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The Sylvester Resultant Matrix and Image Deblurring

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Curves and Surfaces (Curves and Surfaces 2014)

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

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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 (\(\text {GCD}\)) 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.

References

  1. Barnett, S.: Polynomials and Linear Control Systems. Marcel Dekker, New York (1983)

    MATH  Google Scholar 

  2. Chin, P., Corless, R.M.: Optimization strategies for the approximate GCD problem. In: Proceeding of International Symposium Symbolic and Algebraic Computation, pp. 228–235, Rostock, Germany (1998)

    Google Scholar 

  3. Corless, R.M., Watt, S.M., Zhi, L.: QR factoring to compute the GCD of univariate approximate polynomials. IEEE Trans. Signal Process. 52(12), 3394–3402 (2004)

    Article  MathSciNet  Google Scholar 

  4. Cornelio, A., Piccolomini, E., Nagy, J.: Constrained numerical optimization methods for blind deconvolution. Numer. Algor. 65, 23–42 (2014)

    Article  MATH  Google Scholar 

  5. Danelakis, A., Mitrouli, M., Triantafyllou, D.: Blind image deconvolution using a banded matrix method. Numer. Algor. 64, 43–72 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fulkerson, D., Wolfe, P.: An algorithm for scaling matrices. SIAM Rev. 4, 142–146 (1962)

    Article  MathSciNet  Google Scholar 

  7. Golub, G.H., Van Loan, C.F.: Matrix Computations. John Hopkins University Press, Baltimore (2013)

    MATH  Google Scholar 

  8. Gonzalez, R.C., Woods, R.E., Eddins, S.L.: Digital Image Processing Using Matlab. Gatesmark Publishing, Knoxville (2009)

    Google Scholar 

  9. Gunturk, B., Li, X.: Image Registration: Fundamentals and Advanves. CRC Press, Florida (2013)

    Google Scholar 

  10. Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring Images: Matrices, Spectra, and Filtering. SIAM, Philadelphia (2006)

    Book  Google Scholar 

  11. Kaltofen, E., Yang, Z., Zhi, L.: Structured low rank approximation of a Sylvester matrix. In: Wang, D., Zhi, L. (eds.) Trends in Mathematics, pp. 69–83. Birkhäuser Verlag, Basel (2006)

    Google Scholar 

  12. Karmarkar, N.K., Lakshman, Y.N.: On approximate GCDs of univariate polynomials. J. Symb. Comput. 26, 653–666 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kundur, D., Hatzinakos, D.: Blind image deconvolution. IEEE Signal Process. Mag. 13(3), 43–64 (1996)

    Article  Google Scholar 

  14. Li, B., Yang, Z., Zhi, L.: Fast low rank approximation of a sylvester matrix by structured total least norm. J. Jpn Soc. Symb. Algebraic Comp. 11, 165–174 (2005)

    Google Scholar 

  15. Li, Z., Yang, Z., Zhi, L.: Blind image deconvolution via fast approximate GCD. In: Proceedings of International Symposium Symbolic and Algebraic Computation, pp. 155–162 (2010)

    Google Scholar 

  16. Liang, B., Pillai, S.: Blind image deconvolution using a robust 2-D GCD approach. In: IEEE International Symposium Circuits and Systems, pp. 1185–1188 (1997)

    Google Scholar 

  17. Nagy, J., Palmer, K., Perrone, L.: Iterative methods for image deblurring: a Matlab object-oriented approach. Numer. Algor. 36, 73–93 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nash, S.G., Sofer, A.: Linear and Nonlinear Programming. McGraw-Hill, New York (1996)

    Google Scholar 

  19. Pillai, S., Liang, B.: Blind image deconvolution using a robust GCD approach. IEEE Trans. Image Process. 8(2), 295–301 (1999)

    Article  Google Scholar 

  20. Premaratne, P., Ko, C.: Retrieval of symmetrical image blur using zero sheets. IEE Proc. Vis. Image Signal Process. 148(1), 65–69 (2001)

    Article  Google Scholar 

  21. Rosen, J.B., Park, H., Glick, J.: Total least norm formulation and solution for structured problems. SIAM J. Mat. Anal. Appl. 17(1), 110–128 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rosen, J.B., Park, H., Glick, J.: Structured total least norm for nonlinear problems. SIAM J. Mat. Anal. Appl. 20(1), 14–30 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Satherley, B.L., Parker, C.R.: Two-dimensional image reconstruction from zero sheets. Optics Lett. 18, 2053–2055 (1993)

    Article  Google Scholar 

  24. Triantafyllou, D., Mitrouli, M.: Two resultant based methods computing the greatest common divisor of two polynomials. In: Li, Z., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2004. LNCS, vol. 3401, pp. 519–526. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  25. Winkler, J.R.: Polynomial computations for blind image deconvolution (2015) Submitted

    Google Scholar 

  26. Winkler, J.R., Hasan, M.: A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix. J. Comput. Appl. Math. 234, 3226–3242 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Winkler, J.R., Hasan, M.: An improved non-linear method for the computation of a structured low rank approximation of the Sylvester resultant matrix. J. Comput. Appl. Math. 237(1), 253–268 (2013)

    Article  MathSciNet  Google Scholar 

  28. Winkler, J.R., Hasan, M., Lao, X.Y.: Two methods for the calculation of the degree of an approximate greatest common divsior of two inexact polynomials. Calcolo 49, 241–267 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Zarowski, C.J., Ma, X., Fairman, F.W.: QR-factorization method for computing the greatest common divisor of polynomials with inexact coefficients. IEEE Trans. Signal Process. 48(11), 3042–3051 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zeng, Z.: The approximate GCD of inexact polynomials. Part 1: A Univariate Algorithm (2004) (Preprint)

    Google Scholar 

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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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