Abstract
This paper discusses the solution of large-scale linear discrete ill-posed problems arising from image restoration problems. Since the scale of the problem is usually very large, the computations with the blurring matrix can be very expensive. In this regard, we consider problems in which the coefficient matrix is the sum of Kronecker products of matrices to benefit the computation. Here, we present an alternative approach based on reordering of the image approximations obtained with the global Arnoldi–Tikhonov method. The ordering of the intensities is such that it makes the image approximation monotonic and thus minimizes the finite differences norm. We present theoretical properties of the method and numerical experiments on image restoration.
Similar content being viewed by others
References
Aghazadeh, N., Akbarifard, F., Cigaroudy, L.S.: A restoration-segmentation algorithm based on flexible Arnoldi–Tikhonov method and Curvelet denoising. In: SIVP, pp. 10–935 (2016)
Ponti Jr., M.P., Mascarenhas, N.D.A., Ferreira, P.J.S.G., et al.: Three-dimensional noisy image restoration using filtered extrapolation and deconvolution. SIViP 7(1), 1–10 (2013)
Welk, M., Raudaschl, P., Schwarzbauer, T., et al.: Fast and Robust linear motion deblurring. SIViP 9(5), 1221–1234 (2015)
Moallem, P., Masoumzadeh, M., Habibi, M.: A novel adaptive Gaussian restoration filter for reducing periodic noises in digital image. SIVP 9, 1179 (2015)
Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Surv. Math. Ind. 3, 253–315 (1993)
Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21, 215–222 (1979)
Wohlberg, B., Rodriguez, P.: An iteratively reweighted norm algorithm for minimization of total variation functionals. IEEE Signal Process. Lett. 14, 948951 (2007)
Gazzola, S., Nagy, J.G.: Generalized Arnoldi–Tikhonov method for sparse reconstruction. SIAM J. Sci. Comput. 36(2), B225–B247 (2014)
Akbarifard, F., Aghazadeh, N.: Tikhonov regularization method based on global-GMRES and global-FGMRES methods for image restoration (submitted) (2014)
Kamm, J., Nagy, J.G.: Kronecker product and SVD approximations in image restoration. Linear Algebr. Appl. 284, 177–192 (1998)
Kamm, J., Nagy, J.G.: Kronecker product approximations for restoration image with reflexive boundary conditions. SIAM J. Matrix Anal. Appl. 25(3), 829–841 (2004)
Adluru, G., DiBella, E.V.R.: Reordering for improved constrained reconstruction from undersampled k-space data. Int. J. Biomed. Imaging 1–9 (2008)
Novati, P., Russo, M.R.: Adaptive Arnoldi–Tikhonov regularization for image restoration. Numer. Algoritm. 65, 745–757 (2014)
Messaoudi, A., Jbilou, K., Sado, H.: Global FOM and GMRES algorithms for matrix equations. Appl. Numer. Math. 31, 49–63 (1999)
Novati, P., Russo, M.R.: A gcv based Arnoldi–Tikhonov regularization method. BIT 54, 501521 (2014)
Hansen, P.C.: Regularization tools version 4.0 for matlab 7.3. Numer. Algoritm. 46, 189–194 (2007)
Acknowledgements
Here, we would like to acknowledge constructive comments and fruitful discussions that made by Dr. Farideh Akbarifard during this project.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Siadat, M., Aghazadeh, N. & Öktem, O. Reordering for improving global Arnoldi–Tikhonov method in image restoration problems. SIViP 12, 497–504 (2018). https://doi.org/10.1007/s11760-017-1185-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11760-017-1185-5