Area and Lenght Preserving Geometric Invariant Scale-Spaces

Abstract

In this paper, area preserving geometric multi-scale representations of planar curves are described. This allows geometric smoothing without shrinkage at the same time preserving all the scale-space properties. The representations are obtained deforming the curve via invariant geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant. The flows are geometrically intrinsic to the curve, and exactly satisfy all the basic requirements of scale-space representations. In the case of the Euclidean heat flow for example, it is completely local as well. The same approach is used to define length preserving geometric flows. The geometric scalespaces are implemented using an efficient numerical algorithm.