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International Conference on Curves and Surfaces

Curves and Surfaces 2014: Curves and Surfaces pp 461–490Cite as

The Sylvester Resultant Matrix and Image Deblurring

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9213))

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 (\(\text {GCD}\)) computations and polynomial divisions, enable the point spread function to be calculated and an image to be deblurred. The \(\text {GCD}\) 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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Notes

  1. 1.

    This matrix will, for brevity, henceforth be called the Sylvester matrix.

  2. 2.

    The degree r of \(\hat{p}(y)\) is not related to the degree r in y of H(x, y), which is defined in (3).

  3. 3.

    The integers r and s are not related to the degrees of the polynomials \(\hat{p}(y)\) and \(\hat{q}(y)\), and polynomials derived from them, that are introduced in Sect. 5.

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Author information

Authors and Affiliations

  1. Department of Computer Science, The University of Sheffield, Regent Court, 211 Portobello, Sheffield, S1 4DP, UK

    Joab R. Winkler

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  1. Joab R. Winkler
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Correspondence to Joab R. Winkler .

Editor information

Editors and Affiliations

  1. INRIA, Sophia Antipolis, Sophia Antipolis, France

    Jean-Daniel Boissonnat

  2. École Normale Supérieure, Paris, France

    Albert Cohen

  3. Arts Et Metiers ParisTech Lab de Sciences de l'Information, Paris, France

    Olivier Gibaru

  4. INSA de Rouen, LMI, Saint-Étienne-du-Rouvray, France

    Christian Gout

  5. Department of Mathematics, University of Oslo, Oslo, Norway

    Tom Lyche

  6. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  7. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Winkler, J.R. (2015). The Sylvester Resultant Matrix and Image Deblurring. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2014. Lecture Notes in Computer Science(), vol 9213. Springer, Cham. https://doi.org/10.1007/978-3-319-22804-4_32

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  • DOI: https://doi.org/10.1007/978-3-319-22804-4_32

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22803-7

  • Online ISBN: 978-3-319-22804-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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