2. Problem formulation

The objective is to cluster a set of observed data f g xn; n ¼ 1; 2; ::; N where each data point is an <sup>M</sup> � dimensional real-vector called the feature or the pattern vector, i.e., xn <sup>∈</sup> R1�<sup>M</sup>. For gray-scale image data, f g xn; n ¼ 1; 2; ::; N is a row-wise concatenation of a 2-D image Xpq; <sup>p</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; ::; <sup>P</sup>; <sup>q</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; ::; <sup>Q</sup> � �. That is <sup>n</sup> <sup>¼</sup> ð Þ <sup>p</sup> � <sup>1</sup> <sup>Q</sup> <sup>þ</sup> <sup>q</sup> and the intensity-feature xn is a single-dimensional real-value, i.e., M ¼ 1. Clustering aims at partitioning theses N observations into C < N divisions, {μ1, μ2,…,μC} called C clusters or segments so as to make the entities or pixels in the same cluster as similar as possible and the ones in different clusters as dissimilar as possible. One approach to cluster these data is to minimize the withinclusters sum of squares of distances (WCSS) and to maximize the between-clusters sum of squares of distances (BCSS).
