The DOI matches its Crossref record; 16 of the 16 references it cites were found in a registry or an index.
Variational problems on flows of diffeomorphisms for image matching
Open accessAbstract
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 ⊂ R 3 . The changes of variable are determined as solutions of the nonlinear Eulerian transport equation \[ d η ( s ; x ) d s = v ( η ( s ; x ) , s ) , η ( τ ; x ) = x , ( 0.1 ) \] with the location η ( 0 ; x ) in the canonical image carried to the location x in the deformed image. The variational problem then takes the form \[ arg min v [ ‖ v ‖ 2 + ∫ G | T o η ( 0 ; x ) − D ( x ) | 2 d x ] , ( 0.2 ) \] where ‖ v ‖ is an appropriate norm on the velocity field v ( ⋅ , ⋅ ) , and the second term attempts to enforce fidelity to the data.
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“This is unsurprising as, similar to existing optical flow approaches [14, 15], directly predicting displacements does not encode deformation smoothness.”
From Quicksilver: Fast predictive image registration – A deep learning approach (2017) · cites this paper as [15]
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Cites
Timoshenko, S. P., & Goodier, J. N. (1934) · Virtual Defense Library (Ministerio de Defensa) · Cited by 13,717
Ordinary Differential Equations
Stewart, J. M. (2017) · Cambridge University Press eBooks · doi:10.1017/9781108120241.010 · Cited by 7,299
Weakly Differentiable Functions
Ziemer, W. P. (1989) · Graduate texts in mathematics · doi:10.1007/978-1-4612-1015-3 · Cited by 2,225
Deformable templates using large deformation kinematics
Christensen, G. E., Rabbitt, R. D., & Miller, M. I. (1996) · IEEE Transactions on Image Processing · doi:10.1109/83.536892 · Cited by 1,146
Self-citationMultiresolution elastic matching
Bajcsy, R., & Kovačič, S. (1989) · Computer Vision Graphics and Image Processing · doi:10.1016/s0734-189x(89)80014-3 · Cited by 1,118
Stochastic Flows and Stochastic Differential Equations
MTW, & Kunita, H. (1998) · Journal of the American Statistical Association · doi:10.2307/2669903 · Cited by 1,014
Representations of Knowledge in Complex Systems
Grenander, U., & Miller, M. I. (1994) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1994.tb02000.x · Cited by 619
Self-citationMathematical textbook of deformable neuroanatomies.
Miller, M. I., Christensen, G. E., Amit, Y., & Grenander, U. (1993) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.90.24.11944 · Cited by 464
Self-citation3D brain mapping using a deformable neuroanatomy
Christensen, G. E., Rabbitt, R. D., & Miller, M. I. (1994) · Physics in Medicine and Biology · doi:10.1088/0031-9155/39/3/022 · Cited by 358
Self-citationStructural Image Restoration through Deformable Templates
Amit, Y., Grenander, U., & Piccioni, M. (1991) · Journal of the American Statistical Association · doi:10.1080/01621459.1991.10475053 · Cited by 325
Self-citationMaximum-likelihood recursive nonlinear filtering
Mortensen, R. E. (1968) · Journal of Optimization Theory and Applications · doi:10.1007/bf00925744 · Cited by 237
Hunter, S. C., & PIPKIN, A. C. (1977) · Journal of Applied Mechanics · doi:10.1115/1.3424204 · Cited by 234
Evaluation of Elastic Matching System for Anatomic (CT, MR) and Functional (PET) Cerebral Images
Dann, R., Hoford, J. D., Kovačič, S., Reivich, M., & Bajcsy, R. (1989) · Journal of Computer Assisted Tomography · doi:10.1097/00004728-198907000-00009 · Cited by 93
A maximum a posteriori estimator for trajectories of diffusion processes
Zeitouni, O., & Dembo, A. (1987) · Stochastics · doi:10.1080/17442508708833444 · Cited by 48
Ordinary Differential Equations
Potter, M. C., & Feeny, B. F. (2023) · Springer eBooks · doi:10.1007/978-3-031-26151-0_1 · Cited by 35
Zeitouni, O., & Dembo, A. (1988) · Stochastics · doi:10.1080/17442508808833490 · Cited by 23
Research works printed but not linked: 3
- Christensen, Gary Edward, 1994, Deformable shape models for anatomy
- 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.
- Hijab, Omar Bakri, 1980, MINIMUM ENERGY ESTIMATION
Cited by
Sorted by citations
Avants, B., Epstein, C. L., Grossman, M., & Gee, J. C. (2007) · Medical Image Analysis · doi:10.1016/j.media.2007.06.004 · Cited by 5,924
Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
Beg, M. F., Miller, M. I., Trouvé, A., & Laurent Younes, B. (2004) · International Journal of Computer Vision · doi:10.1023/b:visi.0000043755.93987.aa · Cited by 1,764
Self-citationDeformable medical image registration: a survey
Sotiras, A., Davatzikos, C., & Paragios, N. (2013) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2013.2265603 · Cited by 1,587
“In this case, the deformation is modeled by considering its velocity over time according to the Lagrange transport equation [35,65,66].”
MethodUnbiased diffeomorphic atlas construction for computational anatomy
Joshi, S. C., Davis, B. C., Jomier, M., & Gerig, G. (2004) · NeuroImage · doi:10.1016/j.neuroimage.2004.07.068 · Cited by 881
Quicksilver: Fast predictive image registration – A deep learning approach
Yang, X., Kwitt, R., Styner, M., & Niethammer, M. (2017) · NeuroImage · doi:10.1016/j.neuroimage.2017.07.008 · Cited by 630
“However, as long as the obtained velocity fields v are finite over the unit interval, i.e., 1 0 v 2 L dt ∞ for an L of at least order 6, we will obtain a diffeomorphic transform [51].”
Background
Metadata
- Authors
- Dupuis, P., Grenander, U., & Miller, M. I.
- Affiliations
- Washington University in St. Louis (US)
- ORCID
- Miller, Michael I.
- Venue
- Quarterly of Applied Mathematics
- Publisher
- American Mathematical Society (AMS)
- Published
- 1998-09
- Volume
- 56
- Issue
- 3
- Pages
- 587-600
- ISSN
- 0033-569X (ISSN-L)
- 1552-4485
- OpenAlex
- W1507084921
- Cited by
- 447
- References
- 16
- Citations by year
- 2026: 7 · 2025: 13 · 2024: 12 · 2023: 10 · 2022: 10 · 2021: 19 · 2020: 22 · 2019: 22 · 2018: 21 · 2017: 26 · 2016: 15 · 2015: 32 · 2014: 20 · 2013: 19 · 2012: 25
- FWCI
- 5.67
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Quarterly of Applied Mathematics (publishedVersion)
- Funders
- National Science Foundation (9403820, 9424264)
- Brown University
- National Institutes of Health (R01-MH525158-01A1)
- Advanced Research Projects Agency
- Office of Naval Research (N00014)
- Army Research Office
- Topics
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Medical Image Segmentation Techniques
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Engineering › Computational Mechanics › Advanced Numerical Methods in Computational Mathematics
- Keywords
- image matching, variational problems, nonlinear transport, optimal control, image registration, 3D image data
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