and combined model and hence, new
posterior probabilities. The process continues in an iterative
manner until the difference with the target image remains
unchanged.
Fig. 2. (a) Mean models fitting errors for with dataset 1 as reference. (b),
(d) Segmentation without multiple mean model, (c), (e) Segmentation with
multiple mean model. The white contour gives the ground truth and the black
contour gives the obtained result. Each row shows a different patient.
errors for the test datasets with dataset 1 as reference (Fig.
2(a)). Consequently, the reference dataset is changed from
2 through 23 and the entire process is repeated for all the
datasets (23 in total). The entire procedure yields 23 graphs of
model fitting errors (one for each dataset). We have analyzed
these 23 model fitting error graphs and have observed that
with less fitting error (< 2000 units) we have higher accuracy
in segmentation (in terms of different validation measures).
This is not surprising considering the fact that the objective
function of our optimization framework tries to minimize the
fitting error between the mean model and the target image
with respect to the pose parameters. Hence, an increase in
fitting error indicates a reduction in segmentation accuracies.
An empirical error value is determined from these graphs,
above which, the segmentation accuracy is reduced (in our case
the threshold value is 1700 units). The reference dataset that
has a fitting error less than the empirical value for maximum
number of test datasets is identified (dataset 1 in our case). The
datasets below this fitting error are grouped together (datasets
1, 6, 8, 10, 15 and 21 (Fig. 2(a))) and are removed from
further grouping. The process is repeated until all the datasets
are grouped. These groups of datasets provide individual mean
models (5 mean models in our case). However, increasing the
D. Multiple Mean Models
The statistical shape and texture model assumes the shape
and the texture spaces to be Gaussians. However, inter patient
prostate shape and their intensities may vary significantly.
In such circumstances, a single Gaussian mean model is
inefficient to capture the variations of shape and texture spaces.
To address this problem, we propose to use multiple Gaussian
mean models. The scheme is as follows; initially the 1st dataset
is chosen as the reference to register datasets 3 to 23 to
produce a mean model of shape and texture. This mean model
is used to test dataset 2. The sum of squared difference of the
posterior probabilities between the mean model and dataset
2 is recorded as the fitting error after the final segmentation.
Likewise, with the fixed reference (dataset 1), we build the
second mean model registering datasets 2 and 4-23 to test on
dataset 3 and record the fitting error. The process is repeated
for all datasets from 4-23. This provides 22 model fitting
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TABLE I
P ROSTATE SEGMENTATION QUANTITATIVE COMPARISON (HD, MAD AND M AX D IN MM , S PEC ., S ENS ., AND ACC ., ARE FOR S PECIFICITY, S ENSITIVITY
AND ACCURACY RESPECTIVELY.)
Method
AAM [9]
Ghose et al. [10]
B-AAM
Our Method
DSC
0.92±0.04
0.94±0.03
0.95±0.06
0.97±0.01
HD
3.80±1.98
2.59±1.21
2.53±1.94
1.24±0.56
MAD
1.26±0.76
0.91±0.44
0.87±1.23
0.49±0.20
MaxD
3.81±2.00
2.64±1.19
2.35±2.10
1.30±0.74
Spec.
0.91±0.04
0.91±0.04
0.92±0.04
0.96±0.01
Sens.
0.98±0.01
0.98±0.01
0.97±0.04
0.99±0.00
Acc.
0.97±0.05
0.97±0.05
0.97±0.03
0.98±0.00
TABLE II
Q UALITATIVE COMPARISON OF PROSTATE SEGMENTATION
Reference
Shen [5]
Ladak [4]
Cosio [7]
Our Method
Area Accuracy
Error 3.98±0.97%
Accuracy 90.1±3.2%
DSC 0.97±0.006
Contour Accuracy
Distance 3.2±0.87 pixels
MAD 4.4±1.8 pixels
MAD 1.65±0.67 mm
MAD 1.82±0.76 pixels / 0.49±0.20 mm
number of mean models (decreasing the fitting error threshold)
improves segmentation accuracy. In Fig. 2(b) and 2(d) we
observe that segmentation error is high with one mean model
in the Bayesian framework. However, segmentation accuracy
improves with multiple mean models in the same framework
(Fig. 2(c), 2(e)).
Datasets
8 images
117 images
22 images
23 datasets / 46 images
Time
64 secs
11 minutes
22 seconds
the segmentation accuracy of an algorithm. In this context,
we obtained better segmentation accuracies compared to [9]
and [10]. As observed in Table I, B-AAM (that uses posterior
probability and a single mean model) produces better results
compared to AAM justifying the use of posterior probability
of the prostate region instead of intensity. However, our model
which uses both posterior probability and multiple mean models, produces superior results compared to B-AAM, suggesting
that the use of both posterior probability and multiple mean
models are essential to improve segmentation accuracies. The
improvement in segmentation accuracy with multiple mean
model is evident from the last two rows in Table I.
A quantitative comparison of different prostate segmentation
methodologies is difficult in the absence of a public dataset
and standardized evaluation metrics. Nevertheless, to have an
overall qualitative estimate of the functioning of our method,
we have compared with some of the 2D segmentation works
of the literature in Table II. Note that we may consider area
overlap and area accuracy equivalent to that of DSC values,
while average distance equivalent to that of average MAD.
Analyzing the results we observe that our mean DSC value
is comparable to area overlap accuracy values of Ladak et al.
[4] and very close to the area overlap error of Shen et al. [5].
However, it is to be noted that we have used more images
compared to Shen et al. Our MAD value is comparable to
[5], [4], and [7]. From these observations we may conclude
that qualitatively our method performs better in overlap and
contour accuracy measures. Qualitative results of our method
is illustrated in Fig. 3.
Our method is implemented in Matlab 7 on an Intel
Core2Duo T5250, 1.5 GHz processor and 2 GB RAM. The
mean segmentation time of the method is 21.97±0.55 seconds.
Our mean segmentation time is better compared to Shen et al.
[5] and Cosio et al. [7] with an un-optimized Matlab code.
Due to off-line optimization of our statistical model of shape
and probability prior the mean fitting time is 0.68 seconds
which may be reduced further in an optimized C++ or GPU
environment. The process is computationally expensive in determining the posterior probability from MRF regularization.
However, near real time implementation of MRF regularization
III. E XPERIMENTAL R ESULTS
We have validated the accuracy and robustness of our
method with 46 axial mid gland TRUS images of the prostate
with a resolution of 348×237 pixels from 23 prostate datasets
in a leave-one-patient-out evaluation strategy. During validation, a test dataset is removed and 5 mean models are built with
the remaining 22 datasets. All the 5 mean models are applied
to segment the test dataset. The mean model with the least
fitting error is selected for accurate segmentation. The ground
truth for the experiments are prepared in a schema similar
to MICCAI prostate challenge 2009 [19], where manual segmentations performed by an expert radiologist are validated by
an experienced urologist. We have used most of the popular
prostate segmentation evaluation metrics like Dice similarity
coefficient (DSC), 95% Hausdorff Distance (HD) [19], mean
absolute distance (MAD) [2], maximum distance (MaxD),
specificity, sensitivity, and accuracy [3] to evaluate our method.
Furthermore, the results are compared with the traditional
AAM proposed by Cootes et al. [9], Ghose et al. [10] and to
B-AAM (that uses posterior probability of the prostate region
and a single mean model for segmentation).
It is observed from Table I that a probabilistic representation
of the prostate texture in TRUS images and the use of multiple
mean models significantly improves segmentation accuracy
when compared to traditional AAM and to [10]. We used
posterior probability information for automatic initialization
and training of our statistical shape and texture model. As
opposed to manual initialization of traditional AAM and
[10], our model is initialized automatically. We achieved a
statistically significant improvement in t-test p-value<0.0001
for DSC, HD and MAD compared to the other two approaches.
A high DSC value and low values of contour error metrics
of HD and MAD are all equally important in determining
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Fig. 3. Performance of our algorithm against shape, size and contrast variations for different patients. The white contour gives the ground truth and the
black contour gives the obtained result. Columns 1, and 3 show segmentations with traditional AAM and 2, and 4 show corresponding segmentations with
our model.
exists [18]. We plan to exploit the GPU environment with an
optimized C++ code to produce close to real time segmentation
of the prostate.
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IV. C ONCLUSION AND F UTURE W ORKS
A novel approach of multiple statistical models of shape and
posterior probability information of prostate region with the
goal of segmenting the prostate in 2D TRUS images has been
proposed. Our approach is accurate, and robust to significant
shape, size and contrast variations in TRUS images compared
to traditional AAM and some existing work in literature. While
the proposed method is validated with prostate mid gland
images, effectiveness of the method against base and apical
slices will be validated in our future work.
Acknowledgements
This research has been funded by VALTEC 08-1-0039 of
Generalitat de Catalunya, Spain and Conseil Régional de
Bourgogne, France.
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