Computational and Mathematical Methods in Medicine

Volume 2015 (2015), Article ID 349874, 16 pages

http://dx.doi.org/10.1155/2015/349874

## Dependence of Shape-Based Descriptors and Mass Segmentation Areas on Initial Contour Placement Using the Chan-Vese Method on Digital Mammograms

Department of Medical Physics, University of the Free State, P.O. Box 339, Bloemfontein 9300, South Africa

Received 16 February 2015; Revised 14 July 2015; Accepted 30 July 2015

Academic Editor: Chuangyin Dang

Copyright © 2015 S. N. Acho and W. I. D. Rae. 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

Variation in signal intensity within mass lesions and missing boundary information are intensity inhomogeneities inherent in digital mammograms. These inhomogeneities render the performance of a deformable contour susceptible to the location of its initial position and may lead to poor segmentation results for these images. We investigate the dependence of shape-based descriptors and mass segmentation areas on initial contour placement with the Chan-Vese segmentation method and compare these results to the active contours with selective local or global segmentation model. For each mass lesion, final contours were obtained by propagation of a proposed initial level set contour and by propagation of a manually drawn contour enclosing the region of interest. Differences in shape-based descriptors were quantified using absolute percentage differences, Euclidean distances, and Bland-Altman analysis. Segmented areas were evaluated with the area overlap measure. Differences were dependent upon the characteristics of the mass margins. Boundary moments presented large percentage differences. Pearson correlation analysis showed statistically significant correlations between shape-based descriptors from both initial locations. In conclusion, boundary moments of digital mass lesions are sensitive to the placement of initial level set contours while shape-based descriptors such as Fourier descriptors, shape convexity, and shape rectangularity exhibit a certain degree of robustness to changes in the location of the initial level set contours for both segmentation algorithms.

#### 1. Introduction

Breast masses are one of the most common indications of breast cancer. They are frequently identified on mammograms, due to their saliency relative to the surrounding regions and also to comparable regions on the mammograms with the same projection of the opposite breast [1]. Computer Aided Detection algorithms for breast mass classification exploit suitable shape-based descriptors derived from the mass boundary which are powerful enough to differentiate between benign and malignant masses. Segmentation algorithms are necessary for mass contouring in direct digital mammography. However, in this imaging modality, mass margins are embedded in complex backgrounds of overlying and underlying tissues which creates missing boundary information and local minima where a deforming contour can be entrapped and as a consequence produces an undesirable segmentation outcome. Moreover, the wide dynamic range of flat panel detector systems of direct digital mammography units records small differences between the attenuation coefficients of structures or regions present in a mass lesion and they are clearly distinguishable over a wide range of densities, whereas in film screen mammography the exposure latitude of the film limits the dynamic range of information captured on the film. Hence, masses which may have appeared as dense structures without significant topographical relief features on film screen mammograms can emerge following digital imaging, as regions with varying densities on soft copy display. Enhancement of these variations, following postprocessing by the processing algorithms of the manufacturer, may also be present. Usually, small differences in densities may sometimes appear as low signal areas which can act as local minima for contour entrapment each time an evolving curve determines its path within the mass lesion. Consequently, local minima and missing boundary information render deformable contours susceptible to their initial locations.

A geometric active contour is a deformable contour based approach for image segmentation. In breast mass segmentation, an initial contour is deformed and driven by a partial differential equation (PDE) towards the boundary of the candidate mass. It is categorized into two groups: edge based models [2–4] and region based models [5–13]. Both models make use of a stopping term which reduces the speed of the evolving contour as it approaches the boundary of the object and finally reaches a steady state at the boundary. In edge based models, the stopping term utilizes an edge indicator function modelled on the image gradient; consequently, objects with weak and noisy boundaries may present some difficulties to this segmentation model [14, 15].

The Chan-Vese region based algorithm models energy functionals as a competition of regional statistical information [16]. They defined the stopping term as a competition of the first moments of the local intensity distribution of the foreground and the background within a narrowband, which takes into consideration only pixels which will influence the propagation of the interface (zero level set function) between these two regions. The energy functionals drive the initial contour from its initial location toward a desirable local minimum, which in principle should correspond to the delineated boundary of an expert radiologist. However, these are determined by localized statistics; hence, the evolution of the curve becomes sensitive to the location of the initial level set contour and segmentation results will depend on the placement of this contour, especially when tuning parameters for an arbitrary collection of masses are fixed. This becomes evident during segmentation of direct digital masses with obscured or ill-defined margins and low signal areas within.

The active contours with selective local or global segmentation model [9] are a region based energy functional formulated as a signed pressure force function which propagates the initial contour by modulating the signs of the pressure forces inside and outside the region of interest. These pressure forces are derived from the means of the local intensity distributions of the foreground and the background. The algorithm penalizes the level set function to be binary and regularizes it with a Gaussian smoothing kernel. It can effectively handle images with weak edges and interior intensity inhomogeneity.

In most segmentation problems, the initial contour is either drawn by the operator or estimated from other segmentation algorithms [17, 22–25] and this may place the initial level set contour on different locations within the mass. Any variation in segmentation outcomes will cause changes in shape-based descriptors and the area occupied by the segmented mass. Variations in segmentation outcomes which are due to the placement of the initial level set contours in complicated images have been mentioned [11]. Mass lesions on mammograms are complicated image domains for curve evolution and variations in mass lesion segmented areas and their influences on shape-based feature vectors due to changes in the placement of the initial level set contours are not found in the literature.

Understanding these inconsistencies can improve the choice of tuneable parameters and initial contour locations for curve evolution either for a data set of mass lesions with labelled margin characteristics or unlabelled margin characteristics. Shape-based descriptors [26–28] are feature vectors in training sets for binary classification of mass lesions in mammography and changes in these descriptors can play a role in determining the interclass separability measures, the choice of margin hyperplanes, and hence the classification efficiencies of these algorithms.

In this study, we investigate changes in one-dimensional shape-based descriptors and the segmented areas of masses in direct digital mammograms due to changes in the location of the initial level set contours with the implementation of the Chan-Vese segmentation method and the active contours with selective local or global segmentation model. Two groups of masses are considered in this study, one with obscured or ill-defined margins and low signal areas within and the other with well-defined and distinct margins. We consider a contour which encloses the mass lesion and is propagated towards the margin of the lesion. We propose a semiautomatic method which derives the initial contour as a curve connecting points with maximum gradient in the radial direction, representing an optimum curve characterizing the intrinsic shape of the mass lesion, and then assess the differences in the segmentation results.

#### 2. Background to Mathematical Methods

In mammography, smoothed images present topological surfaces that can be thresholded into multiple layers to obtain topographical relief maps of dominant structures found on the images. Mammograms are filtered with edge-preserving denoising methods such as weighted total variation (TV) scale-space smoothing technique [29, 30] to remove noise and fine details while preserving dominant edge characteristics through different degrees of smoothing.

##### 2.1. Weighted Total Variation Scale-Space Smoothing Technique

Suppose denotes an image and the image domain. The variational approach for image denoising for this model involves the minimization of the following energy functional:where is the noisy input image and its regularized approximation. is the Lagrange multiplier indicating the scale of detail desired in the smoothed image. Bresson et al. proposed a modified model [30] in which the -norm square of Rudin et al.’s model is replaced with an -norm to preserve image contrast [31] and in addition the TV norm of is multiplied with a function, , which is an edge indicator function. This represents the weighted TV model with an -norm as a data fidelity measure. The energy functional for minimization is given aswithwhere is a constant >0 and is a Gaussian kernel with standard deviation, . The minimization of results in the following weighted TV flow equation:For small values of , the degree of image smoothing increases and edge is preserved; therefore, the global boundary information which is essential for segmentation algorithms can be modelled as the initial contour for the gradient descent flow equation of the level set. This contour will depend on the boundary properties of a given mass lesion.

##### 2.2. Chan-Vese’s Piecewise Constant Model for Binary Segmentation

Suppose is an evolving curve that partitions the image domain into the foreground, , and the background, . The Chan-Vese model [16] seeks an optimal contour, representing the boundary of an object by minimizing the following energy functional: where represents the regional term guiding the contour in the image domain and is given byin which, , and and are positive constants while the average image intensities of regions inside and outside the contour are and , respectively. In level set formulation, the interface of the foreground and background is embedded as the zero level set of a Lipschitz function, : with for pixel positions in and for pixel positions in whilst on the curve . Using the Heaviside step function, , can be expressed as Minimizing with respect to yields the following gradient descent flow:where is the Dirac function.

##### 2.3. Active Contours with Selective Local or Global Segmentation Model

The signed pressure force function [9] is derived from the means of regions inside and outside the contour and it is defined aswhere and are defined in (8). The active contour with selective local or global segmentation model utilizes the geodesic active contour to formulate the level set equation asUsing the Gaussian filtering process to regularize the level set function, the above equation can be written as follows:where is a tuneable parameter.

#### 3. Method

##### 3.1. Data Set Description

Direct digital mammograms were acquired from a Hologic Selenia Dimensions system with an image receptor consisting of a 70 *μ*m pixel pitch selenium direct-capture detector. Ninety mammograms with mass lesions were selected for this study. Forty mammograms had masses with low signal areas within the mass and margins described as obscured, or ill-defined, while the others had masses with well-defined or distinct margins. On each mammogram, the region of interest containing the mass lesion was cropped and then resized to a matrix to create a submammogram. Each submammogram was denoised and thresholded to localize the initial level set contour.

##### 3.2. Search Space for Localizing the Initial Level Set Contour

The weighted total variation scale-space smoothed breast mass region is represented as a topological surface in which the grey level value of each pixel is the height of the surface. Let denote a smoothed image and the image domain. The image domain is thresholded into multiple regions with an ordered set of equally spaced grey level threshold values within the intensity range of the image domain [32–34]. Suppose = the maximum grey level intensity in the image domain; = minimum grey level intensity; , a finite sequence of equally spaced partition weights in ascending order; = number of threshold values; and , an ordered set of equally spaced grey level threshold values; then,with and .

The subregions in the image domain with grey level intensities less than or equal to the threshold value, , are given asand the iso-level contours ’s of these regions are boundaries of . The iso-level contour map of the image domain represents the set of all for . A graph-based representation of the iso-level contour map evaluates the enclosure relationship between an iso-level contour and its nearest neighbour, to identify the path to the base contour that delineates the mass. Details of this method can be found in the literature [32, 33]. In our implementation, the boundary region of the breast mass is the region around the base contour with a dense nested pattern of iso-level contours, indicating the search space for the actual boundary of the mass and the placement of the initial level set contour. The dense nested pattern of iso-level contours is extracted and superimposed on the gradient map of the smoothed image.

##### 3.3. Placement of the Initial Level Set Contour

A set of uniformly spaced radial lines, , are generated from a point close to the centre of mass of the innermost iso-level contour, defining the search space on the gradient map of the mass as shown in Figure 1(d). Let this point be the reference point. The gradient strength is noted at every point of intersection of the nested iso-level contours and radial lines. Along each radial line, , for , the coordinates of the point of intersection with the greatest gradient strength are noted and the radial distance from this point to the reference point is calculated and noted as .