© 2005 SFoCM DOI: 10.1007/s10208-004-0128-z Found. Comput. Math. 173–198 (2005) The Journal of the Society for the Foundations of Computational Mathematics FOUNDATIONS OF COMPUTATIONAL MATHEMATICS Metamorphoses Through Lie Group Action Alain Trouv´ e 1 and Laurent Younes 2 1 CMLA ENS de Cachan 61, avenue du Pr´ esident Wilson 94235 Cachan CEDEX, France trouve@cmla.ens-cachan.fr 2 Department of Applied Mathematics and Statistics and Center for Imaging Science The Johns Hopkins University 3400 N-Charles Street Baltimore, MD 21218-2686, USA younes@cis.jhu.edu Abstract. We formally analyze a computational problem which has important applications in image understanding and shape analysis. The problem can be sum- marized as follows. Starting from a group action on a Riemannian manifold M, we introduce a modification of the metric by partly expressing displacements on M as an effect of the action of some group element. The study of this new structure relates to evolutions on M under the combined effect of the action and of resid- ual displacements, called metamorphoses. This can and has been applied to image processing problems, providing in particular diffeomorphic matching algorithms for pattern recognition. Contents 1. Notation 174 2. A New Metric on M 178 2.1. General Form 178 2.2. Examples 179 Date received: March 9, 2004. Final version received: August 20, 2004. Date accepted: September 23, 2004. Communicated by Peter Olver. Online publication February 11, 2005. AMS classification: 68T45, 53B21, 58E10, 58E40. Key words and phrases: Groups of diffeomorphisms, Infinite-dimensional shape spaces, Geodesics, Shape recognition, Image registration.