Solution methods for linear discrete ill-posed problems for color image restoration
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This work discusses four algorithms for the solution of linear discrete ill-posed problems with several right-hand side vectors. These algorithms can be applied, for instance, to multi-channel image restoration when the image degradation model is described by a linear system of equations with multiple right-hand sides that are contaminated by errors. Two of the algorithms are block generalizations of the standard Golub–Kahan bidiagonalization method with the block size equal to the number of channels. One algorithm uses standard Golub–Kahan bidiagonalization without restarts for all right-hand sides. These schemes are compared to standard Golub–Kahan bidiagonalization applied to each right-hand side independently. Tikhonov regularization is used to avoid severe error propagation. Numerical examples illustrate the performance of these algorithms. Applications include the restoration of color images.
KeywordsGolub–Kahan bidiagonalization Block Golub–Kahan bidiagonalization Global Golub–Kahan bidiagonalization Tikhonov regularization Ill-posed problem Multiple right-hand sides Color image restoration
Mathematics Subject Classification6510 65F22
The authors would like to thank Lars Eldén and the referee for carefully reading the manuscript and for comments that improved the presentation. The authors also would like to thank Alessandro Buccini for comments. Research by L.R. is supported in part by NSF Grants DMS-1729509 and DMS-1720259.