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 \[ \frac {{d\eta \left ( s; x \right )}}{{ds}} = v\left ( \eta \left ( s; x \right ),s \right ), \qquad \eta \left ( \tau ; x \right ) = x, \qquad \left ( 0.1 \right )\] 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.
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G. E. Christensen, R. D. Rabbitt, and M. I. Miller, 3D brain mapping using a deformable neuroanatomy, Physics in Medicine and Biology 39, 609–618 (1994)
R. Dann, J. Hoford, S. Kovacic, M. Reivich, and R. Bajcsy, Evaluation of Elastic Matching Systems for Anatomic (CT, MR) and Functional (PET) Cerebral Images, Journal of Computer Assisted Tomography 13(4), 603–611 (July/August 1989)
O. Zeitouni and A. Dembo, A maximum a-posteriori estimator for trajectories of diffusion processes, Stochastics 20, 211–246 (1987) Erratum, p. 341
O. Zeitouni and A. Dembo, An existence theorem and some properties of maximum a-posteriori estimation of trajectories of diffusion, Stochastics 22, 197–218 (1988)
U. Grenander and M. I. Miller, Representations of knowledge in complex systems, Journal of the Royal Statistical Society B 56(3), 549–603 (1994)
O. Hijab, Minimum Energy Estimation, Ph.D. Dissertation, University of California, Berkeley, 1980
S. C. Hunter, Mechanics of Continuous Media, Ellis Horwood Limited, Chichester, England, 1976
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1990
M. I. Miller, G. E. Christensen, Y. Amit, and U. Grenander, Mathematical textbook of deformable neuroanatomies, Proceedings of the National Academy of Science 90(24) (December 1993)
R. E. Mortensen, Maximum-likelihood recursive nonlinear filtering, JOTA 2(6), 386–394 (1968)
S. Timoshenko, Theory of Elasticity, McGraw-Hill, New York, 1934
A. Trouvé, Habilitation à diringer les recherches, Technical Report, University Orsay, 1996
W. P. Ziemer, Weakly Differentiate Functions, Springer-Verlag, New York, 1989
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© Copyright 1998
American Mathematical Society