305.

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Page 73

types and measuring the distance to the shells rather than to the cluster centre. The shell clustering approach provides a choice to solve the hitherto unsolved problem of simultaneously fitting multiple circles/planes to segment data.

The practical value (or usefulness) of the vacuum shell clustering approach may seem to be of doubtful value in labelling the pixels forming a remotely sensed image. However, it could be argued that the existence of the method does draw attention to the often-unspoken assumption that meaningful cluster types should always be compact in nature.

Coray (1981) seems to have been the first to suggest the application of the vacuum shell idea to find circular clusters. Later, Dave’s (1990) Fuzzy c-shells (FCS) algorithm and the adaptive fuzzy c-shells (AFCS) algorithm derived by Dave and Bhaswan (1992) have proven to be successful in detecting circular and elliptical shapes. It should be noted that these algorithms are computationally intensive, since it is necessary to solve non-linear equations at every iteration in order to update the shell parameters. Krishnaparam et al. (1991, 1993) suggest a computationally simpler algorithm, which can be used to detect more general quadratic shapes. This algorithm assumes that each cluster resembles a hyperquadratic surface, which can be characterised by the following functional form:

(2.13)

where P_i is the coefficient vector for cluster i, and Q is the vector of variables. Both P_i and Q are defined as:

(2.14)

and s denotes the total number of entries in P_i and can be calculated as:

(2.15)

Note that Equations (2.13) to (2.15) are used to describe a function that characterises a shell shape.

The algebraic distance from a point x_j to a prototype S_i may therefore be defined as:

(2.16)

where

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