Iterative tensor factorization based on Krylov subspace-type methods with applications to image processing
| Autor: | UGWU, UGOCHUKWU OBINNA |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Applied Mathematics
Inverse problems Iterative tensor decomposition Krylov subspaces Image and video processing Large-scale linear discrete ill-posed problems Tikhonov regularization Truncated iterations Tensor Arnoldi process Tensor Golub-Kahan bidiagonalization Tensor Lanczos process tensor SVD Randomized tensor SVD t-product Invertible linear transform |
| Druh dokumentu: | Text |
| Popis: | This work is concerned with structure preserving and other techniques for the solution of linear discrete ill-posed problems with transform-based tensor-tensor products, e.g., the t-product and the invertible linear transform product. Specifically, we focus on two categories of solution methods, those that involve flattening, i.e., reduce the tensor equation to an equivalent equation involving a matrix and a vector, and those that preserve the tensor structure by avoiding flattening. Various techniques based on Krylov subspace-type methods for solving third order tensor ill-posed problems are discussed. The data is a laterally oriented matrix or a general third order tensor. Regularization of tensor ill-posed problem by Tikhonov's approach and truncated iterations are considered. Golub-Kahan bidiagonalization-type, Arnoldi-type, and Lanczos-type processes are applied to reduce large-scale Tikhonov minimization problems to small-sized problems. A few steps of the t-product bidiagonalization process can be employed to inexpensively compute approximations of the singular tubes of the largest Frobenius norm and the associated left and right singular matrices. A less prohibitive computation of approximations of eigentubes of the largest Frobenius norm and the corresponding eigenmatrix by a few steps of the t-product Lanczos process is considered. The interlacing of the Frobenius norm of the singular tubes is shown and applied. The discrepancy principle is used to determine the regularization parameter and the number of iterations by a chosen method. Several truncated iteration techniques, e.g., SVD-like, and those based on the above processes are considered. Solution methods for the weighted tensor Tikhonov minimization problem with weighted global and non-global bidiagonalization processes are discussed. The weights on the fidelity and regularization parts of this problem are suitably defined symmetric positive definite (SPD) tensors. The computation of the inverse of the SPD tensor that defines the weight on the regularization part is considered intractable. The use of weights in Tikhonov regularization is seen to improve the quality of the reconstructed images and is more appropriate when the noise in the data has a known covariance tensor that is different from the identity. The structure of the weights allows for their application in the spatial domain to reduce computing time. The use of randomization when solving tensor ill-posed problems can improve computational time and the quality of the computed solution. Applications to image and video restorations are discussed. Solution methods that tensorize are seen to be generally superior to the solution methods that involve flattening. |
| Databáze: | Networked Digital Library of Theses & Dissertations |
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