Variational problems on flows of diffeomorphisms for image matching

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 ⊂ {R^3} . The changes of variable are determined as solutions of the nonlinear Eulerian transport equation \[ d η ( s ; x ) d s = v ( η ( s ; x ) , s ) , η ( τ ; x ) = x , ( 0.1 ) {{dη ( s; x )}}{{ds}} = v ( η ( s; x ),s ), η ( τ ; x ) = x, ( 0.1 ) \] with the location η ( 0 ; x ) η ( 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