Summary of Computer Vision Session

Research on computer vision is an example of a successful two-way street between mathematics and computer science: interesting mathematics has been used by vision researchers to make practical tools and to develop commercially important innovations such as the JPEG standard for image storage. In turn, the work of some vision researchers in using anisotropic diffusion, for instance, led mathematicians to study the differential equation model developed by those researchers. Stanley Osher of UCLA noted another example of rich interaction, wherein a paper5 that developed axioms for morphology-based image processing led to the analysis of motion of level sets by affine mean curvature. Also, L. Rudin's PhD thesis in computer science (from the California Institute of Technology) introduced shock wave theory and BV (bounded variation) spaces to image restoration.

The mathematical foundations of computer vision are well recognized. Geometry has contributed the basis for projections and for the level set method, and topology aids in the understanding of deformable surfaces. Probability and statistics are used in estimation under uncertainty. Tools such as wavelet analysis and methods for solving diffusion problems and evolving vector fields have been contributed by analysis and partial differential equations, while an understanding of properties such as reflectance and radiosity comes from mathematical physics. Tools from harmonic analysis, for example, have been basic to the development of the JPEG standard for storing images. Algebraic geometry has been useful in work on curve recognition. Computational math is essential to developing fast algorithms, while many problems are best understood in terms of graph theory.

Despite the richness of past interactions, there is a potential for much more collaboration. Jitendra Malik noted that the connection between theoretical computer science and combinatorics has not yet been fully exploited. In order to make progress in computer vision, researchers need to jump from continuous formulations to discrete ones and back, giving an even greater opportunity for interplay.

5  

Alvarez, L., Guichard, F., Lions, P.L., and Morel, J.M., Axioms and fundamental equations of image processing, Archive for Rational Mechanics and Analysis, Vol. 16 (1993), pp. 199-257.



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