Malignant tumors due to breast cancer and masses due to benign disease appear in mammograms with different shape characteristics: the former usually have rough, spiculated, or microlobulated contours, whereas the latter commonly have smooth, round, oval, or macrolobulated contours. Features that characterize shape roughness and complexity can assist in distinguishing between malignant tumors and benign masses. In spite of the established importance of shape factors in the analysis of breast tumors and masses, difficulties exist in obtaining accurate and artifact-free boundaries of the related regions from mammograms. Whereas manually drawn contours could contain artifacts related to hand tremor and are subject to intra-observer and inter-observer variations, automatically detected contours could contain noise and inaccuracies due to limitations or errors in the procedures for the detection and segmentation of the related regions. Modeling procedures are desired to eliminate the artifacts in a given contour, while preserving the important and significant details present in the contour. This book presents polygonal modeling methods that reduce the influence of noise and artifacts while preserving the diagnostically relevant features, in particular the spicules and lobulations in the given contours. In order to facilitate the derivation of features that capture the characteristics of shape roughness of contours of breast masses, methods to derive a signature based on the turning angle function obtained from the polygonal model are described. Methods are also described to derive an index of spiculation, an index characterizing the presence of convex regions, an index characterizing the presence of concave regions, an index of convexity, and a measure of fractal dimension from the turning angle function. Results of testing the methods with a set of 111 contours of 65 benign masses and 46 malignant tumors are presented and discussed. It is shown that shape modeling and analysis can lead to classification accuracy in discriminating between benign masses and malignant tumors, in terms of the area under the receiver operating characteristic curve, of up to 0.94. The methods have applications in modeling and analysis of the shape of various types of regions or objects in images, computer vision, computer graphics, and analysis of biomedical images, with particular significance in computer-aided diagnosis of breast cancer. Table of Contents: Analysis of Shape / Polygonal Modeling of Contours / Shape Factors for Pattern Classification / Classification of Breast Masses

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Denise Guliato is an Associate Professor with the Faculdade de Computação at the Universidade Federal de Uberlândia, Uberlândia, Minas Gerais, Brazil. She received the Bachelor of Computer Science degree from the Universidade Federal de São Carlos, São Carlos, Brazil, the Master's degree in Computer Science from the Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil, and the Ph.D. degree in Electrical Engineering from Universidade de São Paulo, São Paulo, Brazil. Her research interests include digital image processing, image analysis, pattern recognition, medical imaging and image analysis, computer[1]aided diagnosis, content-based image and video retrieval, e-learning systems, and Web services. Her current research projects are on feature extraction, analysis and classification of mammograms; computer-aided diagnosis of breast cancer; and content-based image and video retrieval using bags of visual features. Rangaraj M. Rangayyan is a Professor with the Department of Electrical and Computer Engineering, and an Adjunct Professor of Surgery and Radiology, at the University of Calgary, Calgary, Alberta, Canada. He received the Bachelor of Engineering degree in Electronics and Communication in 1976 from the University of Mysore at the People's Education Society College of Engineering, Mandya, Karnataka, India, and the Ph.D. degree in Electrical Engineering from the Indian Institute of Science, Bangalore, Karnataka, India, in 1980. His research interests are in the areas of digital signal and image processing, biomedical signal analysis, biomedical image analysis, and computer-aided diagnosis. He has published more than 140 papers in journals and 220 papers in proceedings of conferences. His research productivity was recognized with the 1997 and 2001 Research Excellence Awards of the Department of Electrical and Computer Engineering, the 1997 Research Award of the Faculty of Engineering, and by appointment as a "University Professor" in 2003, at the University of Calgary. He is the author of two textbooks: Biomedical Signal Analysis (IEEE/ Wiley, 2002) and Biomedical Image Analysis (CRC, 2005); he has coauthored and coedited several other books. He was recognized by the IEEE with the award of the Third Millennium Medal in 2000, and he was elected as a Fellow of the IEEE in 2001, Fellow of the Engineering Institute of Canada in 2002, Fellow of the American Institute for Medical and Bio[1]logical Engineering in 2003, Fellow of SPIE: the International Society for Optical Engineering in 2003, Fellow of the Society for Imaging Informatics in Medicine in 2007, Fellow of the Canadian Medical and Biological Engineering Society in 2007, and Fellow of the Canadian Academy of Engineering in 2009. He has been awarded the Killam Resident Fellowship thrice (1998, 2002, and 2007) in support of his book-writing projects.