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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Y. Amit, U. Grenander, and M. Piccioni, Structural image restoration through deformable templates. J. American Statistical Association 86 (414), 376-387 (June 1991)
  • [2] V. I. Arnol′d, Ordinary differential equations, The M.I.T. Press, Cambridge, Mass.-London, 1973. Translated from the Russian and edited by Richard A. Silverman. MR 0361233
  • [3] R. Bajcsy and S. Kovacic, Multiresolution Elastic Matching, Computer Vision, Graphics, and Image Processing 46, pp. 1-21, 1989
  • [4] Gary Edward Christensen, Deformable shape models for anatomy, ProQuest LLC, Ann Arbor, MI, 1994. Thesis (D.Sc.)–Washington University in St. Louis. MR 2691771
  • [5] G. E. Christensen, R. D. Rabbitt, and M. I. Miller, Deformable templates using large deformation kinematics, IEEE Transactions on Image Processing, 5(10), 1996, pp. 1435-1447.
  • [6] G. E. Christensen, R. D. Rabbitt, and M. I. Miller, A deformable neuroanatomy textbook based on viscous fluid mechanics, In Jerry Prince and Thordur Runolfsson, editors, Proceedings of the Twenty-Seventh Annual Conference on Information Sciences and Systems, pp. 211-216, Baltimore, Maryland, March 24-26, 1993. Department of Electrical Engineering, The Johns Hopkins University.
  • [7] 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)
  • [8] 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)
  • [9] O. Zeitouni and A. Dembo, A maximum a posteriori estimator for trajectories of diffusion processes, Stochastics 20 (1987), no. 3, 221–246. MR 878313, https://doi.org/10.1080/17442508708833444
    O. Zeitouni and A. Dembo, Erratum: “A maximum a posteriori estimator for trajectories of diffusion processes”, Stochastics 20 (1987), no. 4, 341. MR 885878
  • [10] O. Zeitouni and A. Dembo, An existence theorem and some properties of maximum a posteriori estimators of trajectories of diffusions, Stochastics 23 (1988), no. 2, 197–218. MR 928355, https://doi.org/10.1080/17442508808833490
  • [11] Ulf Grenander and Michael I. Miller, Representations of knowledge in complex systems, J. Roy. Statist. Soc. Ser. B 56 (1994), no. 4, 549–603. With discussion and a reply by the authors. MR 1293234
  • [12] Omar Bakri Hijab, MINIMUM ENERGY ESTIMATION, ProQuest LLC, Ann Arbor, MI, 1980. Thesis (Ph.D.)–University of California, Berkeley. MR 2631198
  • [13] S. C. Hunter, Mechanics of continuous media, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons Inc.], New York-London-Sydney, 1976. Mathematics & its Applications. MR 0445984
  • [14] Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990. MR 1070361
  • [15] 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)
  • [16] R. E. Mortensen, Maximum-likelihood recursive nonlinear filtering, J. Optim. Theory Appl. 2 (1968), no. 6, 386–394. MR 1551293, https://doi.org/10.1007/BF00925744
  • [17] S. Timoshenko, Theory of Elasticity, McGraw-Hill, New York, 1934
  • [18] A. Trouvé, Habilitation à diringer les recherches, Technical Report, University Orsay, 1996
  • [19] William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 49J20, 58E25

Retrieve articles in all journals with MSC: 49J20, 58E25


Additional Information

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


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website