Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices

We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges

A Tensor-Based Dictionary Learning Approach to Tomographic Image Reconstruction

We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that a) we use a third-order tensor representation for our images and b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.Comment: 29 page

"Plug-and-Play" Edge-Preserving Regularization

In many inverse problems it is essential to use regularization methods that preserve edges in the reconstructions, and many reconstruction models have been developed for this task, such as the Total Variation (TV) approach. The associated algorithms are complex and require a good knowledge of large-scale optimization algorithms, and they involve certain tolerances that the user must choose. We present a simpler approach that relies only on standard computational building blocks in matrix computations, such as orthogonal transformations, preconditioned iterative solvers, Kronecker products, and the discrete cosine transform -- hence the term "plug-and-play." We do not attempt to improve on TV reconstructions, but rather provide an easy-to-use approach to computing reconstructions with similar properties.Comment: 14 pages, 7 figures, 3 table

CAUCHY-LIKE PRECONDITIONERS FOR 2-DIMENSIONAL ILL-POSED PROBLEMS

Ill-conditioned matrices with block Toeplitz, Toeplitz block (BTTB) structure arise from the discretization of certain ill-posed problems in signal and image processing. We use a preconditioned conjugate gradient algorithm to compute a regularized solution to this linear system given noisy data. Our preconditioner is a Cauchy-like block diagonal approximation to an orthogonal transformation of the BTTB matrix. We show the preconditioner has desirable properties when the kernel of the ill-posed problem is smooth: the largest singular values of the preconditioned matrix are clustered around one, the smallest singular values remain small, and the subspaces corresponding to the largest and smallest singular values, respectively, remain unmixed. For a system involving $np$ variables, the preconditioned algorithm costs only $O(np (\lg n + \lg p))$ operations per iteration. We demonstrate the effectiveness of the preconditioner on three examples

Symmetric Cauchy-like Preconditioners for the Regularized Solution of 1-D Ill-Posed Problems

The discretization of integral equations can lead to systems involving symmetric Toeplitz matrices. We describe a preconditioning technique for the regularized solution of the related discrete ill-posed problem. We use discrete sine transforms to transform the system to one involving a Cauchy-like matrix. Based on the approach of Kilmer and O'Leary, the preconditioner is a symmetric, rank $m^{*}$ approximation to the Cauchy-like matrix augmented by the identity. We shall show that if the kernel of the integral equation is smooth then the preconditioned matrix has two desirable properties; namely, the largest $m^{*}$ magnitude eigenvalues are clustered around and bounded below by one, and that small magnitude eigenvalues remain small. We also show that the initialization cost is less than the initialization cost for the preconditioner introduced by Kilmer and O'Leary. Further, we describe a method for applying the preconditioner in $O((n+1) \lg (n+1))$ operations when $n+1$ is a power of 2, and describe a variant of the MINRES algorithm to solve the symmetrically preconditioned problem. The preconditioned method is tested on two examples