split-and-merge image segmentation based on localized feature analysis and statistical tests

CVGIP: GRAPHICAL MODELS AND IMAGE PROCESSING

Vol. 53, No. 5. September, pp. 457-475, 1991

Split-and-Merge Image Segmentation Based on Localized Feature Analysis and Statistical Tests*

SHIUH-YUNG CHEN AND WEI-CHUNG LINT

Depurtment of Elrctricul Engineering and Computer Science, Northwesiem University. Evwzsion, Illinois 60208

AND

CHIN-TU CHEN

Department of Radiology, The University of Chicago, Chicago, Illinois, 60637

Communicated by John W. Woods

Received April 18, 1990; accepted December 18, 1990

In this paper, an adaptive split-and-merge image segmentation algorithm based on characteristic features and a hypothesis model is proposed. The analysis of characteristic features provides the requisite parameters that serve as constraints in the hypothesis model. The strength of the proposed method lies in the fact that the parameters in the algorithms are computed automatically and depend only on the context of the image under analysis. One of the key processes, the determination of region homogeneity, is treated as a sequence of decision problems in terms of predicates in the hypothesis model. Experimental results on natural scene pictures and medical images are included to demonstrate the robustness of the algorithm. 0 1991 Academic Press, Inc.

1. INTRODUCTION

Image segmentation has been the subject of extensive research in the areas of computer vision and pictorial pattern recognition over the past decades. The primary goal of image segmentation is to divide an image into regions that are meaningful for their correspondence to physical objects or their parts. Despite considerable ef- forts over a long period, there is no “perfect” segmentation algorithm available. According to Marr [I], there are two reasons why the theory and practice of segmentation remain primitive: (1) the exact goals of segmentation are, in most cases, impossible to formulate precisely, and (2) regions that have “semantic” importance do not always have any particular visual distinction. Therefore, in most cases, image segmentation results in regions that cover semantically distinct visual entities or regions that are

* This project is supported by the Whitaker Foundation. f TO whom correspondence should be sent.

fragmented. Without the domain-specific knowledge about the objects in an image, it is almost impossible to obtain a satisfactory result.

To rectify this drawback, an image analysis system often adopts a two-stage process consisting of low-level processing that segments the image into regions roughly corresponding to objects, or parts of objects, and high- level processing that is devoted to splitting or merging these regions on the basis of domain-specific knowledge [2-51. It is clear that at the low-level processing stage, domain-specific knowledge should not be incorporated in the algorithms since they arc intended to be universally applicable. In other words, the segmentation process at the low-level stage should not include a priori knowledge about the objects in a scene and can employ only general- purpose and domain-independent models. Within this framework, the goal of the initial image segmentation process is to provide adequate information to support high-level reasoning. The information flow in this process is basically bottom-up. During high-level processing, a knowledge (or model)-based top-down process is invoked to refine and improve the results from initial segmentation. Of course, the better the results obtained from the initial segmentation process, the less effort required in the refinement process.

Recently, various expert systems for image processing (ESIP) were proposed [2-121 to facilitate the development of image understanding systems (IUSs). The objective of an ESIP is to implement effective image analysis processes by making full use of available image processing primitives (e.g., edge detection and region growing operators), while an IUS is developed to implement intel- ligent machines with visual perception capability. By using an ESIP as a module in an IUS, the flexibility and

457

1049-9652/91 $3.00 Copyright 0 1991 by Academic Press, Inc.

All rights of reproduction in any form reserved.

458 CHEN, LIN, AND CHEN

capability of image analysis in an IUS can be drastically increased. However, most of the existing image processing primitives, either purely statistical or purely structural, require human operators to adjust the parameters in the algorithms and the ability to select appropriate parameters is usually crucial to the success of extracting meaningful features, In general, it is very difficult to estimate a priori the performance of a primitive on a given image and one has to proceed by trial and error in order to determine the optimal parameters. The situation gets more unmanageable in the case where a combination of several primitives is needed.

In this paper, we propose an adaptive split-and-merge image segmentation algorithm based on characteristic features and a hypothesis model. The analysis of characteristic features provides the requisite parameters serving as constraints in the hypothesis model. The proposed method is strong because the parameters in the algorithms are computed automatically (rather than provided by users) and depend only on the context of the image under analysis. One of the key processes, the determination of region homogeneity, is treated as a sequence of decision problems in terms of predicates in the hypothesis model. This new algorithm, which is free from user intervention, is used to perform the data-driven processing at the low-level stage in order to support the goal- driven processing at the high-level stage in a medical image understanding system currently under development [4]. The system is designed to identify the anatomically meaningful entities in an X-ray computed tomography (CT) and in a magnetic resonance (MR) image and to identify the functionally meaningful entities in a positron emission tomography (PET) scan.

In the next section, the main principle and properties of the split-and-merge algorithm are reviewed. A comparative study of testing region homogeneity using features analysis and statistical inference is reviewed in Section 3. In Section 4, the proposed algorithm is described. Fi- nally, several experimental results are given to demonstrate the capability of the proposed method.

2. THE FRAMEWORK OF THE SPLIT-AND-MERGE IMAGE SEGMENTATION ALGORITHM

LetN={1,2,. . . , n}, (x, y), x, y E N, be the spatial coordinates of a pixel in a digitized image, and G = (0, 1, . . . , Z} be the set of nonnegative integers representing the gray levels of the pixels. Let the domain of an image with n rows and it columns be denoted by I, i.e., Z = N x N. Then, an image function can be defined as the mapping f: N x N + G. The gray level of a pixel at (x, y) is denoted f(x, y), A region R of an image is defined as a connected homogeneous subset of Z with respect to some criterion such as gray tone or texture. Let P denote a

logical predicate for homogeneity measure defined on the R such that

1

true if X(R) E !&?I, P(R) =

false otherwise,

where %? R -+ 5!Zj is a function for evaluating the homogeneity of a region R and 9 is a predefined subset of range 9. A segmentation of an image is a partition of Z into several homogeneous regions Ri, i = 1, . . . , m. The relationships among these regions are defined as follows:

1. Z = ‘JE, Ri, 2. Ri II Rj = 0, 1 5 i, j I m, and i f j, 3. P(R;) = true, for all i, 4. P(Ri U Rj) = false, 1 5 i, j 5 m, i f j, and Ri, Rj

are adjacent.

The region-based segmentation algorithms in the litera- ture can be classified into three categories: pure merging [13-161, pure splitting [17, 181, and split-and-merge schemes [19-231. In the first scheme, the picture is first divided into many small primitive regions (or even pixels) which are then merged to form larger regions on the basis of certain homogeneous criterion. This is achieved by starting with a partition that satisfies (3) and proceeding to fulfill (4). In contrast, a pure split method views the entire image as the initial segmentation. Then, it succes- sively splits each current segment into quarters if the segment is not homogeneous enough. The processing starts in a condition satisfying (4) and proceeds to fulfill (3). In split-and-merge scheme, the efficiency of processing is improved by first partitioning the image into square subregions and then splitting them further if they are not homogeneous. Afterwards, a merging process is applied to those adjacent segments that satisfy some uniformity criterion.

Since these algorithms are region-based, as opposed to edge-based or pixel-based, some useful features such as texture and other spatial or geometric measures can be easily incorporated in the algorithms. A typical split-and- merge algorithm has four specific processing phases and requires several input parameters, including the regular and the relaxed predicates, P, and P,, and the initial cut set size. The predicates are used to test for region homogeneity. Pyramid (or quad tree) is employed as the basic data structure. The initial cut set size specifies the level of the quad tree at which leaf nodes are formed. Figure lb illustrates a tree construction corresponding to a segmented picture in Fig. la, where level 2 is assigned as the initial cut set. The four phases are designed to operate on different abstract levels of a quad tree and are detailed as follows.

SPLIT-AND-MERGE IMAGE SEGMENTATION 459

Level 0 A.

a b

221 222 223 224 231 232 233 234 321 322 323 324 331 332 333 334

FIG. 1. (a) Example of a region segmentation: (b) segmentation represented by tree structure.

Phase 1: Splitting the Quad Tree. The predicate P, is invoked to determine the homogeneity of a given leaf in the quad tree. If the predicate results in falsity, a split is performed to create four children quad nodes of the same size. Otherwise, the split is made permanent. This is formulated as

U:=l Rij if P,(Ri) is false, Ri 3

Ri otherwise.

The predicate P, is defined as

Pn (Ri)

true =

if ,y,“~“~ [fk ~11 - cxy;z Lfk ~11 5 cl, x, I . , false otherwise,

where f(x, y) is the gray level of the pixel at location (x, y), and cl is a threshold. Each new leaf node is in turn considered for additional splitting. Such extension of tree continues until all the leaf nodes satisfy the uniformity criterion.

Phase 2: Merging Quad Siblings in a Branch. Fol- lowing the quad tree splitting, the quad siblings are merged if they are all leaves and the uniformity predicate applied to the union of the four quads is true:

+ Ri = U:=j Rij if P,,(U;=l Rij) is true,

(Rii, Ri2, Ri3y Rid) otherwise.

Phase 3: Grouping Adjacent Siblings in Different Branches. A grouping process is performed to merge the quad tree leaves which are spatially adjacent but re-

side on different branches of the tree. This step is necessary due to the fact that the artificial cutting of the image may segment a single object into different parts embed- ded in the quad tree. After the processing, the segmentation can be represented as a region adjacency graph which is transformed from the quad tree structure. The test of homogeneity for two adjacent leaves is defined as

(Ri, Rj) + Rk = Ri U Rj if P,(Ri U R,i) is true,

CR;, Rj) otherwise,

where Ri and Rj are adjacent.

Phase 4: Eliminating Small Regions. The final step of the split-and-merge algorithm is to eliminate the small regions. In this step, small regions are merged with an adjacent larger region. There are two reasons for the occurrence of these small regions. The first is due to the existence of narrow transition zones between large regions. The second reason is the existence of high-frequency noises. The homogeneity test here is very similar to that of phase 3 but adopts a different predicate as follows,

(Ri 3 Rj) 3 Rk = Ri U Rj if P,(Ri U Rj) is true,

(Ri, Rj) otherwise,

where Ri and Rj are adjacent. The predicate P, is defined as

Pr (Ri U Rj)

true =

if 1 C fk Y)ISi - $zER ./lx’, yYS,j( 5 82, (x.yER, .I’ ,

false otherwise,

460 CHEN,LIN,ANDCHEN

where f(x, y) is the gray level of the pixel at location (x, y), ~2 is a predefined threshold, and Si and &y)ER, f(x, y)/ Si denote the size and the average intensity of region Ri, respectively.

Segmentation of an image is usually accomplished by applying the operators described in phases 1 to 4. The two predicates, P, and P, , are easy to implement and are effective for most of the images. However, segmentation for a complicated image may not be satisfactory when just using such a global measurement for testing the region homogeneity. In addition, the determination of thresholds cl and c2 that is crucial to the success of the algorithm is done manually in [19]. In this paper, an improvement on the measure of region homogeneity is proposed in Section 4. In the next section, a review of region homogeneity analysis is given and the performances are also evaluated.

3. REVIEW OF REGION HOMOGENEITY ANALYSIS

The test of region homogeneity is a crucial step to any region-based algorithm. An improperly selected criterion often results in over- or undersegmentation. In this section, we give a review of two types of methodologies which are widely used for evaluating region homogeneity, i.e., feature analysis and statistical hypothesis.

3.1. Characteristic Feature Analysis

Let T denote the domain of a feature and M be the set of nonnegative integers. Histogram analysis is one of the popular tools for analyzing the characteristics of images. A histogram H of an image is a mapping from T to M which provides the frequencies of the various values of the desired feature (e.g., gray-level distribution or local texture measure). If an object in some area of the image possesses uniform intensity, the histogram may have a peak (or mode) around the intensity levels of the pixels in that area. When the histogram consists of a small number of distinct modes, it is likely that the image can be segmented into several regions with similar properties. These regions are formed by grouping the pixels whose feature values cluster around the same mode.

The thresholding technique is widely used to separate distinct modes in a histogram. In general, the objective of a thresholding method is to determine the set of threshold values Tk E T, k = 0, 1, 2, 3, . . . , m, where m is the number of distinct modes in the histogram such that

S(x, y) = k’

if Tk,-, % g(x, y) < TV, k’ = 1, 2, . . . , m,

where S(x, y) and g(x, y) denote the segmented and the characteristic feature (e.g., gray level) of the pixel at the

coordinates (x, y) before and after thresholding, respectively [50]. These desired thresholds can be determined by a function 9 defined as

Tk = s”(X, Y, 2(X, Y), dx, Y)),

where 2(x, y) denotes the local property of the pixel at (x, y). When B depends only on g(x, y), the technique is referred to as the “global point-dependent thresholding” method. If 3 depends on both g(x, y) and 2(x, y), it is referred to as the “global region-dependent thresholding” technique [52]. Some popular global point-dependent thresholding techniques are as follows.

l Mode method [24]. Many types of images contain objects that occupy a different range of gray levels than their background. If these ranges are sufficiently separated, the object and background subpopulations give rise to distinct peaks on the image’s histogram. A threshold can be chosen at the bottom of the valley between these peaks.

l Ostu’s method [25]. This technique is based on discriminant analysis [26-291. The threshold operation is regarded as the partitioning of gray-level distribution of an image into two classes C, = (0, 1, . . . , t} and Ci = {t + 1, t + 2, . . . , l} at gray level t E G = (0, 1, . . . ) I}. The optimal threshold t* can be determined by minimizing the criterion function defined on the two classes:

d t*=min - i 1 2 1EG uo

where

4 = $6 - Uo12Pi 7

and ni and IZ are the number of pixels with gray level i and the total number of pixels in the image, respectively.

l Histogram concavity analysis [30]. For histograms with overlapping gray-level subpopulations, there may be no distinct valley for threshold selection. In such a case, the threshold is selected at the “shoulder,” i.e., the inter- section of the two distributions peaked respectively at background and object. Since both the valleys and the shoulders correspond to the concavities in the histogram, the analysis of histogram concavities is performed to determine the optimal threshold.

l Entropic method. Let G = (0, 1,2, . . . , 1} be a set of nonnegative integers representing the gray levels. Let


the number of pixels with gray level i be ni and the total number of pixels be n. The histogram is considered as an l-symbol source with observed occurrence probability of each symbol i E G defined as

piA. n

The average information obtained for each symbol source is defined by

On the average, the information (or entropy) over the whole symbols can be defined as follows:

H = i p; log, . i=O

On the basis of the concept of entropy, the object and background gray-level distributions can be described by the respective entropy functions H,(t) and Hb(t). The optimal threshold t* is obtained by maximizing the evaluation function $(t) as

t* = max $(t), IEG

where

$0) = H,,(f) + H/,(t).

Several variations of this technique can be found in 131- 331.

l Moment-preserving method [34]. The threshold values are calculated deterministically on the basis of the observation that the moments of an image to be thresholded are preserved in the output image. The ith moment mi is defined as

mi = k tzc t’h(t), i = 1, 2, 3, ,

where G is the set of gray levels, h denotes the intensity histogram function, n is the total number of pixels in the image, and the 0th moment m. is defined to be 1. The threshold t* is obtained from the gray-level histogram of the image by choosing t* as the p-tile, where p is given by

z. - ml P = cc: - 4co)112

and

m1m3 - rn: m711m2 - mom3 co =

m2 - rn: ’ m2 - rnf ’

<c: - 4c#‘2 - Cl z= 2 *

l Minimum error method [35]. They gray-level histogram is regarded as an estimate of the probability density function p(g) of the mixture population composed of the gray levels of the object and the background pixels. It is usually assumed that each of the two components p(g 1 i) of the mixture is normally distributed with mean pi and standard deviation 6i and a priori probability Pi ; that is,

P(g) = 2 piP(g I 9,

where

P(Rli)=$--siew - i (g - Pi) 2sf 1 .

The threshold value can be selected by solving the qua- dratic equation

(g - Pd2 s: + log, s: - 2 log, P, = (I: ;;2)2

2

+ log, s: - 2 log, P2.

The techniques for global thresholding based on region-dependent techniques are as follows.

l Histogram transformation [36-401. The histogram is first transformed into one with deeper valleys and sharper peaks so that the mode method can be applied to determine the threshold. A common approach to obtain the new histogram is to weight the pixels of the image on the basis of their local properties.

l Second-order gray-level statistics [43]. The co- occurrence matrix is introduced by Haralick ef al. [41] for texture analysis and generalized in [42]. In the original definition, a co-occurrence matrix M(d,+) is one whose (i, j) entry is the relative frequency of occurrence for two neighboring pixels with gray levels i and j, separated by distance d and with orientation 4. On the basis of the matrix, two new histograms can be obtained:

(1) A histogram based on the near-diagonal entries of M. It always has a deep valley between the object and the background gray levels.

(2) A histogram based on the off-diagonal entries of M. It often has a sharp peak between the object and the background gray levels.


The threshold in which the valley in (1) overlaps with the peak in (2) is chosen.

l Relaxation method [44, 451. Each pixel is first probabilistically classified as members of multiple classes on the basis of its gray levels. Then, the probability assigned to each pixel is adjusted according to the probabil- ities of its neighboring pixels in terms of a compatibility function. This adjustment process is iterated until a certain criterion is satisfied. One of the attractive features of these methods is that they are suitable for parallel imple- mentation. However, it is difficult to determine the appropriate compatibility functions.

The major drawback in the global thresholding methods is that the valleys between the peaks of the histogram are long and flat, which makes threshold selection difficult. Furthermore, one cannot expect to use a single threshold to detect all or even most of the object boundaries in a scene without clear foreground-background separation (such as images of natural outdoor scenes, or complicated medical images). With such a concern, it is more appropriate to adopt a dynamic thresholding scheme [50] such as local thresholding or multithresholding techniques, which depend on the coordinates (x, y), gray level f(x, y), and local property 2(x, y) to determine the thresholds.

l Local thresholding [46-481. In this technique, the image is divided into several subimages, in which the characteristic features are analyzed. A set of thresholds is first determined on the basis of the analysis of each subimage. Afterwards, thresholds for each pixel are obtained by a mapping from the selected thresholds.

l Multithresholding. Many multithreshold selection techniques are the extension of global threshold methods. Basically, the global selection method is first applied to the image in which pixels are classified into two initial classes. The procedure is then applied to each class re- cursively until a certain criterion is satisfied.

A detailed comparative study of the aforementioned thresholding algorithms can be found in [49-521. Since these approaches are based on the assumption that different classes of segments of an image are represented by distinct “modes” in the distribution of suitably chosen features extracted from the image, the technique will fail if this assumption is not valid. Furthermore, most of the features are generally image-dependent and it is not clear how these features should be defined to produce good segmentation results.

3.2. Statistical Hypothesis Test

From the point of view of statistics, the image segmentation process can be regarded as a sequence of hypothesis tests [54-621. A region of arbitrary size is tested for

(1)

uniformity and if the result turns out to be not uniform it is further subdivided into smaller regions. On the other hand, a small uniform region tries to expand as much as possible as long as the hypothesis is not violated.

Assume that the image is a two-dimensional discrete random field which is a collection of random variables 1531. Each random variable, supposed to have a Gaussian distribution and to be stochastically independent, represents the gray level of a pixel in the image. The random field is separated into two parts RI and R2 which are formed by the sets of random variables {Xi, i = 1, 2, . . . ) m}, with mean 81 and variance 03, and { Yj, i = 1,2, . . . ) n}, with mean 13~ and variance &, respectively. The process of determining region homogeneity can be formulated as a hypothesis test of the gray-level distribution of the given region or the union of some regions. Two estimators, mean and variance, are involved in the test. The mean test is defined as

IX - r/ < E,

where

X=$Xi, F=i Yi, i=l i=l

and E is a predefined threshold. If the mean test is valid, the examination of variances is performed. The likelihood ratio test is applied for this purpose to test the hypothesis HO: 03 = 04, 8r and 13~ unspecified, against Hi: 83 f 04, 8, and & unspecified, by

fi

F

= ~~~~ (Xj - X)*&z - 1)

ET=, (I$ - Y)2/(m - 1)’ (2)

This statistic fi, has an F distribution with II - 1 and m - 1 degrees of freedom. The hypothesis Ho is rejected if the computed fi, 5 cl or if the computed fi, 2 ~2. The values of cl and c2 are usually selected so that, if 193 = 194,

where (Y is the desired significance level of this test. Up until this point, the method for testing the homoge-

neity of two regions has been expressed in Eqs. (1) and (2). The extension of homogeneity test for quad regions can be defined as follows. Suppose that each quadrant has K pixels, fi(i, j)‘s are the gray levels of the pixels in region R,, X, is the mean of the tth quadrant, and x is the grand mean of all the pixels in the four quadrants. The mean test for the quad regions is performed such that


lx, - Xl 5 E, t = 1, 2, 3, 4. (3)

Since the intensities of the pixels in the regions are independent and their distributions are assumed to be normal, the hypothesis test of uniformity for the four regions is defined by

A K c;l=, (X, - x)*/3

hF' = Xf=, Xti,j)ER, (f,(i, j) - Xl)*/4(K - 1)’ (4)

The statistic ff, has an F3,4(K-1) distribution. Given a significance level, the region is declared not uniform if fin is too high. Equations (3) and (4) were adopted as uniformity criteria in [57, 581.

In [59, 601, the technique of statistical inference is applied to region analysis within the framework of the sloped-facet model. In this model, each region is assumed to have a gray-tone surface which can be approxi- mated by a sloped plane as

f(r, c) = CXY + pc + y.

If the region is rectangular, its observed and estimated plane indexed by row (r) and column (c) coordinates can be represented by

with error

f(r, c) = &r + fit + y (5)

&“=22’ cir + PC + 3 - f(r, c)]*, (6)

where R and C are index sets for rows and columns, respectively. The coefficients of the estimated plane in Eq. (5) can be solved by minimizing the error in Eq. (6), which results in

rER rEC

Two nonoverlapping and adjacent regions are declared uniform if they are part of the same sloped surface. The F statistic is used to measure the significance of the depar- ture from the hypothesis of same surface as

fig = { i (4, - ai*)* C 2 r* + f (41 - /22)* C C c2

r c i- c

+ i

[(ail + &)(Arl2) + (pi + &(Ac/2) + $ - $I*

2[(Ar/2)*2:, x:, r* + (At/2)*/x, x, c* + I/z, 2,. l]

3

+ [(E: + E;)/(2 z, cc 1 - 6)] ’

where (Ar/2, AC/~) is the center of the joined regions, and aZi, /.?;, $i and Ei, i = 1,2, are the coefficients and the error of the estimated sloped planes, respectively. Given a significance level, the hypothesis is rejected if (~9 is too high.

Another method proposed in [61] consists of a two- stage algorithm using exact statistics to classify different types of regions and edges. At the first stage, the pixel is classified as a member of one of the nine classes Cl with bases VI, 1 = 1, 2, . . . , 9. Each basis Uk is a proper subset of the orthogonal polynomial basis {$ij = ni(x) . ?Tj(Y); i, j = 0, 1, 2, . . . , n - l}, where x and y denote the row and column, respectively, and rk(u) is the invari- ate nth-order orthonormal polynomial of degree k in U. In the absence of noise, the elements of class CL can be specified by the structural variation function

.h(x, Y) = +,zuL Bij * $‘ij(X, Y)*

In the presence of noise, the observed gray level f(x, y) coming from class CL is expressed as

f(X, Y) = ,zzuL bij * $ij(Xy Y) + C bij * 4ij(X9 Y) tJ,ZUL

= h Y) + ri(x, Y),

where f(x, y) and rj(x, y) denote the estimated structural variation and error, respectively. To determine a class CL to which a given pixel may belong, a test that each member of the set

VL = {(bij - B;j)* : +ij @ UL}

estimates the same variance is performed by using the criterion in [62]. To test whether two regions Ni and N2 described by the sloped surface functions

fi(X, Y) = 2 Bl,“$ij(x, Y) (X.>‘)ENI

and

.h(X’, Y’) = ,x, ZEN Bz’+ij(X’, Y’) .a’ 2


represent the same surface, the multiple hypothesis Ho,

X, E Bb:’ - BE’ = 0

X2 E B”’ - B’2’ = 0 10 10

X3 z 2 [Bt’$;j((Ax, AY) - Bf’$;j(-Ax, -AY)] = 0, (i+j)<Z

tion results. Another problem that biases the test is due to the inconsistency of the feature distributions in real samples and those assumed in the mathematical model. As a result, the test may lead to a wrong conclusion. In addition, even if the characteristics of the image being analyzed conform to the assumptions made in the mathe- matic model, it is still very difficult to define the best significant level (Y in advance to get satisfactory results.

is tested by applying exact statistics as

E2/(2n2 - no - 6) ’

where P is the orthonormal transformation from estimators 8 of X = [Xi], i = 1, 2, 3;

= T’ zz Q-12,

T = [T;j] = E TikTjk = 0, i f j,

k-1 1, i=j, $1 . [I 2= 22, a x3

hk, k = 1, 2, 3, are the eigenvalues of RQ’, R = [Rij] = E[Xi . Xj], i, j = 1,2, 3, and Q/denotes the Ith column of Q, 1 = 1, 2, 3; nA denotes the number of nonzero eigenvalues; and d is the improved error estimator used in [59]. The hypothesis Ho is rejected if the value fig is too high at the given significant level.

The method of statistical inference provides us with a theoretical model for decision making in a systematic manner. However, one must address the following issues when adopting statistics as the basis of a homogeneity test:

1. region size; 2. gray-level distribution; 3. selection of a significance level.

Inaccuracy of the test may occur due to insufficient samples. Let RI and R2 denote two adjacent regions with the (01, 02, . . * , %I) E 6J,

statistic measures (mean, standard deviation, size) (176, 9.5,30) and (179,0.63,2), respectively. The test of homo- and geneity for the two regions is approximately 0.08 (or 12.4) if Eq. (2) is used. Although it appears that the two regions should be merged, the low likelihood ratio makes the rejection of the hypothesis highly possible. Accordingly, _ many fragmented regions appear and degrade segmenta- ce,, 02. . . . > en) E R.

4. SEGMENTATION BASED ON HYPOTHESIS MODEL AND CHARACTERISTIC FEATURES

To remedy the drawbacks mentioned in the previous sections, we proposed a new scheme to test region homogeneity on the basis of characteristic features and a hypothesis model. The hypothesis is tested by using statistical or heuristic methods according to the characteristics of the region under examination. All the parameters which serve as the constraints in the hypothesis model are computed on the basis of the characteristic feature analysis. With this new scheme, an adaptive split-and- merge image segmentation algorithm can then be derived.

4.1. Hypothesis Model

In this model, the likelihood ratio test [63] is the backbone for testing a statistical hypothesis. Let Xi, x2, . . * 1 X, denote II mutually independent random variables with the respective probability density functions &(xi; 81, 02, . . . , 0,), i = 1, 2, . . . , n. Let R be the m-dimensional parameter space and w be a subset of a. The test of hypothesis Ho : (&, 02, . . . , 0,) E w against all the alternative hypotheses is defined as the ratio

Uw”) h(x,, x2, . . . ) x,) = A* = - L(fl*) ’

where L(o*) and L(O*) are the maximum of the likelihood functions


The hypothesis Ho is rejected if and only if

Mx,, X2, . . . 3 x,)=h*%Ao<l,

where A,, is a suitably chosen constant. The significance level of the test is given by

a = Pr[h(x,, x2, . . . , x,) 5 AC,; HOI.

On the basis of this principle, the analysis of region homogeneity can then be formulated as a statistical hypothesis test. Let Xl, . . . , X, and Y,, . . . , Y4 be the respective random samples corresponding to the gray levels of two regions. These two regions are assumed to have independent normal distributions N(8,, &) and N(&, &), respectively. Let w = {(&, &, &, 0,); ---co < 8, = e2 < co,0 < e3 = e, < co> and IR = {(e,, e2, e3, e4); --M < et, t$ < ~0, 0 < &, e4 < m}. The likelihood functions defined on the parameter spaces w and fi are, respectively,

and

C;=, (xi - e,)2 + ZL, (Yi - e,)2 2e3 1 (7)

ufv = (&jp”($J

[ E~zl (Xi - e,)2 EY=‘=, (Y; - e2J2

exp - 2e3 - 1 2e4 ’ @I

The maximum likelihood estimators u and w of 0, and e3 in Eq. (7) are, respectively,

P+4 ’

Cycl (Xi - U)* + Cyzl (Yi - U)2 w=

P+9

and the maximum of L(w) is

,-1 (P+qvz

L(o*) = G . ( ) (9)

Similarly, the maximum likelihood estimators for 8, , e2, 03, and e4 of Eq. (8) are, respectively,

Iq=, (X; - u,y w, =

P ’

w2 = x:=1 (Yi - U212

4 .

The maximum of L(LR) is

L(s1”) = (g-y:2(&)4’*. (10)

On the basis of Eqs. (9) and (lo), the likelihood ratio for testing the hypothesis of uniformity Ho: 8, = e2, & = e4 against all alternatives is shown as

[IX;=‘=, (Xi - U1)2/p]p’2 ’ [EYE, (q - U2)*/qlqi2

= {[E~E)=~ (Xi - U)’ + 27~1 (c - U)2]l(p + q)}(p+q)‘2

(11)

The likelihood ratio principle calls for the rejection of HO ifandonlyifA*SA,,< 1.

However, as the sizes of the regions decrease, the con- fidence level of the decision drops, and for this reason the pure statistical method must be supplemented with heuristic criteria [57]. Furthermore, if the gray-level distribution of the region under investigation violates the assumption of the mathematical model, a heuristic is also invoked to evaluate region homogeneity. Two uniformity predicates in terms of heuristic and statistical tests are applied in the proposed method to supervise the initial region growing (phases 1 to 3 of the split-and-merge scheme) and the final region formation (phase 4 of the split-and-merge scheme) processes, respectively. The predicates associated with the two processes are defined as

true if /IN(R) I el, PI(R) = (12)

false otherwise,

and

(true if TA : hc(Ri, Ri) I e2 or PF(Ri U Rj) = ( TB : hF(Ri, Rj) 5 e3, (13)

I false otherwise,


where ek, k = 1, 2, 3, are the thresholds and

hN(Ri) = max Lfk Y)l - min Lfk Y)l., (X..VER, (.r.vER,

hT(Ri 9 Rj) = IUR, - uR,(?

hF(Ri, Rj) = G (x.y)ER, [f(x, y) - uRi12~m}m’2{~~(x’,y’)ER, [f(x’, Y’) - UR,12/n)n’2

G (x..y)ER, [fk y) - u12 + &‘,y’)ER, bfcx’? Y’) - u12)b + n))(m+n)‘2 ’

UR, = (x’ FER f(x’7 Y’Yfb 3 J

u = (m ’ uR, + n . uRj)/(m + n),

where f(x, y) is the gray level at location (x, y), and m and n are the sizes of the two regions, Ri and Rj, being tested, respectively.

By applying Eq. (12) to the initial region creation process, the sampling size problem implicit in statistical inference can be overcome. The evaluation function hN in predicate PI is used to test the variance distribution in terms of maximum gray-level difference within a region. After the initial region creation process, Eq. (13) is used as the basis of the uniformity test in the final region formation process. Since some regions formed in the first process may have uniformly distributed gray levels, the evaluation function hG in test TA is employed. On the other hand, test TB is used to determine the homogeneity by assuming that the gray levels of the two regions have normal distributions. The function hF in TB is a likelihood ratio test based on Eq. (11). The thresholds el , e2, and e3 are evaluated according to the chosen characteristic feature distributions and are detailed in the following sub- section.

4.2. Characteristic Feature Analysis

The selection of parameters from the global histogram might be inaccurate if the amount of overlap of the feature distributions in the histogram is large. Two methods that have been used to remedy the problems in these globally oriented techniques are (1) application of the global method to local subimages and (2) recursive application of the global method to increasingly fine-grained regions. In the proposed method, locality is achieved by decomposing an image into rectangular windows.

The histogram analysis of each window provides the necessary information for making a decision in the split- and-merge process. The windows are formed by dividing the image into square subimages. Typically, the dimen- sion of a window is 64 x 64 pixels for a 256 x 256 image. It is important to note that in the process of computing

the histogram, each window is expanded 25% in each direction to include pixels in the margin. Such a strategy is aimed at overcoming the situation where the boundary of a window happens to divide an area (which should be a single region) into two parts, one small and one large. In such a case, the region corresponding to the small part may be missed in the construction of the localized histogram since it contributes only a small number of points to the histogram.

Each pixel at position (x, y) is associated with a set of atom regions (shown in Fig. 2) in which three feature measurements, i.e., average standard deviation 6’s, gray- level constrast e’s, and likelihood ratio p’s are calculated. The distributions of the feature measurements associated with .every pixel are evaluated to determine the desired threshold used in Eqs. (12) and (13). The standard deviation is a measure of the spread of the variables about the mean. For the interior pixels of an object, the standard deviations of their atom regions should have similar values: On the other hand, the atom regions near the boundaries of an object are expected to have standard deviations larger than those of the interior regions. By thresholding the distribution of standard deviations, the parameter el represents the maximum standard deviation of gray-level distribution to be allowed in a uniform region. Similarly, the gray-level contrast and likelihood ra-

FIG. 2. Discrete realization of masks: four pentagonal, four hexago- nal, and one rectangular mask.


tio associated with the interior atom regions are smaller than those of atom regions near the boundaries. The values of e2 and e3 obtained by thresholding the corresponding distributions give the respective maximum tolerances of average gray-level difference and likelihood ratios when an attempt is made to join two regions.

In the rest of this section, the method of threshold selection crucial to the proposed segmentation algorithm is derived. In order to form a type of histogram more suitable for our objective, a co-occurrence matrix is constructed on the basis of the desired characteristic features of the image. The histogram is then formed by including only the near-diagonal entries of the matrix. By applying the entropic method [33] to analyze the histogram, the optimal threshold can be obtained.

4.2.1. Formation of Co-Occurrence Matrix

The feature measurements, 2s, a’s, and p’s, are histo- gramed from the associated co-occurrence matrices to facilitate the determination of the desired thresholds. In order to obtain these quantities, a preprocessing which computes the standard deviation Sj and the average gray level p of each atom region rj, j = 0, 1, . . . , 8, is performed. The best fitting atom region rk of the pixel (x, y) is selected as

6 X,? = Sk = llliIl(6j), j = 0, 1, . . . , 8.

Meanwhile, the size Sk and the average intensity pk of the best fitting atom region are recorded for further processing; i.e.,

- Px,y = pk = c f-(x’, Y’)lsk

(x’,Y’Erk

&,y = Sk,

where f(x, y) represents the gray level (a value between 0 and I) of pixel (x, y). On the basis of this information, the following quantities associated with the pixel at position (x, y) are defined,

sx,, = median(6j), j = 0, 1, . . . , 8,

I: a Px,y =

(x’,y’EN8(x,y) I& - k,y 1 8 9

P&Y = median[h~@,,, , LY, S,,, , &,,,,, FX,.Y,, Sx,,y,)l, w, Y’) E N8k Y),

where 0 5 &, , ~5 X,Y 5 1,O 4 px,y 5 1, and N*(x, y) denotes the set of the eight neighbors of the pixel at (x, y). The function hb used in the likelihood ratio is actually a variation of hF in Eq. (13) and is expressed

h; =

After the measurements are obtained, the co-occurrence matrices for standard deviation MS, gray-level con- strat Mg , and statistical test Mf can be constructed. Each entry in the matrix represents the relative frequency of co-occurrence of two pixels with the characteristic feature values (e.g., average standard deviation 8, gray-level contrast @, statistical ratio p) i and j respectively, separated by distance d and with orientation 4. Distance from the main diagonal in M corresponds to absolute feature difference and in the upper right (and lower left) corner, the difference reaches its maximum. If a histogram is defined on the basis of the pixels that contribute to the near-diagonal elements of M, it should primarily represent the distribution of the pixels of both objects and background and have a deep valley.

To avoid expensive computation, the co-occurrence matrix M used in this paper is defined as

M = WLO, + Mc1.57/2) + MI,,) + M(l.3~/2);

that is, the (i, j) element of M is the frequency that feature i occurs as a 4-connected neighbor of feature j. Ac- cordingly, the entires of the co-occurrence matrices, MS, Mg , and Mf are formulated in the following,

Mati, j) = c flcs,,, ? s,~,,~> (x.Y):(.~‘.Y’)EN~(x,Y)

M,G, j) = c .h(A,y 3 A,,yf) (x,y):(x’.~‘)EN~(x.y)

Mf(m, 4 = c ki(Px.y, Px’.y’)r (*.v):(x’,y’)EN”(x,y)

where

fl<s*.,~ s,f.J = 1, if sl(s,,,) = i and 9(6,,,,,) = j, o

3 otherwise

2

468 CHEN, LIN, AND CHEW

mLr7 fL,.Y,) = 1, if 9(&,) = i and .9(fix,.rr) = j, o

9 otherwise

3

1, if 9(X * px,,,) = m and

h(Px.y , Px,.yf) = $w- - P,‘.y’) = n,

0, otherwise;

and .!+ is a function returning the integer part of a variable, ZV4(x, y) denotes the 4-connected neighbors of pixel (x, y), 0 5 i, j I 1, and 0 I m, IZ I X. In matrix Ma, the entry (i j) contains the relative frequency of co-occurrence of average standard deviations i and j of the atom regions. Matrices MR and Mf are similarly defined by features of gray-level contrast and likelihood ratio, respectively. Since the value of px,Y varies from 0 to 1, a scaling factor X is used to transform it to an appropriate range for matrix formation.

4.2.2. Histogram Formation and Thresholds Selection

On the basis of the information of co-occurrence matrices, the histograms for standard deviation Hs, gray-level contrast Hz, and statistical test HJ can be formed by pro- jecting these elements perpendicularly onto the main diagonal of M as

&s(T) = c J;<i, j, T) (ii) H,(T) = 2 .fi(i, j, T)

(id

H#) = c $di, j, T), (id where

Mdi, 3, ifi=TandjsT+Zrl, fiGi j, T) =

0, otherwise;

.fdL j, T) = 1

M,(i, j), if i = T and j 5 T + W 5 1,

0, otherwise;

f3k j, 7’) = M,G, 3, ifi= Tandjs T-t WlX,

0, otherwise;

and W is the width for extracting the near-diagonal entries of a matrix.

If the distribution of histogram is bimodal, the threshold can then be chosen at the bottom of the valley. How- ever, a histogram is not always bimodal and methods other than valley seeking are required to solve this problem in case of nonbimodal histograms. In this paper, the algorithm proposed in [33] is used to compute the optimal

threshold t*, which is defined as

t* = max{Gb(t) + G,,,(t)},

where

(14)

Gb(t) = - i: ; log, (;I, ;=o t f

Gm = - ;=$, fi log, (&), f f

p, = &I;, i=O

pi = ni n’

and Iti and n denote the numbers of occurrence at the ith entry and the total number of entries in the histogram, respectively. The thresholds ek(wij), k = 1, 2, 3, for each window wij can then be determined by applying Eq. (14) to the histograms H,, Hg, and Hf, respectively.

5. EXPERIMENTAL RESULTS

To demonstrate the capability of our new segmentation scheme, data with distinct characteristics such as natural scenes (houses scene and aerial photographs) and medical images (X-ray CT and MRI) are tested in the experiments.

Figures 3a and 3b show the original house scene and its 16 evenly divided subimages, respectively. Figures 4 to 6 show the distributions of the characteristic features 6, 8, p, the co-occurrence matrices MS, M,, Mf, and the histograms Hs, Hg, Hf for the first subimage (upper-left corner) of Fig. 3b, respectively. Table 1 illustrates the com-

TABLE 1 Optimal Thresholds Selected for Each Window in Fig. 3b

el 26 26 19 38

e2 9 8 7 27

e3 0.19 0.14 0.14 0.29

el 15 15 13 13

e2 20 28 33 36

e3 0.16 0.32 0.29 0.26

el 10 14 12 12

e2 22 41 39 23

e3 0.17 0.29 0.29 0.23

26 24 19 31 20 8 14 14 0.29 0.13 0.16 0.17


FIG. 3. Natural scene: (a) house I image; (b) decomposition by 16 windows; (c) segmentation of (a).

FIG. 4. tion p.

Characteristic feature distributions of the first subimage in Fig. 3b: (a) contrast intensity 6; (b) likelihood ratio $, (c) standard varia-

FIG. 5. Co-occurence matrices of the first subimage in Fig. 3b: (a) matrix ME; (b) matrix Ma; (c) matrix M,

0 I I 0 5 10 15

0 20 25 30 35 40 0 5 10 15 20 25 30 35 40 45 50 55 60 65

c 400 1

I 0 5 10 15 20 25 30 35 40 45 50 55

FIG. 6. Main diagonals of M matrices displayed as histograms for the first subimage in Fig. 3b: (a) histograms Hg; (b) histograms H,; (c) histograms Hf.

SPLIT-AND-MERGE IMAGE SEGMENTATION

FIG. 7. Natural scene: (a) house II image; (b) segmentation of (a).

puted parameters for each window of Fig. 3b by using the entropic method. Interpolation of parameters is performed if there is an attempt to merge two regions across window boundaries. The segmentation result of the house scene is shown in Fig. 3c.

In Fig. 7a, the house scene used in [16] is tested and its segmentation is shown in Fig. 7b. Figures 8b and 9b show the final segmentation results of two aerial photographs, city and highway scenes, in Figs. 8a and 9a, respectively. These images represent different domains of natural scenes. It is noted that the results are generated without incorporating any domain-specific knowledge in the system and without any human intervention.

Figures 10a and lob shows an X-ray CT image and its segmentation, respectively. The final segmentation results for three more medical images are presented in Figs. 11, 12, and 13, respectively. These result demonstrate the robustness and the scope of the proposed algorithm. Although the results are still far from satisfactory, they are adequate to support the subsequent knowledge- based refinement processes.

6. CONCLUDING REMARKS

FIG. 8. Aerial photograph: (a) city image; (b) segmentation of (a).

We have presented a region-based segmentation algorithm which combines the strengths of characteristic feature analysis and a hypothesis mode1 to produce an initial FIG. 9. Aerial photograph: (a) highway image; (b) segmentation of segmentation. All the parameters in the algorithm are (a).


FIG. 10. Medical image: (a) X-ray CT through body of corpus cal- FIG. 12. Medical image: (a) X-ray CT through pituitary gland; (b) losum; (b) segmentation of (a). segmentation of (a).

FIG. 11. Medical image: (a) MRJ scan through body of corpus cal- FIG. 13. Medical image: (a) MRI scan through pituiary gland; (b) losum; (b) segmentation of (a). segmentation of (a).

473

FIG. 14. Comparative study: (a) original MRI with region of interest enclosed by a rectangular window; (b) the 64 by 64 subimage extracted from the window; (c) segmentation using regular decomposition; (d) improvement using decomposition on the region of interest.

computed automatically on the basis of the characteristic features extracted from the windows and depend only on the context of the image under analysis. The computed parameters provide the hypothesis model with appropriate constraints to test the region homogeneity. In the hypothesis model, either a heuristic criterion or a statistical test is a applied according to the feature distribution of the region being analyzed. Using this method, the inconsistency between the empirical data distribution and that assumed in the mathematical model can then be rec- tified. Two processes, initial region creation and final region formation, are implemented within the split-and- merge segmentation scheme based on the hypothesis model. The splitting process in each window is performed independently while the process of merging adjacent regions is applied globally.

Localization is produced by decomposing the image into a set of rectangular windows to reduce the possibility of cluster overlap and hidden clusters. The merits of localization based on the recursive method and regular decomposition have been discussed in [64, 651. It is noted that the strategy of achieving locality by regular decomposition has also been incorporated in other region segmentation methods [14, 16, 17, 461. Since the regular decomposition is done without utilizing any knowledge, some objects may be lost after segmentation due to its smoother contrast variation compared with those of other objects in the same window. This situation can be improved if a suitable window is used so that attention can be focused on that region containing the object. In Fig. 14a, the region of interest is enclosed by a window and its enlaged version is shown in Fig. 14b. Figures 14~ and 14d demonstrate the original segmentation by regular decomposition and the improved segmentation by decomposition on the region of interest, respectively. It is obvious that the result illustrated in Fig. 14d is an improvement on that in Fig. 14~.

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split-and-merge image segmentation based on localized feature analysis and statistical tests

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