Area and Lenght Preserving Geometric Invariant Scale-Spaces

In this paper, area preserving multi-scale representations of planar curves are described. This allows smoothing without shrinkage at the same time preserving all the scale-space properties. The representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant. When the Euclidean geometric heat flow is used, the resulting representation is Euclidean invariant, and similarly it is affine invariant when the affine one is used. 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, it is completely local as well. The same approach is used to define length preserving geometric flows. A similarity (scale) invariant geometric heat flow is studied as well in this work. The geometric scale-spaces are implemented using an efficient numerical algorithm, and their smoothing property is demonstrated by examples as well. *This work was supported in part by grants from the National Science Foundation DMS-8811084 and ECS-9122106, by the Air Force Office of Scientific Research AFOSR-90-0024, and by the Army Research Office DAAL03-91-G-0019. GS is also partially supported by the Rothschild Foundation-Yad Hanadiv, and by the Army Research Office DAAL03-92-G-0115 (Center for Intelligent Control Systems). Part of this work was performed at the Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel 32000.