Laboratoire de Mathématiques Pures et Appliquées
Joseph Liouville

Krylov Subspace Methods for Inverse Problems with Application to Image Restoration


Image restoration often requires the solution of large linear systems of equations with avery ill-conditioned, possibly singular, matrix and an error-contaminated right-hand side.The latter represents the available blur and noise-contaminated image, while the matrix models the blurring. Computation of a meaningful restoration of the available image re-quires the use of a regularization method. We consider the situation when the blurring matrix has a Kronecker product structure and an estimate of the norm of the desired image is available, and illustrate that efficient restoration of the available image can be achieved by Tikhonov regularization based on the global Lanczos method, and by using the connection of the latter to Gauss-type quadrature rules. We also investigate the use of the global Golub-Kahan bidiagonalization method to reduce the given large problem to a small one. The small problem is solved by employing Tikhonov regularization. A regularization parameter determines the amount of regularization. The connection between global Golub-Kahan bidiagonalization and Gauss-type quadrature rules is exploited to in-expensively compute bounds that are useful for determining the regularization parameter by the discrepancy principle. We will also present an efficient algorithm for solving theT ikhonov regularization problem of a linear system of equations with multiple right-handsides contaminated by errors. The proposed algorithm is based on the symmetric block Lanczos algorithm, in connection with block Gauss quadrature rules. We will show how this connection is designed to inexpensively determine a value of the regularization parameter when a solution norm constraint is given. Next, we will present 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 multipler ight-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. Applications include the restoration of color images are given. We will finally give efficient algorithms to solve total variation (TV) regularization of images contaminated by blur and additive noise. The unconstrained structure of the problem suggests that one can solve a constrained optimization by transforming the original unconstrained minimization problem to an equivalent constrained minimization problem. An augmented Lagrangian method is used to handle the constraints, and an alternating direction method (ADM) is used to iteratively find solutions of the subproblems. The solution of these subproblems are belonging to subspaces generated by application of successive orthogonal projections onto a class generalized matrix Krylov subspaces of increasing dimension.

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19/03 Journées Amiens-Calais