Exact shape flattening by means of Weitzenböck geometry and teleparallel distance

Abstract

Flattening shapes without distortion is a problem that has been intriguing scientists for centuries. It is a fundamental problem of high importance in computer vision as many approaches may greatly benefit from its implementation. This paper introduces a new approach that allows flattening without distortion, by transforming the shape from Riemannian geometry to Weitzenböck geometry. This transformation is obtained by calculating the Cholesky frame associated with the Riemannian metric. In the Weitzenböck space, the Riemann tensor is identically zero which means that the Weitzenböck space is entirely flat. The teleparallel equation, which determines distances in the Weitzenböck space, and the geodesic equation, which determines distances in its Riemannian counterpart, are equivalent. The end result is that there is no distortion when passing from Riemannian geometry to Weitzenböck geometry. Given the importance of the heat kernel in computer vision, an analytic solution of the heat kernel in Weitzenböck space is presented.