The Sylvester and B\xe9zout Resultant Matrices for Blind Image Deconvolution

Abstract

Blind image deconvolution (text {BID}) is one of the most important problems in image processing, and it requires the determination of an exact image mathcal {F} from a degraded form of it mathcal {G} when little or no information about mathcal {F} and the point spread function (text {PSF}) mathcal {H} is known. Several methods have been developed for the solution of this problem, and one class of methods considers mathcal {F}, mathcal {G} and mathcal {H} to be bivariate polynomials in which the polynomial computations are implemented by the Sylvester or Bézout resultant matrices. This paper compares these matrices for the solution of the problem of text {BID}, and it is shown that it reduces to a comparison of their effectiveness for greatest common divisor (text {GCD}) computations. This is a difficult problem because the determination of the degree of the text {GCD} of two polynomials requires the calculation of the rank of a matrix, and this rank determines the size of the text {PSF}. It is shown that although the Bézout matrix is symmetric (unlike the Sylvester matrix) and it is smaller than the Sylvester matrix, which has computational advantages, it yields consistently worse results than the Sylvester matrix for the size and coefficients of the text {PSF}. Computational examples of blurred and deblurred images obtained with the Sylvester and Bézout matrices are shown, and the superior results obtained with the Sylvester matrix are evident.