Pattern spectrum and multiscale shape representation

The results of a study on multiscale shape description, smoothing and representation are reported. Multiscale nonlinear smoothing filters are first developed, using morphological opening and closings. G. Matheron (1975) used openings and closings to obtain probabilistic size distributions of Euclidean-space sets (continuous binary images). These distributions are used to develop a concept of pattern spectrum (a shape-size descriptor). A pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals, as well as for discrete images, by developing a discrete-size family of patterns. Large jumps in the pattern spectrum at a certain scale indicate the existence of major (protruding or intruding) substructures of the signal at the scale. An entropy-like shape-size complexity measure is also developed based on the pattern spectrum. For shape representation, a reduced morphological skeleton transform is introduced for discrete binary and graytone images. This transform is a sequence of skeleton components (sparse images) which represent the original shape at various scales. It is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components; in addition, two-way correspondences are established among the degree of shape smoothing via multiscale openings or closings, the pattern spectrum zero values, and the elimination or nonexistence of skeleton components at certain scales