Group Actions, Homeomorphisms, and Matching: A General Framework
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
This paper constructs metrics on the space of images I defined as orbits under group actions G. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouvé (1999, Quaterly of Applied Math.) and Dupuis et al. (1998). Quaterly of Applied Math. Left-invariant metrics are defined on the product G × I thus allowing the generation of transformations of the background geometry as well as the image values. Examples of the application of such metrics are presented for rigid object matching with and without signature variation, curves and volume matching, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another.
- Bajcsy, R., Kovacic, S. (1989) Multiresolution elastic matching. Computer Vision, Graphics, and Image Processing 46: pp. 1-21
- Bajcsy, R. L., ieberson, R., Reivich, M. (1983) A computerized system for the elastic matching of deformed radiographic images to idealized atlas images. Journal of Computer Assisted Tomography 7: pp. 618-625
- Bakircioglu, M., Grenander, U., Khaneja, N., Miller, M. (1998) Curve matching on brain surfaces using frenet distances. Human Brain Mapping 6: pp. 329-332
- Do Carmo, M.P. 1992. Riemannian Geometry. Birkaüser.
- Christensen, G.E. 1999. Consistent linear-elastic transformations for image matching. In XVIth International Conference on Information Processing in Medical Imaging,A. Kuba and M. Samal
- Dann, R., Hoford, J., Kovacic, S., Reivich, M., Bajcsy, R. (1989) Evaluation of elastic matching systems for anatomic (CT, MR) and functional (PET) cerebral images. Journal of Computer Assisted Tomography 13: pp. 603-611
- Davatzikos, C. (1997) Spatial transformation and registration of brain images using elastically deformable models. Comp. Vision and Image Understanding 66: pp. 207-222
- Dengler, J., Schmidt, M. (1988) The dynamic pyramid-a model for motion analysis with controlled continuity. International Journal of Pattern Recognition and Artificial Intelligence 2: pp. 275-286
- Dupuis, P., Grenander, U., Miller, M. (1998) A Variational Formulation of a Problem in Image Matching. Quarterly of Applied Math 56: pp. 587-600
- Gee, J., Briquer, L.L., Haynor, D.R., Bajcsy, R. (1994) Matching structural images of the human brain using statistical and geometrical image features. Visualization in Biomedical Computing 2359: pp. 191-204
- Grenander U. 1993. General Pattern Theory. Oxford Science Publications.
- Grenander, U., Keenan, D.M. (1991) On the shape of plane images. Siam J. Appl. Math 53: pp. 1072-1094
- Grenander, U., Miller, M.I. (1994) Representations and knowledge in complex systems. J. Roy. Stat. Soc 56: pp. 549-603
- Grenander, U., Miller, M., and Srivastava, A. 1998. Hilbert-Schmidt, Lower Bounds for Estimators on Matri Lie Groups for ATR, IEEE PAR I20(8).
- Hagedoorn, M., Veltkamp, C. (1999) R. Reliable and efficient pattern matching using an affine invariant metric. International Journal of Computer Vision 31: pp. 203-225
- Joshi, S. 1997. Large deformation diffeomorphisms and Gaussian random fields for statistical characterization of brain sub manifolds. PhD Thesis, Dept. of Electrical Engineering, Sever Institute of Technology, Washington Univ., St. Louis, MO, Aug.
- Miller, M.I., Christensen, G.E., Amit, Y., and Grenander, U. 1993. Mathematical textbook of deformable neuroanatomies. Proceedings of the National Academy of Science, 90(24).
- Pennec, X., Ayache, N. (1998) Uniform distribution, distance and expectation problems for geometric features processing. Journal of Mathematical Imaging and Vision 9: pp. 49-67
- Rabbitt, R.D., Weiss, J.A., Christensen, G.E., and Miller, M.I. 1995. Mapping of hyperelastic deformable templates using the finite element method. Presented at the International Symposium on Optical Science, Engineering and Instrumentation, July.
- Trouvé, A. 1999. Infinite dimensional group action and pattern recognition Unpublished preprint (Ecole Normale Superioure).
- Younes, L. (1998) Computable elastic distances between shapes. SIAM J. Appl. Math 58: pp. 565-586
- Younes, L. 1999. Optimal matching between shapes via elastic deformations. Image and Vision Computing Journal, to appear.
- Group Actions, Homeomorphisms, and Matching: A General Framework
International Journal of Computer Vision
Volume 41, Issue 1-2 , pp 61-84
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- image warping
- Riemannian metrics
- groups of deformations
- Industry Sectors
- Author Affiliations
- 1. Center for Imaging Science, Whiting School of Engeneering, John Hopkins University, 3400 N. Charles Street, Baltimore, MD, 21218-2686, USA
- 2. CMLA (CNRS, URA 1611), Ecole Normale Superieure de Cachan, 61, Avenue du Président Wilson, F-94 235, Cachan Cedex, France