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 $G \subset {R^3}$. The changes of variable are determined as solutions of the nonlinear Eulerian transport equation \[ \frac {{d\eta \left ( s; x \right )}}{{ds}} = v\left ( \eta \left ( s; x \right ),s \right ), \qquad \eta \left ( \tau ; x \right ) = x, \qquad \left ( 0.1 \right )\] with the location $\eta \left ( 0; x \right )$ in the canonical image carried to the location $x$ in the deformed image. The variational problem then takes the form \[ \arg \min \limits _v {\kern -0.1pt} \left [ {{{\left \| v \right \|}^2} + \int _G {{{\left | {T o \eta \left ( {0; x} \right ) - D\left ( x \right )} \right |}^2}dx} } \right ], \qquad \left ( {0.2} \right )\] where $\left \| v \right \|$ is an appropriate norm on the velocity field $v( \cdot , \cdot )$, and the second term attempts to enforce fidelity to the data.