IngentaConnect Computing Large Deformation Metric Mappings via Geodesic Flows of...

Authors: M. Faisal Beg¹; Michael I. Miller²; Alain Trouvé³; Laurent Younes⁴

Source: International Journal of Computer Vision, Volume 61, Number 2, February 2005 , pp. 139-157(19)

Publisher: Springer

Abstract:

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 I₀, I₁ are given and connected via the diffeomorphic change of coordinates I₀○ PHgr

^-1 = I₁ where PHgr

₁ is the end point at t = 1 of curve phiv

_t, t

[0, 1] satisfying · phiv

_t = v_t ( phiv

_t), t

[0,1] with phiv

₀ = id. The variational problem takes the form

\argmin_{v: \dot \phi_t = v_t (\phi_t)} \Bigg ( \int_0ˆ1 \| v_t\|ˆ2_V \dt+\big \| I_0 \circ \phi_1ˆ{-1} - I_1\big \|_{Lˆ2}ˆ2 \Bigg),

where

v_t

_V is an appropriate Sobolev norm on the velocity field v_t(·), and the second term enforces matching of the images with Verbar

_L² representing the squared-error norm.

In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields v_t, t isin

[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 int

₀¹

v_t

_Vdt on the geodesic shortest paths.

Keywords: Computational Anatomy; Euler-Lagrange Equation; Variational Optimization; Deformable Template; Metrics

Document Type: Research article

DOI: 10.1023/B:VISI.0000043755.93987.aa

Affiliations: 1: Center for Imaging Science & Department of Biomedical Engineering, The Johns Hopkins University, 301 Clark Hall, Baltimore, MD 21218, USA., Email: mfbeg@cis.jhu.edu 2: Center for Imaging Science, department of Biomedical Engineering, Department of Electrical and Computer Engineering and The Department of Computer Science, Whiting School of Engineering, The Johns Hopkins University, 301 Clark Hall, Baltimore, MD 21218, USA., Email: mim@cis.jhu.edu 3: LAGA, Université Paris 13, France., Email: trouve@zeus.math.univ-paris13.fr 4: CMLA, Ecole Normale Supérieure de Cachan, 61, Avenue du President Wilson, F-94 235 Cachan CEDEX, France., Email: younes@cmla.ens-cachan.fr

The full text electronic article is available for purchase. You will be able to download the full text electronic article after payment.

$47.00 plus tax

Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms