Curvature and Tangential Deflection of Discrete Arcs: A Theory Based on the Commutator of Scatter Matrix Pairs and Its Application to Vertex Detection in Planar Shape Data

This paper introduces a new theory for the tangential deflection and curvature of plane discrete curves. Our theory applies to discrete data in either rectangular boundary coordinate or chain coded formats: its rationale is drawn from the statistical and geometric properties associated with the eigenvalue-eigenvector structure of sample covariance matrices. Specifically, we prove that the nonzero entry of the commutator of a piar of scatter matrices constructed from discrete arcs is related to the angle between their eigenspaces. And further, we show that this entry is-in certain limiting cases-also proportional to the analytical curvature of the plane curve from which the discrete data are drawn. These results lend a sound theoretical basis to the notions of discrete curvature and tangential deflection; and moreover, they provide a means for computationally efficient implementation of algorithms which use these ideas in various image processing contexts. As a concrete example, we develop the commutator vertex detection (CVD) algorithm, which identifies the location of vertices in shape data based on excessive cummulative tangential deflection; and we compare its performance to several well established corner detectors that utilize the alternative strategy of finding (approximate) curvature extrema.