Pattern Spectrum and Multiscale Shape Representation

This paper reports the results of a study of multiscale shape description, smoothing and representation. Multiscale nonlinear smoothing filters are first developed by using morphological openings and closings. These filters smooth signals while preserving their edges, and use a definition of scale that agrees with the spatial size of image objects. Matheron used openings and closings to obtain probabilistic size distributions of Eucl idean-space sets (continuous binary images). W e view these distributions as a concept of pattern spectrum (a shapesize descriptor). Then we develop a pattern spectrum for continuous graytone images and arbitrary multilevel signals, as well as for discrete images by using a discrete-size family of patterns. The pattern spectrum quantif ies various aspects of the shape-size content of a signal. Large impulses in the pattern spectrum at a certain scale indicate the existence of major (protruding or intruding) substructures of the signal at that 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 establ ished 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. All the above results provide useful tools for multiscale shape representation and description.