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.
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Dates
| Type | When |
|---|---|
| Created | 8 years, 8 months ago (Dec. 14, 2016, 11:43 p.m.) |
| Deposited | 10 months, 1 week ago (Oct. 25, 2024, 1:35 p.m.) |
| Indexed | 3 months, 1 week ago (May 22, 2025, 3:46 p.m.) |
| Issued | 27 years ago (Sept. 1, 1998) |
| Published | 27 years ago (Sept. 1, 1998) |
| Published Print | 27 years ago (Sept. 1, 1998) |
@article{Dupuis_1998, title={Variational problems on flows of diffeomorphisms for image matching}, volume={56}, ISSN={1552-4485}, url={http://dx.doi.org/10.1090/qam/1632326}, DOI={10.1090/qam/1632326}, number={3}, journal={Quarterly of Applied Mathematics}, publisher={American Mathematical Society (AMS)}, author={Dupuis, Paul and Grenander, Ulf and Miller, Michael I.}, year={1998}, month=sep, pages={587–600} }