Area and Length 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.