Geodesic Warps by Conformal Mappings

TL;DR

This paper develops a Riemannian geometric framework for conformal mappings to generate diffeomorphic image warps, with applications in medical imaging and biology, including numerical methods and analysis of special solutions.

Contribution

It introduces a novel geometric approach to conformal warps using geodesic equations on conformal embedding manifolds, with numerical discretization and analysis of special solutions.

Findings

Derived geodesic equations for conformal warps.

Developed a numerical discretization method.

Identified totally geodesic solutions for scaling and translation.

Abstract

In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications e.g.\ in medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D'Arcy Wentworth Thompson, who demonstrated warps to deform images of one species into another. However, unlike the deformations in modern methods, which are drawn from the full set of diffeomorphism, he deliberately chose lower-dimensional sets of transformations, such as planar conformal mappings. In this paper we study warps of such conformal mappings. The approach is to equip the infinite dimensional manifold of conformal embeddings with a Riemannian metric, and then use the corresponding geodesic equation in order to obtain diffeomorphic warps. After deriving the geodesic equation, a numerical discretisation method is developed. Several examples of geodesic warps are then given. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations.