Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

IMAGE DEBLURRING USING GLOBAL PCG METHOD WITH KRONECKER PRODUCT PRECONDITIONER

  • KIM, KYOUM SUN (Department of Mathematics, Chungbuk National University) ;
  • YUN, JAE HEON (Department of Mathematics, College of Natural Sciences, Chungbuk National University)
  • Received : 2018.04.17
  • Accepted : 2018.08.13
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

We first show how to construct the linear operator equations corresponding to Tikhonov regularization problems for solving image deblurring problems with nearly separable point spread functions. We next propose a Kronecker product preconditioner which is suitable for the global PCG method. Lastly, we provide numerical experiments of the global PCG method with the Kronecker product preconditioner for several image deblurring problems to evaluate its effectiveness.

Keywords

References

  1. A. Bjorck, Numerical methods for least squares problems, SIAM, Philadelphia, 1996.
  2. M. Donatelli, D. Martin and L. Reichel, Arnoldi methods for image deblurring with antire extive boundary conditions, Appl. Math. Comput. 253 (2015), 135-150.
  3. R.W. Freund, G.H. Golub and N.M. Nachtigal, Iterative solutions of linear systems, Acta Numerica 1 (1991), 57-100.
  4. M. Hankey and J.G. Nagy, Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient, Inverse Problems, 12 (1996), 157-173. https://doi.org/10.1088/0266-5611/12/2/004
  5. P.C. Hansen, J.G. Nagy and D.P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM, Philadelphia, 2006.
  6. E. Kreyszig, Introductory functional analysis with applications, John Wiley & Sons. Inc., New York, 1978.
  7. J. Kamn and J.G. Nagy, Optimal kronecker product approximation of block Toeplitz matrices, SIAM J. Matrix Anal. Appl. 22(1) (2000), 155-172. https://doi.org/10.1137/S0895479898345540
  8. A.J. Laub, Matrix Analysis for Scientists and Engineers, SIAM, Philadelphia, 2005.
  9. J.G. Nagy, M.N. Ng and L. Perrone, Kronecker product approximations for image restoration with re exive boundary conditions, SIAMJ. Matrix Anal. Appl. 25(3) (2004), 829-841.
  10. J.G. Nagy, K.M. Palmer and L. Perrone, Iterative methods for image deblurring: A MATLAB object oriented approach, Numerical Algorithms 36 (2004), 73-93. https://doi.org/10.1023/B:NUMA.0000027762.08431.64
  11. C.C. Paige and M.A. Saunders, LSQR. An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), 43-71. https://doi.org/10.1145/355984.355989
  12. D.K. Salkuyeh, CG-type algorithms to solve symmetrics matrix equations, Appl. Math. Comput. 172 (2006), 985-999.
  13. C.F. Van Loan, Computational frameworks for the Fast Fourier Transform, SIAM, Philadelphia, 1992.
  14. J.H. Yun, Performance of $G\ell$-PCG method for image denoising problems, J. Appl. Math. & Informatics 35 (2017), 399-411. https://doi.org/10.14317/jami.2017.399