Invariant Histograms

We introduce and study a Euclidean-invariant distance histogram function for curves. For a sufficiently regular plane curve, we prove that the cumulative distance histograms based on discretizing the curve by either uniformly spaced or randomly chosen sample points converge to our histogram function. Robustness of the curve histogram function under noise and pixelization of the curve is also established. We argue that the histogram function serves as a simple, noise-resistant shape classifier for regular curves under the Euclidean group of rigid motions. Extensions of the underlying ideas to higher-dimensional submanifolds, as well as to area histogram functions invariant under the group of planar area-preserving affine transformations, are discussed.