Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach

Computer vision deals with image understanding at various levels. At the low level, it addresses issues such us planar shape recognition and analysis. Some classical results on differential invariants associated to planar curves are relevant to planar object recognition under different views and partial occlusion, and recent results concerning the evolution of planar shapes under curvature controlled diffusion have found applications in geometric shape decomposition, smoothing, and analysis, as well as in other image processing applications. In this work we first give a modern approach to the theory of differential invariants, describing concepts like Lie theory, jets, and prolongations. Based on this and the theory of symmetry groups, we present a high level way of defining invariant geometric flows for a given Lie group. We then analyze in detail different subgroups of the projective group, which are of special interest for computer vision. We classify the corresponding invariant flows and show that the geometric heat flow is the simplest possible one. This uniqueness result, together with previously reported results which we review in this chapter, confirms the importance of this class of flows. Results on invariant geometric flows of surfaces are presented at the end of the chapter as well.

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