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 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 ⊂ R³. The changes of variable are determined as solutions of the nonlinear Eulerian transport equation dη(s; x)/ds= v(η(s; x), s), η(τ; x) = x, with the location η(0; x) in the canonical image carried to the location x in the deformed image. The variational problem then takes the form arg min _v [∥v∥² + ∫_G |T ○ η(0; x) - D(x)|²dx], where ∥v∥ is an appropriate norm on the velocity field v(., .), and the second term attempts to enforce fidelity to the data. In this paper we derive conditions under which the variational problem described above is well posed. The key issue is the choice of the norm. Conditions are formulated under which the regularity of v(., .) imposed by finiteness of the norm guarantees that the associated flow is supported on a space of diffeomorphisms. The problem (0.2) can be interpreted as a problem in optimal control, in which the superposition of the running cost ∥v∥ and the terminal cost determined by the data is to be minimized. We show that a minimizer v̂ exists, with the optimal smooth change of coordinates denned via (0.1). We also discuss an interpretation of the variational problem in the context of Bayesian estimation.