FEATURE SPACE BASED IMAGE SEGMENTATION VIA DENSITY MODIFICATION Debashis Sen and Sankar K. Pal Center for Soft Computing Research, Indian Statistical Institute 203 B. T. Road, Kolkata, India 700108 E-mail: {dsen t, sankar}@isical.ac.in ABSTRACT Feature space based approaches have been the most popular ones among those used to perform image segmentation. In this paper, a density modification framework is proposed in order to aid feature space based segmentation in images. The framework embeds a position-dependent property associated with each sample in the feature space of an image into the corresponding density map and hence modifies it. The property association and embedding operations in the framework is implemented using a fuzzy set theory based system devised with cue from beam theory of solid mechanics and the appropriateness of this approach is established. Experimental results of segmentation in images are given to demonstrate the effectiveness of the proposed framework. Index Terms— Density Modification, Fuzzy Sets, Beam Theory, Feature Space Analysis, Image Segmentation 1. INTRODUCTION Feature space analysis based approaches have been the most popular ones among those used to perform image segmentation. Initial works resulting in such segmentation techniques were based on finding thresholds in gray-level (single feature) histograms of images [1]. These were followed by the use of clustering techniques in a multidimensional feature space of an image to carry out segmentation, which allowed the consideration of multiple image features simultaneously [2]. Most of the thresholding and clustering systems used for feature space based segmentation consider judicious but speculative formulation of the underlying problem. For example, most histogram thresholding techniques are based on assumptions such as the histogram fits a particular model very well or valleys in the histogram represent class boundaries. Partitional clustering techniques [3] make assumptions about prototypes and shapes of the clusters to be formed, mode seeking algorithms [2] assume local modes (maxima) in the density of the samples in the feature space as cluster prototypes. Assumptions such as the aforesaid ones corresponding to various algorithms may or may not be appropriate for the feature space based segmentation task at hand as the representation of various contents (regions, edges) of an image in a feature space corresponding to it is unclear. For example, it is very difficult to ascertain that clusters formed in an image feature space using the c-means algorithm, where cluster means are considered as cluster prototypes, would correspond to the natural regions in the underlying image. Although it is difficult to ascertain the appropriateness of a thresholding or clustering system for a feature space based segmentation task, one way of improving its performance could be by providing additional useful information in a suitable manner. Density modification techniques, which modify the density map of the samples in a feature space by incorporating certain useful information [4], has seldom

978-1-4244-5654-3/09/$26.00 ©2009 IEEE

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been considered in literature in order to aid segmentation in images. Few such techniques were reported in [5], where histogram modification was considered in order to aid valley seeking based threshold determination in gray-level feature spaces. In this paper, we propose a density modification framework to aid feature space based segmentation in images. In the proposed framework, first, fuzzy set theory is used to associate a positiondependent property with each sample in a feature space of an image and then this property is embedded into the corresponding density map resulting in its modification. We propose the use of beam theory from the field of solid mechanics to carry out the property association and embedding processes in the framework and we also establish the appropriateness of the usage. The effectiveness of the proposed density modification framework is demonstrated by comparing the segmentation performances achieved with and without the use of the proposed framework. 2. DENSITY MODIFICATION IN A FEATURE SPACE: CUE FROM BEAM THEORY Let us consider a n-dimensional feature space S (with real coordinates) of an image as a universe of positions and let I be an element (a position) in it, that is, I ∈ S. Let the density of the samples at I be given by f (I) and without loss of generality let us consider that the density map f of the samples in S is the corresponding ndimensional histogram. 2.1. The Framework We suggest that in order to aid a feature space based segmentation task, the samples at every I (in a populated feature space) be associated with a suitable property, which is dependent on (a function of) I. A fuzzy set can be defined in a feature space such that the corresponding membership function associates a property to the positions in a feature space. This property, which provides certain useful information, then be embedded into the density map of the samples in S in a suitable manner resulting in its modification. The aforesaid proposed density modification framework is shown in Figure 1.

Fig. 1. The proposed density modification framework

ICIP 2009

2.2. The Methodology We first normalize f as follows: f (I) , I∈S f¯(I) = f¯(i1 , i2 , · · · , in ) =  z∈S f (z)

(1)

where ik is the value in the kth dimension and each dimension in the space corresponds to a feature. We then treat f¯ as the multivariate PDF of random variables, which are considered to have generated the values of the underlying features at the pixels in the image. In order to carry out density modification in the feature space S, we consider conditional PDFs giving the probabilities corresponding to one feature, when specific values of the other features are given. The conditional PDF giving the probabilities corresponding to a feature, which is represented by i1 , is expressed as f¯(i1 , i2 , · · · , in ) f¯(i1 /i2 , i3 , · · · , in ) = (2) ζ  where ζ = i1 f¯(i1 , i2 , · · · , in ). Note that, the PDF f¯(i1 /i2 , i3 , · · · , in ), for specific values of ik with k = 2, 3, · · · , n, is a function of i1 alone and hence it is an one dimensional quantity. Let us denote this one dimensional quantity by f¯1 . Now, a few concepts are considered from beam theory of solid mechanics in a manner similar to the usage given in [6]. The quantity f¯1 is considered proportional to a solid load P placed on a uniform beam, which is rested on pivots at two ends, and the resulting bending moment B is given as B(i1 ) = R × l − CP (i1 ) × (l − CG(i1 )), l = i1 − ea

(3)

where i1 ∈ [ea , eb ], with ea and eb respectively corresponding to the smallest and largest value of the underlying feature at those pixels in the image where the other features have the considered specific values. The values ea and eb correspond to the two ends of the beam and i1 corresponds to a position on the beam. The symbol CP (i1 ), which stands for the total load between the positions ea and i1 , is given as CP (i1 ) =

i1 

P (z) =

z=ea

i1 

(f¯1 (z) + γ)

(4)

z=ea

 where γ = ( z∈S f (z) × ζ)−1 . The symbol CG(i1 ), which stands for the center of gravity between the positions ea and li , is given as CG(i1 ) =

i1  1 (z − ea )P (z) CP (i1 ) z=e

(5)

shape of the beam is given by f¯1 + γ. Hence, we have c(i1 ) = 0.5 × (f¯1 (i1 ) + γ). However, it is observed that a PDF f¯1 may correspond to solids which when considered in the setup would result in curvature (ρ) values that are very large at a few positions, making the values at other positions insignificant. We find that such situations are unfavorable for density modification in a feature space and in order to circumvent such situations, we consider the following measure instead of ρ: ρ´(i1 ) =

The curvature due to the bending moment B(i1 ) at a position 1) i1 ∈ [ea , eb ] is calculated as ρ(i1 ) = B(i , where ι(i1 ) is the ι(i1 ) moment of inertia at i1 , which is calculated as f¯1 (i1 )+γ

ι(i1 ) =



(k − c(i1 ))2

(7)

k=0

where c(i1 ) denotes the centroid of the beam at position i1 . The solid load and the uniform beam forms a composite beam and the

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(8)

and then normalize ρ´ as follows: ρ¯ ´(i1 ) = 

ρ´(i1 ) z∈[ea ,eb ]

ρ´(z)

, i1 ∈ [ea , eb ]

(9)

The ρ¯ ´(i1 ) values are then used to get a conditional PDF (an one dimensional quantity) as given below:  ρ¯ ´(i1 ) i1 ∈ [ea , eb ] gi1 (i1 /i2 , i3 · · · , in ) = (10) 0 i1 ∈ / [ea , eb ] and then using gi1 (i1 /i2 , i3 · · · , in ), we get: gi1 (i1 , i2 , · · · , in ) = gi1 (i1 /i2 , i3 · · · , in ) × ζ

(11)

where gi1 (i1 , i2 , · · · , in ), for specific values of ik with k = 2, 3, · · · , n, is an one dimensional quantity. When all such one dimensional quantities deducible from the conditional PDF in (2) are obtained by considering all possible values of ik for all k = 2, 3, · · · , n, we get an n-dimensional quantity expressed as gi1 (I) at a position I ∈ S. We also define a set Ωi1 in S having all those positions (I) as elements, where we find i1 ∈ / [ea , eb ] (see (10)) during the calculation of gi1 . Next, we obtain all giz and Ωiz with z = 1, 2, · · · , n, in a manner similar to the  one described above. We then obtain a set, say Ω, in S as Ω = n z=1 Ωiz . It is easily deducible from the above description that giz (I) is positively correlated to the corresponding curvature ρ(iz ). Now, after obtaining all giz with z = 1, 2, · · · , n, we combine them as follows:  g(I) = gi21 (I) + · · · + gi2n (I), I ∈ S, I ∈ /Ω (12) where it is obvious that g(I) is positively correlated to the quadratic mean (root mean square) of the n curvatures (corresponding to the n giz s) associated with the position I. We then normalize g as follows g¯(I) = 

a

In (3), R is the reactive force at the pivot corresponding to the value ea and it is calculated as   L − CG(eb ) R = CP (eb ) × , where L = eb − ea (6) L

B(i1 ) ι(i1 ) + maxz∈[ea ,eb ] ι(z)

g(I) y∈S, y ∈Ω /

g(y)

, I ∈ S, I ∈ /Ω

(13)

The following operation is then carried out: Θ(I) =

max

y∈S, y ∈Ω /

g¯(y) − g¯(I) + C −1 ×

max

y∈S, y ∈Ω /

g¯(y)

(14)

where C ∈ Z+ (set of positive integers). Then Θ is normalized as ¯ Θ(I) = 

Θ(I) y∈S, y ∈Ω /

Θ(y)

, I ∈ S, I ∈ /Ω

(15)

From (14) and (15), we see that larger the value of g at a position, ¯ at that position. As g is positively correlated smaller the value of Θ to the root mean square of the n curvatures, we consider the mea¯ at a position as the ‘chance of sustainability’ against possible sure Θ breakage at that position.

¯ at the various positions in S in order We then use the values of Θ M ¯ to get a PDF, say f , as follows:  ¯ Θ(I) I∈ /Ω M ¯ f (I) = (16) 0 I∈Ω We may say that a beam theory based modification process applied on the PDF f¯ has yielded the PDF f¯M . Now, a density map, say f M , which corresponds to f¯M for all I ∈ S, is obtained as follows: f M (I) = round(f¯M (I) × ξ) (17)  where ξ = C × (maxy∈S, y∈Ω ¯(y))−1 × y∈S, y∈Ω / g / Θ(y). As it will be evident from Section 3, the density map f M in the ndimensional feature space S can be considered as a modified density map obtained using the proposed density modification framework on the density map f of the samples in S. 3. APPROPRIATENESS OF BEAM THEORY BASED DENSITY MODIFICATION FOR SEGMENTATION 3.1. Property Association Consider a quantity μ1 related to the bending moment B in (3) as follows: μ1 (i1 ) =

B(i1 ) , i1 ∈ [ea , eb ] maxz∈[ea ,eb ] B(z)

(18)

It is deduced that the quantity μ1 satisfies the following four properties; 1. Nonnegativity μ1 ≥ 0, 2. Range 0 ≤ μ1 ≤ 1, 3. Vanish identically μ1 (i1 ) = 0 iff i1 = ea or i1 = eb , 4. Concavity. From these properties, we may say that μ1 represents a membership function and it associates a position i1 with a property ‘farness of the position from the nearest pivot’, where the nearest pivot corresponds to either the smallest (ea ) or the largest (eb ) value of the underlying feature at those pixels in the image where the other features have the considered specific values. Therefore, we find that a fuzzy set F = (I, μ(I)), where the membership function μ(I) is given by a combination of n membership functions μz (I), with z = 1, 2, · · · , n, is inherently considered in the methodology explained in Section 2. The membership function μ(I) associates the property ‘farness of the position I in S from all the nearest positions corresponding to all the pairs of smallest and largest values of all the features’ with every position I in S. Intuitively, the aforesaid property represented by μ is an useful one for segmentation, as it represents the farness of every pixel from the nearest among the most discriminable groups of pixels in an image with respect to every feature considered. Therefore, an additional (other than that given by the density map f ) useful information is provided by μ and hence the usage of the fuzzy set F = (I, μ(I)) devised using beam theory for property association in the proposed framework is appropriate. 3.2. Property Embedding The expressions in (12)-(17) and the calculation of all the ρ¯ ´ quantities corresponding to all giz in (12) represent the property embedding process of the proposed framework. In order to ascertain that the embedding process is appropriate, we need to see whether the embedment of the property μ that provides useful information indeed aids feature space based segmentation. In order to do so, let us consider the example of one dimensional gray-level feature based segmentation of the grayscale image given in Figure 2(a). The various types of thresholding and clustering systems mentioned in

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(a) Image

(b) f

(c) f M

(d) Result

Fig. 2. Segmentation in an image with uniform original density map for judging appropriateness of the embedding process Section 1 were applied to the original density map (f ) and the modified density map (f M ) of the grayscale image to perform segmentation. When f was considered, some systems failed to segment the image, whereas the rest gave the result shown in Figure 2(d). However, when f M was considered, all the systems were observed to have increased discriminability and they all gave the result shown in Figure 2(d). This observation suggests that density map based analyzability of feature values (positions) in an image feature space is enhanced by performing the embedment of the property μ, and hence the embedding process may be considered to be appropriate. Note that, in the modified density map obtained in (17), the number of samples corresponding to any feature value (position) within the set Ω is at least one and at most C + 1. The total number of samples in the modified density map might not be equal to that in the density map before modification, and in order to perform segmentation, the outcome of a modified density based analysis performed on a feature space of an image is mapped on to the pixels in the image with respect to the associated feature values. 4. EXPERIMENTAL RESULTS In this section, we consider segmentation of grayscale images using gray-level and local homogeneity features. The local homogeneity measure considered is the angular second moment [7]. The c-means (CM) [3] and mean shift (MS) [8] algorithms are used in two dimensional feature spaces, which are formed considering the two aforesaid features, with original and modified density maps to obtain segmentation results. The results obtained using original density maps are compared to those obtained using modified density maps (considering C = 50) in order to demonstrate the effectiveness of the proposed density modification process. We consider 100 grayscale images from the ‘Berkeley Segmentation Dataset and Benchmark’ [9]. Each one of the 100 images considered are associated with multiple segmentation results hand labeled by multiple human subjects and hence we have multiple segmentation ground truths for every single image. We use the global consistency error (GCE) and the local consistency error (LCE) measures defined in [9] in order to judge the appropriateness of a segmentation result. Both GCE and LCE take values in the range [0, 1], where a smaller value indicates more appropriateness of the segmentation result with reference to the considered ground truth.

(a) GCE - CM

(b) LCE - CM

(c) GCE - MS

(d) LCE - MS

Fig. 3. p-values corresponding to the statistical tests performed

We calculate the GCE and LCE measures corresponding to all the segmentation results, which are obtained by applying the c-means and mean shift algorithms in image feature spaces having original and modified density maps, with reference to all segmentation ground truths available for every image among the 100 images considered. Therefore, for one image and one algorithm, a set of GCE values and a set of LCE values are obtained corresponding to both the use of original and modified density maps. We take the help of statistical hypothesis testing in order to compare the results obtained using the original and modified density maps, when one of the two mentioned algorithms is applied on one of the 100 images. We use the statistical t-test [10] assuming that the GCE values and the LCE values corresponding the use of the two density maps to be compared have come from normally distributed populations with unknown and possibly unequal variances. We perform one-sided t-tests considering the alternative hypothesis (H1 ) that ‘the average segmentation error occurred while using the original density map is greater than the average segmentation error occurred while using the modified density map’. The p-value obtained from such a t-test gives the probability that a superior performance obtained, when the modified density map is used, would have been due to chance alone. Such one-sided t-tests are performed for all the 100 images. The p-values obtained from these tests are shown using dark shaded bars in Figure 3. We also perform onesided t-tests considering the alternative hypothesis (H2 ) that ‘the average segmentation error occurred while using the modified density map is greater than the average segmentation error occurred while using the original density map’. The p-value obtained from such a ttest gives the probability that a superior performance obtained, when the original density map is used, would have been due to chance alone. Such one-sided t-tests are performed for all the 100 images. The p-values obtained from these tests are shown using light shaded bars in Figure 3. The segmentation error considered in the experiments represented by Figures 3(a) and (c) is GCE, where Figure 3(a) and Figure 3(c) correspond to the application of c-means algorithm and mean shift algorithm, respectively. Similarly, the segmentation error considered in the experiments represented by Figures 3(b) and (d) is LCE, where Figure 3(b) and Figure 3(d) correspond to the application of c-means algorithm and mean shift algorithm, respectively. Note that, we have considered two different sets of the required predefined parameters of both the algorithms and hence we see two bars of the same shade in all the illustrations given in Figure 3. As required during the usage of GCE and LCE [9], the predefined parameters are considered such that the number of regions in an image obtained using the algorithms is approximately same as the numbers of regions in the corresponding segmentation ground truths. It is evident from all the illustrations in Figure 3 that the dark shaded bars are in general shorter than the light shaded bars. This signifies that, for most of the 100 images considered, it is less likely that a better segmentation performance observed when the modified density map is used is merely due to chance alone and not due to a real effect, compared to the case when the original density map is used. This observation points to the superiority of segmentation results obtained using the modified density map compared to the use of the original density map. Figure 4 shows the results of segmentation performed on two images using the c-means (CM) and the mean shift (MS) algorithms (with parameters unchanged) on both the original (f ) and modified (f M ) density maps. As can be seen, the contents of the images are better represented in the results obtained when the modified density is considered compared to when the original density is considered.

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(a) Image

(b) MS - f

(c) MS - f M

(d) CM - f

(e) CM - f M

(f) Image

(g) MS - f

(h) MS - f M

(i) CM - f

(j) CM - f M

Fig. 4. Segmentation using original and modified density maps 5. CONCLUSION A density modification framework has been proposed in this paper with cue from beam theory in order to aid feature space based segmentation in images, and the appropriateness of the framework has been established. Experimental results of segmentation in images using the original and modified density maps have been presented to demonstrate the effectiveness of the proposed framework. 6. REFERENCES [1] Nikhil R Pal and Sankar K Pal, “A review on image segmentation techniques,” Pattern Recognition, vol. 26, no. 9, pp. 1277–1294, 1993. [2] A K Jain, M N Murthy, and P J Flynn, “Data clustering: a review,” ACM Computing Surveys, vol. 31, no. 3, pp. 264–323, 1999. [3] Richard O Duda, Peter E Hart, and David G Stock, Pattern Classification, Wiley Interscience, U.S.A, 2nd edition, 2000. [4] K D Cowtan and K Y Zhang, “Density modification for macromolecular phase improvement,” Progress in Biophysics & Molecular Biology, vol. 72, no. 3, pp. 245–270, 1999. [5] Joan S Wezka and Azriel Rosenfeld, “Histogram modification for threshold selection,” IEEE Trans. Syst., Man, Cybern., vol. 9, no. 1, pp. 38–52, 1979. [6] D Sen and S K Pal, “Histogram thresholding using beam theory and ambiguity measures,” Fundamenta Informaticae, vol. 75, no. 1-4, pp. 483–504, 2007. [7] Robert M Haralick, K Shanmugam, and Its’hak Dinstein, “Textural features for image classification,” IEEE Trans. Syst., Man, Cybern., vol. 3, no. 6, pp. 610–621, 1973. [8] Yizong Cheng, “Mean shift, mode seeking, and clustering,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 17, no. 8, pp. 790–799, 1995. [9] 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 8th International Conference on Computer Vision, July 2001, vol. 2, pp. 416–423. [10] E L Lehmann and J P Romano, Testing Statistical Hypothesis, Springer, U.S.A, 3rd edition, 2005.

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