Time Discrete Geodesic Paths in the Space of Images

TL;DR

This paper models the space of images as a Riemannian manifold using metamorphosis, proposing a variational time discretization for geodesic paths that combines image transport and intensity variation, with proven existence and convergence.

Contribution

It introduces a novel variational discretization method for geodesic paths in the space of images, including existence, convergence proofs, and a practical finite element implementation.

Findings

Existence of discrete geodesic paths as minimizers.

Gamma-convergence of discrete to continuous path energy.

Numerical results demonstrate efficiency and qualitative properties.

Abstract

In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach, where the underlying Riemannian metric simultaneously measures the cost of image transport and intensity variation. A robust and effective variational time discretization of geodesics paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals over a set of image intensity maps and pairwise matching deformations. For square-integrable input images the existence of discrete, connecting geodesic paths defined as minimizers of this variational problem is shown. Furthermore, $\Gamma$-convergence of the underlying discrete path energy to the continuous path energy is proved. This includes a diffeomorphism property for the induced transport and the existence of a square-integrable weak material derivative in space and time. A spatial discretization via finite elements combined with an alternating descent scheme in the set of image intensity maps and the set of matching deformations is presented to approximate discrete geodesic paths numerically. Computational results underline the efficiency of the proposed approach and demonstrate important qualitative properties.