The pattern theory proposed by Grenander [17] described a systematic framework to represent shape classes that exhibit a substantial amount of variability but also possess a characteristic structure. Grenander and Keenan [18] formulated a global, pattern-theoretic model of shape which provides a structured method to systematically generate patterns from a class of shapes. In general, the above shape model can be represented by:
The global shape model has been successfully used to model biological shapes, hands, leaves, etc. [2,5,19,18,28]. This shape model can be very versatile because of different choices of the prototype template and the deformation process [18]. For example, Chow et al. [5] used polygons to approximate the contour of human hands. The building blocks in their shape model are the polygonal edges which meet the regularity condition because polygons are simple and connected. Variations in different hands are described by Markov processes on the edges. Chow et al. applied this shape model for hand synthesis and restoration. Similar approaches have also been used to model leaves, 3-D chairs, and 3-D human organs. In another paper on restoration of human hands from noisy greyscale images, Amit et al. [2] used an intensity image to represent a typical human hand. All instances of the class of hands are obtained by applying a number of admissible continuous transformations to the ``ideal'' hand image. Furthermore, this set of continuous mappings is governed by a Gaussian distribution. The observed image is assumed to be corrupted by an additive noise process. The reconstruction is obtained by maximizing the posterior distribution.