Breast cancer is one of the leading causes of death from cancer among women. Early diagnosis can increase the chances of survival. Although mammography is being used for breast cancer screening,3'4 analysis of masses and tumors on mam-mograms is, at times, difficult because developing signs of cancer may be minimal or masked by superimposed tissues, making their visual detection and analysis difficult. Additional diagnostic procedures may be recommended when the original mammogram is equivocal.
Computer-aided image analysis techniques have the potential to improve the diagnostic accuracy of mammography, and reduce the use of adjunctive procedures, morbidity, as well as health-care costs. Computer analysis can facilitate the enhancement, detection, characterization, and quantification of diagnostic features such as the shapes of calcifications and masses, the growth of tumors into surrounding tissues, and distortion caused by developing densities.5 Annotation of mammograms with objective measures may assist radiologists in diagnosis.6
Various segmentation algorithms have been presented in the literature to extract tumor regions from mammographic images (refer to recent papers by Rangayyan et al.7'8 and Mudigonda et al.9'W for reviews on this topic). In general, the reported segmentation techniques attempt to define precisely an ROI, such as a tumor or a mass. However, it is difficult to define a criterion to obtain precise regions on mammograms. The problem is complicated by the fact that most malignant tumors possess fuzzy boundaries with slow and extended transition from a dense core region to the surrounding tissues. Very few works consider the uncertainty present around the ROI boundaries.11'12
Computer-aided detection of breast masses is a challenging problem requiring sophisticated techniques due to the low contrast and poor definition of their boundaries. Classical segmentation techniques attempt to define precisely the ROI, such as a calcification or a mass. Shen et al.13 proposed thresholding and multitolerance region growing methods for the detection of potential calcification regions and the extraction of their contours. Karssemeijer,14 Laine et al.,15 and Miller and Ramsey16
proposed methods for tumor detection based on scale-space analysis. Zhang et al.17 proposed an automated detection method for the initial identification of spiculated lesions based on an analysis of mammographic texture patterns. Matsubara et al.18 described an algorithm based on an adaptive thresholding technique for mass detection. Kupinski and Giger19 presented two methods for segmenting lesions in digital mammograms: a radial-gradient-index-based algorithm that considers both the gray-level information and a geometric constraint, and a probabilistic approach. The difference among the methods lies in the utility function to determine the final lesion area. However, defining criteria to realize precisely the boundaries of masses in mammograms is difficult.
An alternative to address this problem is to represent tumor or mass regions by fuzzy sets.20 The most popular algorithm that uses the fuzzy-set approach is the Fuzzy C-Means (FCM) algorithm.12'21 '22 The FCM algorithm uses iterative optimization of an objective function based on weighted similarity measures between the pixels in the image and each cluster center. The segmentation method of Chen and Lee12 uses FCM as a preprocessing step in a Bayesian learning paradigm realized via the expectation-maximization algorithm for edge detection and segmentation of calcifications and masses in mammograms. However, their final result is based on classical segmentation to produce crisp boundaries. Sameti and Ward11 proposed a lesion segmentation algorithm using fuzzy sets to partition a given mammogram. Their method divides a mammogram into two crisp regions according to a fuzzy membership function and an iterative optimization procedure to minimize an objective function. If more than two regions are required, the algorithm can be applied to each region obtained using the same procedure. The authors presented results of application of the method to mammograms with four levels of segmentation.
In this chapter, we describe two segmentation methods that incorporate fuzzy concepts.23-25 The first method determines the boundary of a tumor or mass by region growing after a preprocessing step based on fuzzy sets to enhance the ROI.23 The method is simple and easy to implement, always produces closed contours, and has yielded good results even in the presence of high levels of noise. The second segmentation method is a fuzzy region growing method that takes into account the uncertainty present around the boundaries of tumors.24 The method produces a fuzzy representation of the ROI, and preserves the uncertainty around the boundaries of tumors. We demonstrate the potential use of features derived from the results of segmentation in pattern classification of the regions as benign masses or malignant tumors.
Given the difficult nature of the problem of detection of masses and tumors in a mammogram, the question arises if the problem could benefit from the use of multiple approaches: How may we combine the results of several approaches — which may be considered to be complementary — so as to obtain a possibly better result?
In generic terms, image segmentation may be defined as a procedure that groups the pixels of an image according to one or more local properties.26 A property of pixels is said to be local if it depends only on a pixel or its immediate neighborhood (for example, gray level, gradient, and local statistical measures). Techniques for image segmentation may be divided into two main categories: those based on the discontinuity of local properties, and those based on the similarity of local properties.27 The techniques based on discontinuity are simple in concept, but generally produce segmented regions with disconnected edges, requiring the application of additional methods (such as contour following). Techniques based on similarity, on the other hand, depend on a seed pixel (or a seed subregion) and on a strategy to traverse the image for region growing. Because different segmentation methods explore distinct, and sometimes complementary, characteristics of the given image (such as contour detection and region growing), it is natural to consider combinations of techniques that could possibly produce better results than any one technique on its own. Although cooperative combination of results of segmentation procedures can offer good results, there are very few publications devoted to this subject.28-34 This is partly due to the difficulty in simultaneously handling distinct local properties, and due to the limitations of the commonly used Boolean set operations in combining different image segmentation results.
Using the theory of fuzzy sets, it is possible to define several classes of fusion operators that generalize Boolean operators. We describe a general fusion operator, oriented by a finite automaton, to combine information from different sources.35'36 In particular, we apply the idea to the problem of mammographic image segmentation for the detection of breast tumors, combining results obtained via contour detection and region growing. The final fuzzy set can classify pixels with more certainty, and preserve more information than either of the individual methods. The results are evaluated using a measure of agreement with reference to the contours of the tumors drawn independently by an expert radiologist specialized in mammography.
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