Robust Image Segmentation with Mixtures of Student's t-Distributions

Abstract

Gaussian mixture models have been widely used in image segmentation. However, such models are sensitive to outliers. In this paper, we consider a robust model for image segmentation based on mixtures of Student's t -distributions which have heavier tails than Gaussian and thus are not sensitive to outliers. The t -distribution is one of the few heavy tailed probability density functions (pdf) closely related to the Gaussian, that gives tractable maximum likelihood inference via the Expectation-Maximization (EM) algorithm. Numerical experiments that demonstrate the properties of the proposed model for image segmentation are presented.

Full text

ROBUST IMAGE SEGMENTATION WITH MIXTURES OF STUDENT’S t-DISTRIBUTIONS
Giorgos Sfikas Christophoros Nikou Nikolaos Galatsanos
University of Ioannina,
Department of Computer Science,
PO Box 1185, 45110 Ioannina, Greece,
{sfikas, cnikou, galatsanos}@cs.uoi.gr
ABSTRACT
Gaussian mixture models have been widely used in image
segmentation. However, such models are sensitive to outliers.
In this paper, we consider a robust model for image segmenta-
tion based on mixtures of Student’s t-distributions which have
heavier tails than Gaussian and thus are not sensitive to out-
liers. The t-distribution is one of the few heavy tailed proba-
bility density functions (pdf) closely related to the Gaussian,
that gives tractable maximum likelihood inference via the Ex-
pectation-Maximization (EM) algorithm. Numerical experi-
ments that demonstrate the properties of the proposed model
for image segmentation are presented.
Index Terms— Image segmentation, clustering, Student’s
t-distribution, mixture model, EM algorithm, segmentation
evaluation.
1. INTRODUCTION
Image segmentation is the process of grouping image pixels
based on the coherence of certain attributes such as intensity,
color or texture. Many approaches have been proposed to
solve the image segmentation problem. For surveys on this
topic the reader may refer to [1]. In this paper, we will focus
our attention to image segmentation methods based on clus-
tering. Clustering is the process of arranging data into groups
having common characteristics and is a fundamental problem
in many fields of science [2]. Thus, image segmentation can
be viewed a special type of clustering. Usually, in image seg-
mentation, our data, the image pixels have spatial locations
associated with them. Thus, apart from the commonality of
attributes such as intensity, color or texture, commonality of
location is an important characteristic of the grouping that we
are seeking in image segmentation.
More specifically, in this paper we will focus our attention
on clustering methods based on the modeling of the probabil-
ity density function (pdf) of the data via finite mixture models
(FMM) [3, 4]. Modeling the pdf of data with FMM is a nat-
ural way to cluster data because it automatically provides a
This work was partially supported by Interreg IIIA (Greece-Italy) grant
I2101005.
grouping of the data based on the components of the mixture
that generated them. More specifically, FMM are based on
the assumption that each datum originates from one compo-
nent of the mixture according to some probability. Thus, this
probability can be used to assign each datum to the compo-
nent that has most likely generated it. Furthermore, the like-
lihood of an FMM is a rigorous measure for evaluating the
clustering performance [4].
FMM based pdf modeling with Gaussian components has
been used successfully in a number of applications ranging
from bioinformatics [5] to image retrieval [6]. The parameters
of Gaussian mixture models (GMM) can be estimated very
efficiently through maximum likelihood (ML) estimation us-
ing the EM algorithm [7]. Furthermore, it can be shown that
Gaussian components allow efficient representation of any
pdf [4]. However, it is well known that GMM are sensi-
tive to outliers and may lead to excessive sensitivity to small
numbers of data points. The problem of providing protec-
tion against outliers in multivariate data is very difficult and
increases with the dimensionality.
In this paper, we apply to the image segmentation prob-
lem mixture models with Student-t pdf components. This pdf
has heavier tails as compared to the exponentially decaying
tails of a Gaussian [8]. Thus, each component in the mixture
originates from a wider class of elliptically symmetric dis-
tributions with an additional parameter called the degrees of
freedom. Hence a more robust model is used than the classi-
cal normal mixture.
In the remainder of this paper, background on standard
GMM is given in Section 2 and the mixture of multivariate
t-distributions and the EM algorithm for parameter estima-
tion are described in Section 3. Results on image segmenta-
tion and comparisons with the standard GMM are presented
in Section 4 and conclusions are drawn in Section 5.
2. BACKGROUND ON STANDARD GAUSSIAN
MIXTURE MODELS
Let X denote the vector of features representing an image
spatial location (pixel). The GMM assumes that the pdf of
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the observation x is expressed by
φ(x; Ω)=
K

i=1
π
i
f(x; μ
i
, Σ
i
) (1)
where Ω is the mixture parameter set Ω =[Ω
1
, Ω
2
, ..., Ω
K
]
with Ω
i
=(π
i
,μ
i
, Σ
i
). For the mixing proportions of the i
th
component π
i
, we have that
0 ≤ π
i
≤ 1,i=1, 2, ..., K,
K

i=1
π
i
=1 (2)
For each component of the model in (1), the Gaussian pdf is
expressed by
f(x; μ
i
, Σ
i
)=
1
(2π)
−
d
2
|Σ
i
|
−
1
2
exp
−
1
2
(x−μ
i
)
T
Σ
−1
i
(x−μ
i
)
(3)
where d is the dimensionality of the vector (e.g. intensity,
location, texture features) and μ
i
, Σ
i
are the mean vector and
covariance matrix respectively.
Training of a GMM, or in other words finding its ML so-
lution, can be performed using the EM algorithm [7]. The
EM algorithm is a well-known numerical method used in a
variety of ML problems. In the case of a GMM, each im-
age pixel x, is associated with a binary hidden variable z of
dimension K, whose k
th
component has a value of 1 if the
observation (i.e. the pixel) was produced by that component
and is zero otherwise. In the E-step of the algorithm, the ex-
pected value of the hidden variables conditioned on the obser-
vation is computed. These expected values give the probabil-
ities that a given datum originates from a different component
of the mixture. Thus, they provide a means for segmenting
the data. In the M-step, the model parameters (mean, covari-
ance and mixing proportions) are computed by maximizing
the log-likelihood of the complete data (hidden variables and
observations). This scheme is repeated iteratively until con-
vergence is achieved.
3. MIXTURE OF STUDENT’S t-DISTRIBUTIONS
AND THE EM ALGORITHM
A d-dimensional random variable X follows a multivariate
t-distribution with mean μ, positive definite, symmetric and
real d × d covariance matrix Σ and has ν ∈ [0, ∞) degrees
of freedom when, given the weight u, the variable X has the
multivariate normal distribution with mean μ and covariance
Σ/u:
X|μ, Σ,ν,u∼ N(μ, Σ/u),
and the weight u follows a Gamma distribution parameterized
by ν:
u ∼ Gamma(ν/2,ν/2).
Integrating out the weights from the joint density leads to the
density function of the marginal distribution:
p(x; μ, Σ,ν)=
Γ

ν+d
2

|Σ|
−
1
2
(πν)
d
2
Γ

ν
2

[1 + ν
−1
δ(x, μ; Σ)]
ν+d
2
(4)
where δ(x, μ;Σ)=(x − μ)
T
Σ
−1
(x − μ) is the Mahalanobis
squared distance and Γ is the Gamma function. It can be
shown that for ν →∞the Student’s t-distribution tends to
a Gaussian distribution with covariance Σ. Also, if ν>1, μ
is the mean of X and if ν>2, ν(ν − 2)
−1
Σ is the covariance
matrix of X. Therefore, the family of t-distributions provides
a heavy-tailed alternative to the normal family with mean μ
and covariance matrix that is equal to a scalar multiple of Σ,
if ν>2 (fig. 1).
Fig. 1. The Student’s t-distribution for various degrees of
freedom. As ν →∞the distribution tends to a Gaussian.
For small values of ν the distribution has heavier tails than a
Gaussian.
A Student’s t-distribution mixture model (SMM) may also
be trained using the EM algorithm [8]. A K-component mix-
ture of t-distributions is given by
φ(x, Ψ) =
K

i=1
π
i
p(x; μ
i
, Σ
i
,ν
i
) (5)
where x =(x
1
, ..., x
N
)
T
denotes the observed-data vector
and
Ψ=(π
1
, ..., π
K
,μ
1
, ..., μ
K
, Σ
1
, ..., Σ
K
,ν
1
, ..., ν
K
)
T
. (6)
are the parameters of the components of the mixture.
Consider now the complete data vector
x
c
=(x
1
, ...x
N
,z
1
, ..., z
N
,u
1
, ..., u
N
)
T
(7)
where z
1
, ..., z
N
are the component-label vectors and z
ij
=
(z
j
)
i
is either one or zero, according to whether the observa-
tion x
j
is generated or not by the i
th
component. In the light
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of the definition of the t-distribution, it is convenient to view
that the observed data augmented by the z
j
, j =1, ..., N are
still incomplete because the component covariance matrices
depend on the degrees of freedom. This is the reason that
the complete-data vector also includes the additional missing
data u
1
, ..., u
N
. Thus, the E-step on the (t +1)
th
iteration
of the EM algorithm requires the calculation of the posterior
probability that the datum x
j
belongs to the i
th
component of
the mixture:
z
t+1
ij
=
π
t
i
p(x
j
; μ
t
i
, Σ
t
i
,ν
t
i
)
K

m=1
p(x
j
; μ
t
m
, Σ
t
m
,ν
t
m
)
(8)
as well as the expectation of the weights for each observation:
u
t+1
ij
=
ν
t
i
+ d
ν
t
i
+ δ(x
j
,μ
t
i
;Σ
t
i
)
(9)
Maximizing the log-likelihood of the complete data pro-
vides the update equations of the respective mixture model
parameters:
π
t+1
i
=
1
N
N

j=1
z
t
ij
,μ
t+1
i
=
N

j=1
z
t
ij
u
t
ij
x
j
N

j=1
z
t
ij
u
t
ij
, (10)
Σ
t+1
i
=
N

j=1
z
t
ij
u
t
ij
(x
j
− μ
t+1
i
)(x
j
− μ
t+1
i
)
T
N

j=1
z
t+1
ij
. (11)
The degrees of freedom for each component are computed as
the solution to the equation:
log

ν
t+1
i
2

− ψ

ν
t+1
i
2

+1− log

ν
t
i
+ d
2

+
+
N

j=1
z
t
ij
(log u
t
ij
− u
t
ij
)
N

j=1
z
t
ij
+ ψ

ν
t
i
+ d
2

=0 (12)
where ψ(x)=
∂(lnΓ(x))
∂x
is the digamma function. A detailed
derivation of the EM algorithm for Student’s t-mixtures is pre-
sented in [8].
4. EXPERIMENTAL RESULTS
In this paper, we employed an 8-dimensional vector as a fea-
ture for each image pixel [9]. The first three components of
Table 1. Number of images (over 30) where the SMM pro-
vides lower quantization error than the GMM for p =1.2.
Noise type K =3 K =5 K =7
noise free 23 17 24
uniform 20 dB 20 19 21
uniform 14 dB 23 17 18
uniform 7 dB 26 15 17
salt pepper 10% 19 15 16
the feature vector are the Lab color coordinates, the next three
components are texture descriptors, namely, the polarity, the
anisotropy and the contrast as described in [9] and the remain-
ing two coordinates are the horizontal and vertical pixel loca-
tions. Prior to model training, each feature vector component
was separately normalized to ensure that no feature dominates
the others.
In order to evaluate the proposed segmentation scheme
and compare it to GMM segmentations we compute a quan-
tization error for 30 images provided by the Berkeley image
segmentation data base [10]. The quantization error, for each
pixel location, is defined as the distance between the image
feature and the mean of the mixture component that gener-
ated the measure (i.e. the component with the larger mixing
proportion). This p-norme distance, between a d-dimensional
feature vector x and the mean vector μ is defined as
D(x, μ)=

d

i=1
(x
i
− μ
i
)
p

1
p
(13)
We have experimented with different values of p, namely 0.7,
1.2 and 2 (Euclidean distance). It is well known that norms
close to 1 measure a quantization error that better corresponds
to human perceptual characteristics. These experiments were
performed by degrading the images by uniform and salt-and-
pepper noise of varying strength. Also, the predefined number
of kernels varied (K =3, 5, 7).
Let us also notice that the experiments were performed us-
ing a variation of the standard EM algorithm, called Greedy
EM [11] providing a segmentation result independent of the
model initialization. The performance of the model is pre-
sented in table 1. A comparison is shown in table 2 where
one can see that the SMM has a slight yet better performance
than the GMM. Some segmentation results are depicted in fig.
2 and 3 where it can be observed that SMM provide smoother
segmentations than the standard GMM.
5. CONCLUSION
We have presented a methodology for image segmentation
based on mixtures of Student’s t-distributions. The model
can account for outliers values and thus provides smoother
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Tabl e 2. Quantization error statistics for 30 images of the
Berkeley segmentation data base for all the configurations of
the uniform noise (see table 1).
K =3 K =5 K =7
GMM SMM GMM SMM GMM SMM
p =0.7
mean 11.54 11.54 10.27 10.23 9.59 9.51
s. d. 0.85 0.92 0.87 0.89 0.88 0.92
p =1.2
mean 3.85 3.83 3.48 3.47 3.27 3.25
s. d. 0.24 0.25 0.25 0.26 0.27 0.28
p =2.0
mean 2.26 2.25 2.07 2.07 1.96 1.96
s. d. 0.12 0.13 0.14 0.14 0.15 0.15
Original GMM SMM
Fig. 2. Segmentation examples using the GMM and the SMM
methods for K =5components.
Original GMM SMM
Fig. 3. Segmentation of a MRI brain image into K =3
classes (white matter, grey matter and cerebrospinal fluid).
segmentations than the standard GMM. However, important
issues for mixture based clustering still need to be addressed.
Such issues are how the number of model components can
be selected automatically and which features should be used.
These are open questions and are subject of current research.
6. REFERENCES
[1] N. Pal and S. Pal, “A review of image segmentation
techniques,” Pattern Recognition, vol. 26, pp. 1277–
1294, 1993.
[2] R. Xu and D. Wunsch II, “Survey of clustering algo-
rithms,” IEEE Transactions on Neural Networks, vol.
16, no. 3, pp. 645–678, 2005.
[3] C. M. Bishop, Pattern Recognition and Machine Learn-
ing, Springer, 2006.
[4] G. McLachlan, Finite mixture models, Wiley-
Interscience, 2000.
[5] K. Blekas, A. Likas, N. Galatsanos, and I. Lagaris, “A
spatially constrained mixture model for image segmen-
tation,” IEEE Transactions on Neural Networks, vol. 16,
no. 2, pp. 494–498, 2005.
[6] H. Greenspan, G. Dvir, and Y. Rubner, “Context-
dependent segmentation and matching in image data-
bases,” Computer Vision and Image Understanding, vol.
93, no. 1, pp. 86–109, 2004.
[7] P. Dempster, N. M. Laird, and D. B. Rubin, “Max-
imum likelihood from incomplete data via EM algo-
rithm,” Journal of the Royal Statistical Society, vol. 39,
no. 1, pp. 1–38, 1977.
[8] D. Peel and G. J. McLachlan, “Robust mixture modeling
using the t-distribution,” Statistics and Computing, vol.
10, pp. 339–348, 2000.
[9] C. Carson, S. Belongie, H. Greenspan, and J. Ma-
lik, “Blobworld: image segmentation using expectation-
maximization and its application to image querying,”
IEEE Transactions on Pattern Analysis and Machine In-
telligence, vol. 24, no. 8, pp. 1026–1038, 2002.
[10] D. Martin, C. Fowlkes, D. Tal, and J. Malik, “A database
of human segmented natural images and its application
to evaluating segmentation algorithms and measuring
ecological statistics,” in Proceedings of the 8th Inter-
national Conference one Computer Vision, July 2001,
vol. 2, pp. 416–423.
[11] N. Vlassis and A. Likas, “A greedy EM algorithm for
Gaussian mixture learning,” Neural Processing Letters,
vol. 15, pp. 77–87, 2002.
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