Clustering of Fuzzy Shapes by Integrating Procrustean Metrics and Full Mean Shape Estimation into K-Means Algorithm

In this paper we propose a generalization of K-means algorithm, which is adapted to integrate Procrustean metrics and full mean shape estimation, with the aim of clustering objects with either multiple or fuzzy contours. First we are concerned with the representation of fuzzy shapes and introduce appropriate shape metrics and descriptors. Next, we discuss Procrustean methods for aligning shapes, finding mutual dissimilarities and estimating shape class centroid. In the case of multiple-contour crisp shapes, we can benefit from the Extended Orthogonal Procrustes method to find mutual distances between shape pairs and from the Generalized Orthogonal Procrustes technique to estimate the Procrustes mean shape of a collection of shapes. On the other hand, dealing with the case of fuzzy shapes needs more advanced Procrustean techniques to consider weighted distances between points placed on level contours with different membership degrees. This leads to solve a Weighted Orthogonal Procrustes problem, which typically needs to introduce a weighting matrix of residuals (distances). As an application, we suggest using such methods to cluster ultrasound images of lymph nodes, which typically appear as double-contour shapes. Keywords— Clustering of fuzzy shapes, Fuzzy shape metrics and descriptors, Procrustes analysis, Mixing K-means algorithm with Procrustean metrics and mean shape estimation. 1 Shape analysis Shapes and textures are extremely important features in human as well as machine vision and understanding systems. Shape analysis is concerned with two main classes of algorithms: boundary-based (when only the shape boundary points are used for the description) and region-based (when the whole interior of a shape is used). There are many imaging applications where image analysis can be reduced to the analysis of shapes, in contrast to texture analysis. However, many shape/edge detection techniques use texture information during the segmentation process. There are several methods for extracting data from shapes, each with their own benefits and weaknesses. These include measurement of lengths and angles, landmark analysis and outline analysis. A landmark is a point of correspondence on each object that matches higher dimensionalities between and within populations. Landmark placement consists of locating a finite number of points on the outline. More advanced techniques have been designed for semiautomatic and automatic feature extractions. Active contour modeling techniques are commonly used for shape analysis and detection. Some of the techniques for texture feature extraction use gray level co-occurrence matrices, fractal dimension, etc. Morphometric analysis aims to describe the shape of an object in a way that removes extraneous information and thereby facilitates comparison between different objects. In these terms, a shape is referred to as an invariant to similarity transformations (such as scaling, rotation and translation). The image fuzzification plays a pivotal role in all image processing systems. Several kinds of image fuzzification can be distinguished: histogram-based grey-level fuzzification (e.g. brightness in image enhancement); local fuzzification (e.g. edge detection); feature fuzzification (scene analysis, object recognition). 2 Representation of fuzzy shapes 2.1 Crisp shapes Crisp shapes represent objects with crisp borders. Furthermore, if a texture is associated with the object, it has to be uniformly represented (e.g. a digitized image, where all pixels are classified as object pixels, or as background pixels). The coordinates of selected landmarks for a crisp shape can be arranged in a p n configuration matrix A , or equivalently on a np×1 configuration vector ) (A vec a . 2.2 Continuous fuzzy shapes This paper primarily focuses on the representation of fuzzy shapes with fuzzy contour, which are commonly obtained through fuzzy segmentation techniques. In particular, we also consider the case of crisp shapes with multiple contours. In the same way as it is convenient to model binary images as crisp objects, it is possible to model grey-level images directly as fuzzy sets. If the grey-level values of an image are scaled to be between 0 and 1, the grey-level of a pixel can be seen as its membership to the set of high-valued (bright) pixels. Fuzziness of an image representation can arise from various reasons, such as limited acquisition conditions (scanning resolution), but also as intrinsic property of the image, which may have imprecise borders. In such cases, pixels close to the border of the object have assigned to them a fuzzy membership value according to the extent of their belongingness to the object. ISBN: 978-989-95079-6-8 IFSA-EUSFLAT 2009