304.

[Cover] [Contents] [Index]

Page 72

where n_i and n_j are the number of pixels in cluster i and j, and μ_i and μ_j are the mean vectors of cluster i and j, respectively. Assume that the number of clusters at the start of the first iteration is k. At each subsequent iteration, the two closest clusters are merged, and the process repeated with k−1, k−2,…clusters until the user-specified minimum number of clusters has been reached. The solution that best meets the needs of the specific project is then selected.

2.3.2 Fuzzy clustering

An alternative approach to unsupervised classification using ISODATA is fuzzy clustering. The main difference between fuzzy clustering and the ISODATA method is that the resulting clusters generated by fuzzy clustering are no longer ‘hard’ or ‘crisp’, but fuzzy. In other words, each pixel may simultaneously belong to two or more clusters and will have a ‘membership value’ for each cluster. In general, the performance of fuzzy clustering methods is superior to that of the corresponding hard versions, and they are less likely to stick in a local minima (Bezdek, 1981).

Most analytic fuzzy clustering algorithms are derived from Bezdek’s (1981) fuzzy c-means (FCM) algorithm. The FCM algorithm and its derivations have been implemented successfully in many applications, such as pattern classification and image segmentation, especially those in which the final goal is to make a crisp decision. The FCM algorithm uses the probabilistic constraint that the membership probabilities of a data point across classes must sum to one. This constraint comes from generalising a crisp c-partition of a data set, and is used to generate membership-update equations for an iterative algorithm based on the minimisation of a least-square type of criterion function. The constraint on membership used in FCM is meant to avoid the trivial solution of all membership probabilities being equal to zero. FCM does provide meaningful results in applications where it is appropriate to interpret memberships as likelihood estimates (Goth and Geva, 1989) although some studies have provided a different perspective on the meaning of membership probabilities (see, for example, Krishnapuram and Keller, 1993, 1996; Krishnapuram et al., 1995). A more detailed description of FCM is provided in Chapter 4.

2.3.3 Vacuum shell detection

Traditionally, clustering methods are used to detect compact clusters in multidimensional feature space. These clusters are typically described in terms of the location of the cluster mean (centre) and the cluster variance-covariance matrix. FCM and its derivations have performed successfully in generating such clusters. This approach to clustering has been further extended to the case of hollow or shell-like clusters by using shell proto

[Cover] [Contents] [Index]