Variational problems on flows of diffeomorphisms for image matching

Abstract

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