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Variational problems on flows of diffeomorphisms for image matching
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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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Fuentes, D., Contreras, J. A., Yu, J., He, R., Castillo, E., Castillo, R., & Guerrero, T. M. (2014) · International Journal of Computer Assisted Radiology and Surgery · doi:10.1007/s11548-014-1128-3 · Cited by 17
3D Orientation-Preserving Variational Models for Accurate Image Registration
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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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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