Geometric Heat Equation and Non-linear Diiusion of Shapes and Images Contents 1 Introduction 1 2 the Shape from Deformation Framework 2 3 Nonlinear Smoothing by Curvature Deformation 4 3.1 Order Preserving Smoothing Annihilation of Extrema and Innection Points

Visual tasks often require a hierarchical representation of shapes and images in scales ranging from coarse to ne. A variety of linear and nonlinear smoothing techniques, such as Gaussian smoothing, anisotropic di usion, regularization, etc. , have been proposed, leading to scalespace representations. We propose a geometric smoothing method based on local curvature in shapes and images. This deformation by curvature, or the geometric heat equation, is a special case of the reaction-di usion framework proposed by [34]. For shapes, the approach is analogous to the classical heat equation smoothing, but with a renormalization by arc-length at each in nitesimal step. For images, the smoothing is similar to anisotropic di usion in that, since the component of di usion in the direction of the brightness gradient is nil, edge location and sharpness is left intact. Curvature deformation smoothing for shape has a number of desirable properties: it preserves inclusion order, annihilates extrema and in ection points without creating new ones, decreases total curvature, satis es the semigroup property allowing for local iterative computations, etc. Curvature deformation smoothing of an image is based on viewing it as a collection of iso-intensity level sets, each of which is smoothed by curvature. The re-assembly of these smoothed level sets into a smoothed image requires that a number of mathematical properties hold; these are discussed and it is shown that the extension from smoothing shapes to smoothing images is mathematically sound due to a number of recent results [19]. A generalization of these results [11] justi es the extension of the entire entropy scale space for shapes [35] to one for images, where each iso-intensity level curve is deformed by a combination of constant and curvature deformation [63]. The scheme has been implemented and is illustrated for several medical, aerial, and range images.