Affine Geometry, Curve Flows, and Invariant Numerical Approximations

Abstract

A new geometric approach to the affine geometry of curves in the plane and to affine-invariant curve shortening is presented. We describe methods of approximating the affine curvature with discrete finite difference approximations, based on a general theory of approximating differential invariants of Lie group actions by joint invariants. Applications to computer vision are indicated.

Geometric Foundations of Numerical Algorithms and Symmetry

“…. , z n ; C) where C denotes the graph of the polynomial 5) and z k = (x k , p n (x k )) ∈ C for k = 0, . .…”

“…Thus, the theory of multi-invariants is the theory of invariant numerical approximations! The basic idea of replacing differential invariants by joint invariants forms the foundation of Dorodnitsyn's approach to invariant numerical algorithms, [10,11], and also the invariant numerical approximations of differential invariant signatures in computer vision, [1,4,5,27].…”

“…The interested reader may find the analysis of the planar equi-affine group w = A z + b, where det A = 1, a good challenge. This group action underlies the (equi-)affine geometry of planar curves, [16], and has been extensively used in computer vision; see [4,5] and the references therein.…”

“…These methods are closely related to the active area of geometric integration of ordinary differential equations on Lie groups, [3,22]. In practical applications of invariant theory to computer vision, group-invariant numerical schemes to approximate differential invariants based on suitable combinations of joint invariants, [27], have been applied to the problem of symmetry-based object recognition, [1,4,5].…”

“…To compare these with the invariant numerical approximations proposed in [5,4,1], we reformulate the divided difference formulae in terms of the geometrical configurations of the four distinct points z 0 , z 1 , z 2 , z 3 on our curve. We find…”