Journal Article

Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

Miller, Michael I., Younes, Laurent, Beg, M. Faisal, Trouvé, Alain

2005

Overview

This paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouvé (1995) in which two images I ^sub 0^, I ^sub 1^ are given and connected via the diffeomorphic change of coordinates I ^sub 0^^sup -1^=I ^sub 1^ where =Φ^sub 1^ is the end point at t= 1 of curve Φ^sub t^, t[0, 1] satisfying ^sup .^Φ^sub t^=v ^sub t^ (Φ^sub t^), t [0,1] with Φ^sub 0^=id. The variational problem takes the form ... where v ^sub t^^sub V^ is an appropriate Sobolev norm on the velocity field v ^sub t^(·), and the second term enforces matching of the images with ·^sub L^ ^sup 2^ representing the squared-error norm. In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields v ^sub t^, t[0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms. We describe the implementation of the Euler equations using semi-lagrangian method of computing particle flows and show the solutions for various examples. As well, we compute the metric distance on several anatomical configurations as measured by ∫^sub 0^ ^sup 1^v ^sub t^^sub V^dt on the geodesic shortest paths.[PUBLICATION ABSTRACT]

• Publisher: Springer Nature B.V
• Subject: Deformation / Studies