Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Variational problems on flows of diffeomorphisms for image matching

Authors: Paul Dupuis, Ulf Grenander and Michael I. Miller
Journal: Quart. Appl. Math. 56 (1998), 587-600
MSC: Primary 49J20; Secondary 58E25
DOI: https://doi.org/10.1090/qam/1632326
MathSciNet review: MR1632326
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Abstract: 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

$\displaystyle \frac{{d\eta \left( s; x \right)}}{{ds}} = v\left( \eta \left( s;... ...),s \right), \qquad \eta \left( \tau ; x \right) = x, \qquad \left( 0.1 \right)$

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

$\displaystyle \mathop {\arg \min }\limits_v {\kern-0.1pt} \left[ {{{\left\Vert ... ... - D\left( x \right)} \right\vert}^2}dx} } \right], \qquad \left( {0.2} \right)$

where $ \left\Vert v \right\Vert$ is an appropriate norm on the velocity field $ v( \cdot , \cdot )$, and the second term attempts to enforce fidelity to the data.

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DOI: https://doi.org/10.1090/qam/1632326
Article copyright: © Copyright 1998 American Mathematical Society

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