Invariant Framework for Diierential Aane Signatures

A framework for generating diierential aane invariant signatures based on the gray level images of planar shapes is introduced. Non trivial invariant signatures and their corresponding arclengths are computed for planar shapes. These signatures are useful for pattern recognition and classiication under partial occlusion. We deal only with implementable signatures, which practically means using up to second order derivatives , and restrict the aane transformation group accordingly. Based on the theory of aane curve evolution, an invariant gradient magnitude along the geometric scale space is deened and used as an invariant edge enhancer. The geometric heat equation for weighted (by the enhancer) aane arclength deenition is shown to yield an invariant selective smoothing algorithm. This algorithm is used for image denoising in cases where we need to clean noisy images before computing invariant features. The denois-ing 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. The relation between the aane curvature and the Euclidean one leads to an eecient method for approximating the aane curvature signature , while the Euclidean curvature itself is used for generating the aane arclength parameter. Both curvatures are computed from the gray level image, using the implicit representation of the object's boundary as it appears in real world images. When the projection invariance assumption of the gray levels is added, robust non-trivial signatures are obtained.