Local symmetries of shapes in arbitrary dimension

Abstract

Motivated by a need to define an object-centered reference system determined by most salient characteristics of shape, many methods have been proposed, all of which directly or indirectly involve an axis about which shape is locally symmetric. Recently, a function /spl upsi/, called the edge strength function, has been successfully used to determine efficiently axes of local symmetries of 2-d shapes. The level curves of /spl upsi/ are interpreted as successively smoother versions of initial shape boundary. The local minima of absolute gradient /spl par//spl nabla//spl upsi//spl par/ along level curves of /spl upsi/ are shown to be a robust criterion for determining shape skeleton. More generally, at an extremal point of /spl par//spl nabla//spl upsi//spl par/ along a level curve, level curve is locally symmetric with respect to gradient vector /spl nabla//spl upsi/. That is, at such a point, level curve is approximately a conic section whose one of principal axes coincides with gradient vector. Thus, locus of extremal points of /spl par//spl nabla//spl upsi//spl par/ along level curves determines axes of local symmetries of shape. In this paper, we extend this method to shapes of arbitrary dimension.