Annals of Mathematical Sciences and Applications

Volume 3 (2018)

Number 1

Special issue in honor of Professor David Mumford, dedicated to the memory of Jennifer Mumford

Guest Editors: Stuart Geman, David Gu, Stanley Osher, Chi-Wang Shu, Yang Wang, and Shing-Tung Yau

Hybrid Riemannian metrics for diffeomorphic shape registration

Pages: 189 – 210



Laurent Younes (Department of Applied Mathematics and Statistics, and Center for Imaging Science, Johns Hopkins University, Baltimore, Maryland, U.S.A.)


We consider the results of combining two approaches developed for the design of Riemannian metrics on curves and surfaces, namely parametrization-invariant metrics of the Sobolev type on spaces of immersions, and metrics derived through Riemannian submersions from right-invariant Sobolev metrics on groups of diffeomorphisms (the latter leading to the “large deformation diffeomorphic metric mapping” framework).We show that this quite simple approach inherits the advantages of both methods, both on the theoretical and experimental levels, and provide additional flexibility and modeling power, especially when dealing with complex configurations of shapes. Experimental results illustrating the method are provided for curve and surface registration.


shape analysis, groups of diffeomorphisms, sub-Riemannian metrics, optimal control, computational anatomy

2010 Mathematics Subject Classification

49N90, 49Q10, 58D05, 68-xx

Partially supported by NIH U19AG033655, R01HL130292 and R01DC016784.

Received 5 June 2017

Published 27 March 2018