Geometric heat equation and nonlinear diffusion of shapes and images

49, 90]. Witkin proposed that the convolution of a signal Visual tasks often require a hierarchical representation of with a Gaussian removes small-scale features, namely, shapes and images in scales ranging from coarse to fine. A zero-crossings, while retaining the more significant ones variety of linear and nonlinear smoothing techniques, such as [88]. Koenderink showed that among the linear operators Gaussian smoothing, anisotropic diffusion, regularization, etc., the heat (diffusion) equation and its associated Gaussian have been proposed, leading to scalespace representations. We kernel is the only sensible way of smoothing images, by propose a geometric smoothing method based on local curvature demanding that the process satisfy the properties of causalfor shapes and images. The deformation by curvature, or the ity, homogeneity, and isotropy [48, 50]. These properties geometric heat equation, is a special case of the reaction– require that ‘‘structure’’ is not created with increasing scale diffusion framework proposed in [41]. For shapes, the approach and that operations are homogeneous in space and direcis analogous to the classical heat equation smoothing, but with a renormalization by arc-length at each infinitesimal step. For tion; see also [9]. Yuille and Poggio presented scaling theoimages, the smoothing is similar to anisotropic diffusion in rems for zero-crossings [89], and Hummel and Monoit that, since the component of diffusion in the direction of the showed that zero-crossings, when supplemented with grabrightness gradient is nil, edge location is left intact. Curvature dient data along the zero-crossing boundaries, are suffideformation smoothing for shape has a number of desirable cient to reconstruct the original signal. Recently Florack et properties: it preserves inclusion order, annihilates extrema and al. have provided a physical motivation and mathematical inflection points without creating new ones, decreases total basis for a scale-space representation [25], arriving at the curvature, satisfies the semigroup property allowing for local Gaussian family of filters without an explicit requirement iterative computations, etc. Curvature deformation smoothing for causality. In the discrete domain, Lindeberg has formuof an image is based on viewing it as a collection of iso-intensity lated a scale-space by discretizing the underlying diffusion level sets, each of which is smoothed by curvature. The reassembly of these smoothed level sets into a smoothed image follows equation [52, 53]. The kernel of the discretized equation a number of mathematical properties; it is shown that the is related to modified Bessel functions of integer order. extension from smoothing shapes to smoothing images is mathFor a recent review of linear scale-space theory see [54, 55]. ematically sound due to a number of recent results [21]. A In addition to image intensity, scale-spaces have also generalization of these results [14] justifies the extension of the been constructed for shapes. Asada and Brady [8] smooth entire entropy scale space for shapes [42] to one for images, the curvature function to obtain a hierarchy of features. where each iso-intensity level curve is deformed by a combinaMokhtarian and Mackworth [60] smooth the coordinates tion of constant and curvature deformation. The scheme has by a Gaussian filter. Horn and Weldon [34] point to the been implemented and is illustrated for several medical, aerial, shrinkage problems of this method and propose instead and range images.  1996 Academic Press, Inc. to filter the extended circular image of the curve with a Gaussian filter. This method avoids the shrinkage problem, but appears to be applicable only to convex curves. Lowe,