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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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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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