Polynomial computations for blind image deconvolution

Winkler, Joab R.

doi:10.1016/j.laa.2015.10.010

Cited by 5 publications

(8 citation statements)

References 34 publications

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“…A separable Gaussian PSF is used in [20,21], and comparison of the Sylvester and Bézout matrices for BID therefore requires that a PSF that satisfies this property be used. This property is not realistic for practical problems, but it is the simplest form of a PSF because a deblurred image can be calculated from one blurred image [30,31], and the computations are simplified because the Fourier transform is not required. By contrast, two blurred images are required for the computation of a non-separable PSF, and the Fourier transform is used to reduce the two-dimensional BID problem to two sets of GCD computations on univariate polynomials.…”

Section: Resultant Matrices and Image Deblurringmentioning

confidence: 99%

“…The function deconvblind.m is different because only an estimate of the PSF, rather than the exact PSF, need be specified as an input argument, but the computed PSF is very dependent on the estimate of its size, and less by the entries of the matrix that defines it. A comparison of these four functions and the Sylvester resultant matrix for the computation of a deblurred image is in [30,31], and it is noted in [15] that the best deblurred image obtained from the four functions requires visual inspection of several deblurred images that differ in the values of the input arguments, for example, the number of iterations and the noise power.…”

Section: Introductionmentioning

confidence: 99%

“…It therefore follows that if the pixel values of F, G and H are the coefficients of their polynomial forms, f (x, y), g(x, y) and h(x, y), respectively, then g(x, y) = f (x, y)h(x, y). The polynomial computations are implemented by resultant matrices, of which there are several types, but only the Bézout matrix [12,20,21] and the Sylvester matrix [11,22,24,30,31] have been used for image deblurring. Section 2 contains a comparison of these matrices for greatest common divisor (GCD) computations, which are required for the determination of the size of the PSF and the solution of the problem of BID.…”

Section: Introductionmentioning

confidence: 99%

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The Sylvester and Bézout Resultant Matrices for Blind Image Deconvolution

Winkler

Halawani

2018

J Math Imaging Vis

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Blind image deconvolution (BID) is one of the most important problems in image processing, and it requires the determination of an exact image F from a degraded form of it G when little or no information about F and the point spread function (PSF) H is known. Several methods have been developed for the solution of this problem, and one class of methods considers F, G and 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 BID, and it is shown that it reduces to a comparison of their effectiveness for greatest common divisor (GCD) computations. This is a difficult problem because the determination of the degree of the GCD of two polynomials requires the calculation of the rank of a matrix, and this rank determines the size of the 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 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.

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Section: Resultant Matrices and Image Deblurringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Sylvester and Bézout Resultant Matrices for Blind Image Deconvolution

Winkler

Halawani

2018

J Math Imaging Vis

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“…Consider Stage 1 of the calculation of an AGCD ofp(w, θ 0 ) and α 0q (w, θ 0 ), which is, as noted in Section 5, the determination of its degree t. Two methods for the calculation of t that use the singular value decomposition (SVD) of the Sylvester matrix ofp(w, θ 0 ) and α 0q (w, θ 0 ), and its subresultant matrices, are described in [28], and it is shown in more recent work [25] that t can also be computed from the QR decomposition of these matrices. Since the subresultant matrix S k+1 (p, α 0q ) is formed by the deletion of two columns and one row from the subresultant matrix S k (p, α 0q ), it follows that the update formula of the QR decomposition allows efficient computation, and it is therefore preferred to the SVD, whose update formula is complicated, for the calculation of t.…”

Section: The Sylvester Resultant Matrixmentioning

confidence: 99%

The Sylvester Resultant Matrix and Image Deblurring

Winkler

2015

Curves and Surfaces

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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.

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“…The second method is based on considering the change in the error between two estimates of the GCD of the polynomials as a function of its degree. The properties and the application of the Sylvester matrix to GCD computations is also studied in [27].…”

Section: Implementation Of the Gcd Computation Through Matrix Factorimentioning

confidence: 99%

Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials

2017

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This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coe cients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical e ciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of the GCD of sets of several polynomials with real coe cients is provided. Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.

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Polynomial computations for blind image deconvolution

Cited by 5 publications

References 34 publications

The Sylvester and Bézout Resultant Matrices for Blind Image Deconvolution

The Sylvester and Bézout Resultant Matrices for Blind Image Deconvolution

The Sylvester Resultant Matrix and Image Deblurring

Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials

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