An improved non-linear method for the computation of a structured low rank approximation of the Sylvester resultant matrix

Winkler, Joab R.; Hasan, Madina

doi:10.1016/j.cam.2012.07.023

Cited by 14 publications

(29 citation statements)

References 21 publications

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“…. , min(m, n), in order to avoid computational problems that may arise [33][34][35]. Two of these operations introduce the parameters α 0 and θ 0 , whose initial values are computed from the solution of a linear programming problem.…”

Section: Resultsmentioning

confidence: 99%

“…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]. In this paper, the method of SNTLN is applied to the Sylvester matrix in order to compute an AGCD of two inexact (noisy) polynomials.…”

Section: The Computation Of An Agcdmentioning

confidence: 99%

“…The application of Theorem 4.1 to the inexact polynomials f (y) and g(y) requires that they be preprocessed by three operations [33][34][35], where the second and third operations introduce the parameters α 0 > 0 and θ 0 > 0 respectively, such that y = θ 0 w where w is the new independent variable, that is, an AGCD of the polynomialsf (w) = f (θ 0 w) and α 0ḡ (w) = α 0 g(θ 0 w) is computed. The values α 0 and θ 0 are refined in the iterative procedure for the calculation of an AGCD off (w) and α 0ḡ (w), and if θ * is the value of θ 0 at the termination of this procedure, then the inverse transformation w = y/θ * is applied to the results of the AGCD computation in order that the results be expressed in the same independent variable, y, as the specified polynomials f (y) and g(y).…”

Section: The Degree Of An Agcdmentioning

confidence: 99%

“…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,…”

Section: The Coefficients Of An Agcdmentioning

confidence: 99%

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Polynomial computations for blind image deconvolution

Winkler

2016

Linear Algebra and its Applications

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Section: Resultsmentioning

confidence: 99%

Section: The Computation Of An Agcdmentioning

confidence: 99%

Section: The Degree Of An Agcdmentioning

confidence: 99%

Section: The Coefficients Of An Agcdmentioning

confidence: 99%

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Polynomial computations for blind image deconvolution

Winkler

2016

Linear Algebra and its Applications

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“…It is shown in [26,27] that the entries of x are the coefficients of the coprime polynomialsū(w, θ 0 ) andv(w, θ 0 ), which are of degrees r − t and s − t respectively, and that they satisfȳ…”

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 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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An improved non-linear method for the computation of a structured low rank approximation of the Sylvester resultant matrix

Cited by 14 publications

References 21 publications

Polynomial computations for blind image deconvolution

Polynomial computations for blind image deconvolution

The Sylvester Resultant Matrix and Image Deblurring

The Sylvester and Bézout Resultant Matrices for Blind Image Deconvolution

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