Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 891692, 16 pages

http://dx.doi.org/10.1155/2015/891692

## A Spatial Shape Constrained Clustering Method for Mammographic Mass Segmentation

^{1}School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China^{2}Department of Computing Science, Institute of High Performance Computing, A^{*}STAR, Singapore 138632^{3}Department of Psychiatry, Vanderbilt University Medical Center, Nashville, TN 37232, USA

Received 26 September 2014; Revised 21 December 2014; Accepted 12 January 2015

Academic Editor: William Crum

Copyright © 2015 Jian-Yong Lou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A novel clustering method is proposed for mammographic mass segmentation on extracted regions of interest (ROIs) by using deterministic annealing incorporating circular shape function (DACF). The objective function reported in this
study uses both intensity and spatial shape information, and the dominant dissimilarity measure is controlled by two weighting
parameters. As a result, pixels having similar intensity information but located in different regions can be
differentiated. Experimental results shows that, by using DACF, the mass segmentation results in digitized mammograms are improved
with optimal mass boundaries, less number of noisy patches, and computational efficiency. An average probability of segmentation
error of 7.18% for well-defined masses (or 8.06% for ill-defined masses) was obtained by using DACF on MiniMIAS database, with 5.86% (or 5.55%) and 6.14% (or 5.27%) improvements as compared to the standard DA and fuzzy *c*-means methods.

#### 1. Introduction

Image segmentation is a process which divides an image into several meaningful areas such that the segmented image can be further analyzed and interpreted. A segmentation algorithm, in a mammographic context, is an algorithm used to detect something, usually the whole breast or a specific kind of abnormalities like microcalcifications or masses. In the digitized mammograms with low contrast, masses are embedded in various breast tissues with fuzzy margins. This variability introduces a challenge for breast mass segmentation and causes the false positive detection rate to increase as well as decreasing the sensitivity.

In the past decades, a number of image processing techniques have been developed to segment masses from their surrounding breast tissues in digitized mammograms, as reviewed in [1–4]. Among them, clustering methods are one of the most commonly used techniques for image segmentation [5] as well as for mass detection and/or segmentation [4]. Partitioning clustering and hierarchical clustering are two main approaches to clustering. -means [6] and fuzzy -means (FCM) [7] algorithms are widely used partitioning techniques by the researchers in many real world applications. For mass segmentation, -means has been used in [8, 9] to generate initial segmentation results and in [10, 11] to refine an initial detection from adaptive thresholding. FCM was also used for mass segmentation with different objectives: while [12] used it to group pixels with similar grey-level values in the original images, [13] used it over the set of local features extracted from application of a multiresolution wavelet transform and Gaussian Markov random fields analysis. In contrast to -means and FCM, which are sensitive to data initialization and converge to local optimal solutions, deterministic annealing (DA) clustering [14] is a global minimisation algorithm by incorporating randomness into the energy function to be minimized, such that it is independent of the choice of the initial data configuration and has the ability to avoid poor local optima. The DA approach has also been used for mass segmentation in [15, 16].

Most clustering algorithms (including -means, FCM, and DA) perform image segmentation directly from the intensity (or color) space with an intensity filter to enlarge the difference between normal and abnormal breast tissue. The processing time is a prominent advantage of these algorithms. However, the intensity-based methods cannot satisfactorily outline the boundary of the mass region when the image contrast and signal noise ratio are low and therefore lead to poor segmentation results. Markov random field technique was used in mass segmentation [17] to exploit the spatial continuity in order to improve the performance of segmentation algorithm. It has the ability to reduce segmentation error caused by intensity noise; however, the computational cost is high. Reference [18] proposed a fuzzy clustering algorithm incorporating an elliptic shape function for lip image segmentation. The pitfall is that the convergence time increases as the weighting parameter that controls the spatial shape information increases.

In this paper, we propose a novel clustering algorithm based on DA approach to overcome the problems of most existing clustering techniques. In the standard DA clustering [14, 19] for image segmentation, the dissimilarity measure in the objective function is defined merely based on Euclidean distances between the image intensity and the intensity centroids without knowledge of the spatial shape information. Solely using the intensity or intensity related information is hard to differentiate pixels with the same intensity information but located in unconnected regions. As a result, large number of subregions in the same cluster that contains a mass may lead to heavy computational load. Additionally, it is hard to find the fuzzy boundary when the image contrast is low. To handle these challenges, a new dissimilarity measure for DA clustering incorporating a circular shape function (DACF) is proposed. Since both intensity and spatial information are used in the optimization process, the DACF algorithm offers two advantages. First, it is robust against noise and cluster number; that is, pixels having similar intensity information but located in different regions can be differentiated, with just two clusters for the entire images. Second, it is computationally efficient. The convergence time decreases as the difference between the two weighting parameters increases. Experimental results have demonstrated the advantages of the DACF algorithm.

The main contribution of our current work includes the following: (1) the geometry shape is integrated into the intensity feature space for mass segmentation in terms of dynamically fitted circular shape function; (2) the proposed method can differentiate the pixels with the same intensity values but located in different (mass and nonmass) regions, which cannot be achieved by standard clustering methods like FCM and DA; (3) the proposed method achieves better segmentation performance than FCM and DA in terms of segmentation accuracy and computational time; (4) the proposed method is general, which can be integrated into other segmentation algorithms and applicable for other biomedical applications.

The rest of this paper is organized as follows. Section 2 briefly reviews the standard DA clustering approach and derives the formulation and implementation of the proposed DACF algorithm. The experimental results and related discussions on real mass images are given qualitatively and quantitatively in Section 3. The conclusion is given in Section 4.

#### 2. The Proposed Method

##### 2.1. A Brief Review of Standard Deterministic Annealing Approach

Suppose there are input vectors , , which are partitioned into clusters with mass center at . The DA clustering algorithm [14, 19] aims to minimize the following Lagrangian formulation: where is the Lagrange multiplier, which is analogous to the temperature in statistical mechanics, is the cost function, is the Shannon entropy, is the source distribution (equal to in [19]), is the association probability (distribution) relating input point with cluster center , and is the squared Euclidian distance between and defined by It turns out [19] that the resultant distribution is the titled distribution given by where is the partition function and is the mass probability of cluster. Plugging (3) back into (1), the effective cost to be minimized becomes the free energy (a well-known concept in statistical mechanics [14]) as follows: The expression of cluster center is then derived by minimizing (4) with respect to ; that is Alternatively updating (3) and (5) with phase transition gives the DA algorithm. The DA approach to clustering has demonstrated to be independent of the data initialization and has ability to avoid poor local optima, as discussed in [14, 19].

##### 2.2. Deterministic Annealing Clustering Incorporating Circular Function (DACF)

Consider an image with pixels, whose locations are denoted by , where and . Let us define the new dissimilarity measure of the proposed DACF by where are predefined regulated parameter, stands for the Euclidean distance between the pixel and the centroid of the cluster, as and represents the shape information, given by circular function as where is a unique clique, and , are the physical - coordinate of the center of a mass region. The dissimilarity measure consists of a measure of the intensity dissimilarity between the th pixel and the centroid in the intensity feature space, and the spatial distance between the pixel (located at ) and the center (denoted by ) of the targeted mass region. With the inclusion of circular shape information, the pixels with similar intensity but located in disjointed region will be differentiated. The purpose of the inclusion of the shape function is to obtain a large membership for the cluster associated with mass region. In order to achieve it, the weighting parameter is defined as the weight of the spatial distance against the intensity feature. According to the dissimilarity definition of the Euclidean distance, the closer a pixel belongs to a cluster, the smaller the distance is. Therefore, the shape distance between the location of a pixel and a specific cluster center is small if the pixel belongs to the cluster; otherwise the distance is larger if it belongs to other clusters.

The expected distortion or objective function of the DACF incorporating spatial information is then defined as where is the distortion measure as in the original DA method defined by and is the distortion measure of spatial information, as We recast the optimization problem as seeking the distribution which minimizes subject to a specified level of randomness that is measured by Shannon entropy The optimization is reformulated as minimization of the Lagrangian Minimizing with respect to the probability of leads to the titled distribution [14, 19] where the normalized factor is given by Taking the partial derivative on with respect to cluster center, we have It can be seen that the partial derivative of the objective function with the new dissimilarity measure with respect to is identical to that of DA. Hence, the formula for computing centroids of DACF in the intensity feature space is the same as in DA; that is,

The partial derivative of with respect to is given by that is, Substituting (8) into (19), the spatial parameters can be obtained as Alternatively updating and according to (14) and (17) as well as and according to (20) gives the proposed DACF algorithm.

The titled distribution (14) is the membership of each pixel belonging to different clusters. Generally, the intensities of the center part of a mass region are higher than those locating outside of mass region. For pixels inside a mass region, the intensity dissimilarity is in dominant position, while spatial information plays a major role in dissimilarity measure for pixels outside the mass region. Therefore, the pixels with the same intensity values but locate in different positions in an image will be differentiated, which makes DACF yield better performance for both mass and nonmass related regions.

#### 3. Experimental Results

Thirty-six mammograms from MiniMIAS database [20] that contain thirty-nine masses with various backgrounds (fatty, fatty glandular, and dense-glandular breast tissues) were examined. The mammograms mdb005, mdb132, and mdb144 each contain two mass regions. The two masses in mammogram mdb005 were heavily overlapped, so they were processed together as a single one. The two mass regions in mdb132 and mdb144 were processed independently. Therefore, thirty-eight regions of interest (ROIs) were analyzed. Instead of automatic extraction, in this study, the ROIs were taken from the mammographic image based on the information provided by the database. The size of each extracted ROI, as well as the center and radius of each mass are listed in the appendix at the end of this paper.

According to the information of “class of abnormality” provided by the database, the thirty-eight ROIs were classified into two categories: well-defined masses (twenty-three cases) and ill-defined masses (fifteen cases). The ROIs including well-defined masses are illustrated in Figure 1, while the ROIs including ill-defined masses are shown in Figure 2. Gaussian filter (kernel size and standard deviation 1.0) and image equalization are used to phase out noisy points and enhance mass regions in the image preprocessing step. In all examples, we fix the fuzziness degree for the FCM algorithm, and the annealing factor alpha = 0.9 for the standard DA and proposed DACF algorithms.