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This paper studies a variational formulation of the image matching problem. We consider a scenario in which a canonical representative image T T is to be carried via a smooth change of variable into an image that is intended to provide a good fit to the observed data. The images are all defined on an open bounded set G ⊂ R 3 G R³. The changes of variable are determined as solutions of the nonlinear Eulerian transport equation \ d η (s ; x) d s = v (η (s ; x), s), η (τ ; x) = x, (0. 1) {d (s; x) }{ds} = v ( (s; x), s), (; x) = x, (0. 1) \ with the location η (0 ; x) (0; x) in the canonical image carried to the location x x in the deformed image. The variational problem then takes the form \[ arg min v [ ‖ v ‖ 2 + ∫ G
Dupuis et al. (Tue,) studied this question.