GEOMETRIC HEAT-EQUATION AND NONLINEAR DIFFUSION OF SHAPES AND IMAGES

Visual tasks often require a hierarchical representation of shapes and images in scales ranging from coarse to fine. A variety of linear and nonlinear smoothing techniques, such as Gaussian smoothing, anisotrop ic diffusion, regularization, etc., have been proposed, leading to sca lespace representations. We propose a geometric smoothing method based on local curvature for shapes and images. The deformation by curvatur e, or the geometric heat equation, is a special case of the reaction-d iffusion framework proposed in [41]. For shapes, the approach is analo gous to the classical heat equation smoothing, but with a renormalizat ion by are-length at each infinitesimal step. For images, the smoothin g is similar to anisotropic diffusion in that, since the component of diffusion in the direction of the brightness gradient is nil, edge loc ation is left intact. Curvature deformation smoothing for shape has a number of desirable properties: it preserves inclusion order, annihila tes extrema and inflection points without creating new ones, decreases total curvature, satisfies the semigroup property allowing for local iterative computations, etc. Curvature deformation smoothing of an ima ge is based on viewing it as a collection of iso-intensity level sets, each of which is smoothed by curvature. The reassembly of these smoot hed level sets into a smoothed image follows a number of mathematical properties; it is shown that the extension from smoothing shapes to sm oothing images is mathematically sound due to a number of recent resul ts [21]. A generalization of these results [14] justifies the extensio n of the entire entropy scale space for shapes [42] to one for images, where each iso-intensity level curve is deformed by a combination of constant and curvature deformation. The scheme has been implemented an d is illustrated for several medical, aerial, and range images. (C) 19 96 Academic Press, Inc.