A Bayesian Segmentation Methodology for Parametric Image Models.
AbstractRegion-based image segmentation methods require some criterion for determining when to merge regions. This paper presents a novel approach by introducing a Bayesian probability of homogeneity in a general statistical context. Our approach does not require parameter estimation and is therefore particularly beneficial for cases in which estimation-based methods are most prone to error: when little information is contained in some of the regions and, therefore, parameter estimates are unreliable. We apply this formulation to three distinct parametric model families that have been used in past segmentation schemes: implicit polynomial surfaces, parametric polynomial surfaces, and Gaussian Markov random fields. We present results on a variety of real range and intensity images.
Introduction
The problem of image segmentation, partitioning an image into a set of homogeneous regions,
is a fundamental problem in computer vision. Approaches to the segmentation problem can be
grouped into region-based methods, in which image subsets are grouped together when they share
some property (e.g., [26]); edge-based methods, in which dissimilarity between regions is used to
partition the image (e.g., [9]); and combined region- and edge-based methods (e.g., [22]). In this
paper, we present a new, Bayesian region-based approach to segmentation.
A standard approach to region-based segmentation is to characterize region homogeneity using
parameterized models. With this approach, two regions are considered to be homogeneous if they
can be explained by a single instance of the model, i.e., if they have a common parameter value.
For example, in range image applications, object surfaces are often modeled as being piecewise
algebraic (e.g., [30]). The parameters of such a surface are the coefficients of the corresponding
polynomial. Two regions are homogeneous, and thus should be merged, if they belong to a single
polynomial surface (i.e., if the coefficients for their corresponding polynomials are the same).
In practice, a region's parameters cannot be observed directly, but can only be inferred from the
observed data and knowledge of the imaging process. In statistical approaches, this inference is
made using Bayes' rule and the conditional density, p(y k ju k ), which expresses the probability that
certain data (or statistics derived from the data), y k , will be observed, given that region k has the
parameter value u k . In typical statistical region merging algorithms (e.g., [27]) point estimates
in the parameter space are obtained for different regions, and merging decisions are based on the
similarity of these estimates. Often the maximum a posteriori (MAP) estimate is used, which is
obtained by maximizing p(y k ju k ).
An inherent limitation of nearly all estimation-based segmentation methods reported to date
is that they do not explicitly represent the uncertainty in the estimated parameter values, and
therefore, are prone to error when parameter estimates are poor (one notable exception to this
is the work of Szeliski [29], in which both optimal estimates and the variance in the estimates
are computed). To overcome this problem, we present a Bayesian probability of homogeneity that
directly exploits all of the information contained in the statistical image models, as opposed to
computing point parameter estimates.
The probability of homogeneity is based on the ability to formulate a prior probability density
on the parameter space, and assess homogeneity by taking the expectation of the data likelihood
over a posterior parameter space. This type of expectation was also used by Cohen and Fan to
formulate a data likelihood for segmentation, applied to the Gaussian Markov random field model
[5]. In their work, segmentations are defined by a space of pixel labelings, and through window-based
iterative optimization, a segmentation is determined that maximizes the data likelihood. By
considering the region-based probability of homogeneity, we introduce a different decomposition
and prior on the space of segmentations.
Our probability of homogeneity can also be considered as a function of the Bayes factor from
recent statistical literature [1, 15, 23, 28], which has been developed for statistical decision making,
such as model selection. A detailed description of our model and the derivation of the Bayesian
probability of homogeneity are given in Section 2.
In addition to providing an explicit accounting of the uncertainty associated with a segmentation
(which could feasibly be used in higher level vision processes, such as recognition), our
method extends in a straightforward way to allow application of multiple, independent image
models. Furthermore, our framework does not require the specification of arbitrary parameters
(e.g., threshold values), since context dependent quantities can be statistically estimated.
We have applied our Bayesian probability of homogeneity to segmentation problems using three
popular model families: implicit polynomial surfaces, in Section 3; parametric (explicit) polynomial
surfaces, in Section 4; and Gaussian Markov random fields for texture segmentation, in Section 5.
In Section 7 we present experimental results from each of the model families. These results were
obtained using the algorithm described in Section 6.
Further, we have developed special numerical computation methods for directly computing the
probability of homogeneity using the parametric models presented in this paper [17], without using
large data set, asymptotic assumptions. For this reason, we were able to consider small region
sizes for the implicit polynomial results presented in Section 7. Previous techniques that obtain
expectations over the parameter space have used some form of this assumption [3, 5, 27].
In principle, our Bayesian probability of homogeneity could be applied to most region-based
segmentation algorithms. In related work we have used the probability of homogeneity as a key
component for generating probability distributions of alternative segments and segmentations [18].
2 The General Probability of Homogeneity
This section provides the formulation and derivation of the general probability of homogeneity.
The version presented here determines the probability that the union of two regions is homoge-
neous; probabilistic treatment of more general region sets appears in [16]. Section 2.1 defines the
random variables and densities used in our general statistical context. In Section 2.2 we derive
expressions for the probability of homogeneity.
2.1 General Model Definitions
The elements of an image, D, are arranged in a 2D array. A given point D[i; j] will have a set of
neighbors. Using standard four-neighbors, this set is:
A region, R k , is some connected subset of D. Two regions, R 1 and R 2 , will be called adjacent if
there exists some D[i 1 are neighbors.
It is often profitable to begin with some initial partition of the image into small regions, and
to construct new segmentations by combining these regions. This is a standard approach taken
in the region merging paradigm. For instance, Sabata et al. initially generate an image of sels,
which corresponds to regions that have near-constant differential properties [26], and Silverman
and Cooper begin with an initial grid of small regions [27]. We denote the initial set of regions as
R, which represents a partition of D.
For each R k 2 R we associate the following: a parameter space, an observation space, a degradation
model, and a prior model (see Table 1). The parameter space directly captures the notion of
homogeneity: every region has a parameter value (a point in the parameter space) associated with
it, which is unknown to the observer. The observation space defines statistics that are functions of
the image elements, and that contain information about the region's parameter value. We could
use the image data directly for the observation, or could choose some function (possibly a sufficient
statistic, depending on the application) that increases the efficiency of the Bayesian computations.
Although the parameter values are not known in general, a statistical model is introduced which
uses two probability density functions (pdf's), yielding the prior model and the degradation model.
The prior model is represented by a density on the parameter space (usually uniform), before any
Parameter space A random vector, U k , which could, for instance, represent a
space of polynomial surfaces.
Observation space A random vector, Y k , which represents the data or functions of
the data x 2 R k .
Degradation model A conditional density, p(y k ju k ), which models noise and
uncertainty.
Prior Model An initial parameter space density, p(u k ).

Table

1. The key components in our general statistical framework.
observations have been made. The degradation model is represented by a conditional density on
the observation space, for each given parameter value, and can be considered as a model of image
noise. These components have been used in similar contexts for image segmentation [8, 29].
In order to determine the probability of homogeneity, it will be necessary to consider a statement
of the form H(R 1 [R 2 which corresponds to the condition that R 1 [R 2 is homogeneous,
which corresponds to the condition that R 1 [R 2 is not homogeneous. We
will use H to represent the condition H(R 1 [R 2 and :H to represent
Note that if H is true then R 1 and R 2 share the same parameter value.
2.2 Probability of Homogeneity Derivation
In this section we derive an expression for the Bayesian probability of homogeneity, given observations
from R 1 and R 2 . The result is an expression requiring three integrations on the parameter
space, given by (2) and (5). The vectors Y 1 and Y 2 represent the observation spaces of R 1 and R 2
respectively. In other words, the random vector Y 1 corresponds to applying functions to the data
variables, D[i; j], which belong to R 1 . Similarly, Y 2 is obtained from R 2 . The observations serve
as the evidence used to determine the Bayesian probability of homogeneity, which is represented
as We can apply Bayes' rule to obtain
The denominator of (1) is the standard normalizing factor from Bayes' rule, over the binary sample
space, fH; :Hg. The expression P (H) represents the prior probability of homogeneity, i.e., the
probability that two adjacent regions should be merged, when y 1 and y 2 have not been observed,
and in practice we usually take P This represents a uniform distribution
over the binary sample space. The implications of this and other prior distributions is discussed
in [18].
We can write (1) as
in which
This utilizes the reasonable assumption that p(y 1 which is further discussed
in [16]. The - 0 and - 1 (y decomposition of the factors contributing to the
posterior probability of homogeneity. When either of these ratios takes on the value of 1, it
essentially does not bias the posterior probability of homogeneity.
Using a common prior density p(u 12 ), and an assumption that the observations y 1 and y 2
are independent when given the common parameter value, u 12 , we can write the denominator of
marginal with respect to U
Z
Z
Using (4), and the marginal over U k for each term of the numerator, we obtain:
-Z
-Z
Z
The ratio above (and similar forms) has appeared recently in work from the statistics literature,
and is termed a Bayes factor. Smith and Speigelhalter used a similar ratio for model selection
between nested linear parametric models [28]. Aitkin has developed a Bayes factor for model
comparison that conditions the prior model on the data [1]. Kass and Vaidyanathan present and
discuss some asymptotic approximations and sensitivity to varying priors of the Bayes factor [15].
Pettit also discusses priors, but with concern for robustness with respect to outliers [23].
Our approach extends in a straightforward way to the case in which we have m independent
observation spaces and parameter spaces. In this case, the posterior probability of homogeneity
can be expressed as [16]:
Y
l (y l
in which - l (y l
2 ) is similar to (5).
3 Implicit Polynomial Surfaces for Range Data
Surface models that correspond to the solution sets of implicit algebraic equations are treated
in this section, and parametric (or explicit) polynomial models are treated in Section 4. Bolle
and Cooper have modeled objects appearing in ranges images with patches of planes, spheres, and
cylinders for position estimation [3]. Faugeras and Hebert have used implicit quadric and planar
models for object modeling, segmentation, and recognition [7]. Taubin and Cooper have developed
an efficient estimation procedure for implicit polynomial curves and surfaces of arbitrary order,
with application to object recognition [30].
For this model D[i; j] represents a point in ! 3 , specified by [x coordinates. For simplicity
of notation, we will denote some element of the image D by x instead of D[i; j]. Rather than using
the indices, we will index points that belong to some region R k by x
3.1 The parameter manifold
An implicit polynomial equation is represented as
The constants a j , b j , and c j are positive integers, representing the exponents of each variable. The
used here indicates that we have an implicit function with x as the variables.
With the present formulation, there are redundant representations of the solution sets (i.e.,
there are many parameter vectors that describe the same surface in ! 3 ). It is profitable to choose
some restriction of the parameter space that facilitates the integrations in (5), but maintains full
expressive power. We use the constraints to constrain the parameter space
to a half-hypersphere, \Sigma N , termed the parameter manifold.
3.2 The observation space
The observation considered here is a function of the signed distances of the points x 2 R k from
the surface determined by u k , termed displacements. Define ffi(x; OE(\Delta; u k )) to be the displacement of
the point x to the surface described by the zero set fx : OE(x; 0g. The function ffi(x; OE(\Delta; u k
takes on negative values on one side of the surface and positive on the other.
We consider the following observation space definition (others are mentioned in [16]), and distance
approximation [3, 30]:
Note that we use y k instead of y k when the observation space is scalar. We chose to use the sum of
squares since we obtained improved integration efficiency with similar segmentation results when
experimentally compared to using the displacements directly. The distance approximation is good
for small displacements, and for our approach good approximations are only required for small
displacements. Large displacement errors will not cause difficulty because of the approximately
zero tail values of the chi-square pdf, which will be presented in Section 3.3.
3.3 The degradation model
To define the degradation model, we first need to express the density corresponding to the
displacement of an observed point from a given surface. We use a probability model for range-
scanning error used and justified in [3], and also used in [30]. The model asserts that the density,
p(ffiju), of the displacement of an observed point from the surface, OE(x; u), is a Gaussian random
variable with zero mean and known variance, oe 2 .
This degradation model is merely chosen as a representative of possible models that can be used.
In practice, for different imaging systems, other models may be more appropriate. For instance,
Mirza and Boyer use a t-distribution to model the degradation for robustness with respect to
outliers [20]. Ikeuchi and Kanade provide a detailed discussion of the modeling of a variety of
range-imaging sensors [12].
Since taking the sum of squares of independent standard Gaussian random variables yields a
chi-squared random variable, the degradation density using (8) is
Here y k is the sum-of-squares for a given region, R k , and parameter value u k , given by (8). Also,
\Gamma(\Delta) is the standard gamma function and m (the number of elements in R k ). The variance
oe 2 is estimated, and considered as part of the specified degradation model.
3.4 The prior model
Since the parameter space has been restricted to a bounded set, we can define the prior pdf to
have equal value everywhere on the parameter manifold. This captures the notion of uniformity
due to the lack of information; however, it is important to note that our choice of parameter
manifold affects the prior distribution on the space of surfaces. If other constraints were used on
the parameter space, and we assumed a constant-valued pdf, the distribution would be somewhat
different from the one we have selected here. Once some information is present, i.e., some observed
data points, this distinction becomes less important.
Since the density over the parameter manifold must integrate to one, the uniform density is just
the inverse of the surface area of the half hypersphere that defines the parameter manifold. The
prior model is p(u k
N , in which AN represents the area of the N parameter manifold, which
can be obtained through straightforward integration techniques [16].
4 Parametric (Explicit) Polynomial Surfaces
Parametric polynomial surfaces have been used in past segmentation work to model surface
patches in range imagery and sets of intensities in intensity images. In early work by Haralick
and Watson, the facet model was introduced in which intensity image subsets were approximated
by polynomials, representing an idealized image [10]. Besl and Jain have used polynomials of
variable order for segmentation and select the best model by analyzing fitting-error signs and the
mean-square error [2]. Leonardis et al. have also used bivariate polynomials of variable order
and select an appropriate image description through a cost/benefit objective function to obtain
a segmentation [19]. Silverman and Cooper have used explicit quadric and planar equations to
model surfaces patches in intensity images for clustering-based segmentation [27]. Sabata et al.
have used parametric polynomials to model surfaces in a hierarchical range image segmentation
scheme [26].
The general form of the parametric polynomial model is
um
in which am and b m are positive integers.
The observation space, Y k , represents a vector of point-to-surface displacements of the intensities
in R k , given a parameter value u k . For degradation, we use an additive Gaussian iid zero-mean
noise model, as considered in [27]. Hence, there is a Gaussian pdf associated with each element
of the observation space vector. Due to the independence of the noise model, the joint density is
obtained by taking the product of the individual displacement densities:
We define the prior model by assigning a uniform density to a compact portion of the parameter
space. The problem of selecting bounds for a uniform prior has been known to lead to difficulty
in Bayesian analysis, referred to as Lindley's paradox [28]. As the volume over which the unfiorm
density is defined increases, the ratio (5) decreases. We select P (H) in our experiments to appropriately
cancel the effects of the volume; however, it must be understood that the choice of prior
in this case significantly affects the probability of homogeneity. For our purposes, a problem of
this type only changes - 1 by some scaling factor, leading to the correct ordering of likely merges,
but an ambiguous termination criterion.
5 Texture Segmentation using a Gaussian MRF Model
Models of texture have been used extensively for segmentation. In this section, we consider the
application of our general probability of homogeneity to a Gaussian Markov random field (GMRF)
model for the problem of unsupervised texture segmentation. This problem has been considered in
numerous contexts, and an extensive survey that covers fractal models, operator models, structural
texture methods, and frequency domain techniques is provided in [25]. For our parameter space,
we use a special MRF formulation known as the SAR model, which is described in [14]. This
model has been applied to texture segmentation of intensity images in [4, 5, 27], and has recently
been extended to texture modeling and segmentation of color images [21]. In particular, Cohen
and Fan have considered maximizing likelihoods formulated through the integration on the GMRF
parameter space [5], which is similar to the approach taken here; however, we are interested in
iteratively merging region pairs that maximize the probability of homogeneity.
An image element, D[i; j] represents a single intensity, X[i; j], treated as a random variable. We
have an N-dimensional parameter space, which represents the interaction of a pixel with a local
set of neighboring pixels. The order of an MRF indicates the size of the local neighborhood that is
considered. In a first order MRF, 4, corresponding to interactions of X[i; j] with X[i
1]. For any general order of MRF interactions, the image element
of the l th parameter interaction is denoted by T l (x). Hence, in general at some point X[i;
the model is
We could also consider the intensity mean in R k , - k , as part of the parameter space; instead,
we chose to estimate the mean using the region data for our experiments.
The observation space, Y k , is defined as a vector that corresponds to all of the intensity data,
x[i; j], in some region R k . Hence, the dimension of Y k is equal to the number of pixels in R k .
We assume that the noise process that occurs in the linear prediction (12) is Gaussian. The
joint density that we use over the points in R k is not a proper pdf; however, it has been considered
as a reasonable approximation and used in previous segmentation schemes [4, 21, 27]. We obtain
the degradation model by taking the product of the density expressions over each of the individual
pixels:
in which oe 2
k represents the variance over R k . The variance could also be considered as part of the
parameter space; however, we estimate the variance for each region.
For the texture model we used that same prior that was used in Section 4.
6 Computation Issues
We provide an outline of the algorithm that was used to generate the experiments presented in
Section 7. Our algorithm resembles agglomerative clustering [27]; however, the standard metric-based
merging criterion is replaced by our probability of homogeneity:
1. For each pair of adjacent regions R i store the result
in a priority queue with elements sorted by probability.
2. Remove the first pair from the queue, Rm 1
, and update R by adding Rm j Rm 1
and
removing Rm 1
and Rm 2
3. For each R i adjacent to Rm , compute P (H(R i [Rm )jy i ; ym ), and insert the result into the priority
queue.
4. If the probability of the first pair in the queue is less than P c (or alternatively, the number of
regions in R is c) then terminate
5. Go to 2
With regard to Line 4, many clustering algorithms require the specification of the number of final
clusters. Some recent work has been done specifically addressing the problem of determining the
number of clusters (known as cluster validation), in the context of image segmentation applications
[13, 32].
The integrals arising from (5) were computed using a specialized Monte Carlo-based technique
for the implicit surface model, and an ellipsoidal decomposition technique on the parameter spaces
for parametric polynomials and MRFs. These computation methods are discussed in [17].
7 Experiments
This section presents experimental results using the models presented in Sections 3-5. For each of
the three models, we have performed segmentation experiments on dozens of real images.

Figure

shows eight range image results, using either an implicit planar or quadric model. For
each result, we first show the intensity image (or a synthetic rendering) for the range image, and
then the segmentation result. Figure 1.a also shows the initial region set, R.
The variance in the degradation model was only estimated once for a given range image set. To
yield accurate placement of points that are close to segment boundaries, we performed maximum-likelihood
supervised clustering on the segmentation output from our merging algorithm. We first
discard very small final segments, and then for each image point, x, we choose the region label, l,
such that
in which R a represents the set of regions containing the points that are adjacent to x, and - u k is
a least-squares parameter estimate in region R k .
The initial region set, R, was obtained by combining a small grid with the edge map produced by
running the Canny edge detector on the corresponding intensity image. When building an initial
region set we would like as few nonhomogeneous initial regions as possible, and the application of
the edge detector provides slightly improved performance near boundaries. The edge detector was
applied to synthetic renderings of the range data (generated by the method discussed by Sabata
et al. [26]) for the images in Figures 1a-c, and a true, corresponding intensity image was used for
the remaining range images. We performed no parameter tuning with the Canny edge detector;
therefore, there are many missed edges and extra edges in the initial region maps.
The first three images presented in Figure 1 belong to the MSU range image set, which is
often used for evaluation of segmentation algorithms. Hoffman and Jain performed smoothing on
this data, and then iteratively clustered the range points based on position and surface normal
estimation [11]. A conservative clustering is obtained, and additional merging occurs after surface
type classification and boundary analysis is performed. Sabata et al. also provide some results on
this imagery [26]. They also perform smoothing on the data, use a pyramidal clustering algorithm
on synthetic renderings of the range data, and finally merge regions using a squared-error criterion.
We obtained the remaining images in Figure 1 in our lab using the K 2 T GRF range scanner
setup. This range image set is more suitable for demonstrating our framework, since typical
segments have approximately 400-500 points, while the typical segment sizes in the MSU images
have around 4000-5000 points (the noise levels for the two sets are comparable). This leads to
greater uncertainty in the probabilities; however, good segmentations were obtained.
The next two images show the application of the parametric polynomial model to the segmentation
of intensity images. We do not necessarily propose parametric polynomials as the most
appropriate model for intensity-based segmentation, but instead are demonstrating the success of
our methodology for this given model family, which has been considered previously [27]. Figure
2.a is an image of a tape dispenser, and Figure 2.b shows a plastic slinky. Since the model family
is not as accurate (i.e., the images are not likely to have an underlying polynomial model with
a.
f.
h.
c.
g.
i.

Figure

1. Some of our range image segmentation results.
additive Gaussian noise) the number of classes is more difficult to select; therefore, we show results
in this section for user-specified, fixed class numbers.

Figure

2.c-f show four texture results, which were obtained using a third-order MRF. Figure 2.c
is composed of Brodatz textures. Figure 2.d is a texture image that was constructed for testing
texture segmentation algorithms [6]. Figure 2.e shows a four-class texture result from an image
that was obtained by piecing together photographs of different quilts. Figure 2.f is an image of
NASA Magellan space probe data of Venusian terrain. Some recent discussion and comparisons
of models for texture segmentation can be found in [6, 24, 25], and some texture segmentation
experiments on similar imagery appear in [4, 27, 31, 32].
We have only considered coarse segmentations, which are obtained from an initial region set, R,
that is formed by partitioning the image into square blocks. We present these coarse segmentations
a. b. c.
d. e. f.

Figure

2. Some of our parametric polynomial and texture segmentations.
since: 1) some minimal region size in needed before the model contains useful information [25]; 2)
supervised methods exist for providing good boundary localization, given a coarse segmentation
[4, 5, 6].
Conclusions
In this paper, we have presented a new approach to region-based segmentation. The key to our
approach is a new formulation for the probability that the union of two regions is homogeneous.
Our approach does not require parameter estimation, and is therefore particularly beneficial for
cases in which estimation-based methods are most prone to error. Our experiments provide strong
support for our Bayesian formalism based on the quality of the segmentation results, and the broad
class of models considered, which indicates the general applicability of our methods. The segmentations
that we obtained with the highest-probability-first algorithm are good, in the context of
other recent segmentation results for each of the model types. Further, our Bayesian formalism
has been used in algorithms that generate probability distributions of alternative segments and
segmentations on real imagery [18].

Acknowledgements

We thank Ken Moroney for his work on some experiments. We also thank Pat Flynn, the MSU
Pattern Recognition and Image Processing Lab, Hans du Buf, Becky Casta~no, Fang Liu, and
Narendra Ahuja, and the anonymous reviewers. This work was sponsored by NSF #IRI-9110270.



--R

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Segmentation through variable-order surface fitting
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A facet model for image data.
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Integrating region growing and edge detection.
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bayes factor;range images;statistical image segmentation;bayesian methods;markov random field;texture segmentation;likelihoods