Affine differential signatures for gray level images of planar shapes

A framework for generating differential affine invariant signatures based on the gray level images of planar shapes is introduced. Invariant signatures and their corresponding arclengths are computed for planar shapes. These signatures are useful for pattern recognition and classification under partial occlusion. We deal with implementable signatures, which practically means using up to second order derivatives. An approximation of the affine curvature signature is introduced. In this case the Euclidean curvature is used for generating the affine arclength. Both curvatures are computed from the gray level image, using the implicit representation of the object´s boundary as it appears in the image. We also present robust signatures when `projection invariance´ of the gray levels is assumed. An invariant gradient magnitude along the geometric scale space is defined and used as an invariant edge enhancer. The geometric heat equation for weighted (by the enhancer) affine arclength definition is shown to yield an invariant denoising algorithm. It is used to clean noisy images before computing invariant features. The denoising operation deforms the geometry of the object in a predictable invariant way, unlike traditional image denoising algorithms, so that the mapping between planar shapes after the denoising is preserved