The Sylvester Resultant Matrix and Image Deblurring

Abstract

Abstract. This paper describes the application of the Sylvester resultant matrix to image deblurring. In particular, an image is represented as a bivariate polynomial and it is shown that operations on polynomials, specifically greatest common divisor (GCD) computations and polynomial divisions, enable the point spread function to be calculated and an image to be deblurred. The GCD computations are performed using the Sylvester resultant matrix, which is a structured matrix, and thus a structure-preserving matrix method is used to obtain a deblurred image. Examples of blurred and deblurred images are presented, and the results are compared with the deblurred images obtained from other methods.

Polynomial computations for blind image deconvolution

“…Polynomial computations can be used for image deblurring because the convolution operation defines the multiplication of two polynomials and the formation of a blurred image by a spatially invariant PSF [12][13][14]21,22,25,32]. The formation of a blurred image by a spatially invariant PSF in the presence of additive noise is defined in (1), and these polynomial computations require that F, G, H and N be represented as polynomials.…”

“…It is shown in [32] that the coefficients h c (k) and h r (l) can be obtained by computing an AGCD of two arbitrary columns, and an AGCD of two arbitrary rows, respectively, of G, where the pixel values of each row and each column are the coefficients of a polynomial. The deblurred image is then obtained by deconvolving H from G [32].…”

“…It is shown in [32] that the coefficients h c (k) and h r (l) can be obtained by computing an AGCD of two arbitrary columns, and an AGCD of two arbitrary rows, respectively, of G, where the pixel values of each row and each column are the coefficients of a polynomial. The deblurred image is then obtained by deconvolving H from G [32]. Many methods for the computation of an AGCD of two polynomials have been developed, including methods based on the QR decomposition of the Sylvester matrix [11,37], methods based on the singular value decomposition (SVD) of the Sylvester matrix [10,15], optimisation methods [9,38] and methods that exploit the structure of the Sylvester matrix [3,4,33,34].…”

“…It is shown in [32][33][34] that the computation of the coefficients of an AGCD, as described above, requires the solution a non-linear constrained minimisation problem. The linearisation of this problem leads to a least squares minimisation with an equality constraint, the LSE problem, at each iteration,…”

“…It is assumed in this paper that the PSF is separable, and it is shown in [22,25,32] that the extension of the work described in this paper from a separable PSF to a non-separable PSF requires more computations of the same type as are required for a separable PSF, and extra computations that are not required when a separable PSF is used.…”