Solution of some image processing problems using group representations
Prof. Harish Parthasarathy
S.C. Bose Institute of Technology, New Delhi.
This lecture will introduce three important problems in statistical image analysis that can be solved using the Peter-Weyl theory of Fourier Series on a Compact Group. (1) Construction of the Wiener Filter for image deblurring when the images are random fields on the sphere obtained by focussing a spherical camera in all directions. The random fields are assumed to have rotation-invariant autocorrelation functions. The solution of the convolutional Wiener-Hopf integral equation on SO(3) is achieved by using the Peter-Weyl theory. (2) Construction of the matched filter for image detection when the images are isotropic random fields on S^2 (the unit sphere). The optimum filter is obtained by maximizing a ratio of quadratic forms for the filter function. The quadratic forms for continuous functions are transformed into quadratic forms for discrete sequences by the aid of the Peter-Weyl theorem and maximized using the Cauchy -Schwartz inequality. (3) Pattern classification using invariants of SO(3) x R_+ x T group (rotation- scale-translation group). The group acts on patterns and each pattern orbit is classified by an invariant constructed using projections on the irreducible subspaces arising in the decomposition of the group action. This enables one to identify the prototype pattern from the transformed one.