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Split Bregman Isotropic and Anisotropic Image Deblurring with Kronecker Product Sum Approximations using Single Precision Enlarged-GKB or RSVD Algorithms to provide low rank truncated SVDs

Abdulmajeed Alsubhi, Rosemary Renaut

TL;DR

The reported numerical tests demonstrate the effectiveness of applying the approximate single precision Kronecker product expansion for A, combined with either isotropic or anisotropic regularization implemented using the split Bregman algorithm, for the solution of image deblurring problems.

Abstract

We consider the solution of the $\ell_1$ regularized image deblurring problem using isotropic and anisotropic regularization implemented with the split Bregman algorithm. For large scale problems, we replace the system matrix $A$ using a Kronecker product approximation obtained via an approximate truncated singular value decomposition for the reordered matrix $\mathcal{R}(A)$. To obtain the approximate decomposition for $\mathcal{R}(A)$ we propose the enlarged Golub Kahan Bidiagonalization algorithm that proceeds by enlarging the Krylov subspace beyond either a given rank for the desired approximation, or uses an automatic stopping test that provides a suitable rank for the approximation. The resultant expansion is contrasted with the use of the truncated and the randomized singular value decompositions with the same number of terms. To further extend the scale of problem that can be considered we implement the determination of the approximation using single precision, while performing all steps for the regularization in standard double precision. The reported numerical tests demonstrate the effectiveness of applying the approximate single precision Kronecker product expansion for $A$, combined with either isotropic or anisotropic regularization implemented using the split Bregman algorithm, for the solution of image deblurring problems. As the size of the problem increases, our results demonstrate that the major costs are associated with determining the Kronecker product approximation, rather than with the cost of the regularization algorithm. Moreover, the enlarged Golub Kahan Bidiagonalization algorithm competes favorably with the randomized singular value decomposition for estimating the approximate singular value decomposition.

Split Bregman Isotropic and Anisotropic Image Deblurring with Kronecker Product Sum Approximations using Single Precision Enlarged-GKB or RSVD Algorithms to provide low rank truncated SVDs

TL;DR

The reported numerical tests demonstrate the effectiveness of applying the approximate single precision Kronecker product expansion for A, combined with either isotropic or anisotropic regularization implemented using the split Bregman algorithm, for the solution of image deblurring problems.

Abstract

We consider the solution of the regularized image deblurring problem using isotropic and anisotropic regularization implemented with the split Bregman algorithm. For large scale problems, we replace the system matrix using a Kronecker product approximation obtained via an approximate truncated singular value decomposition for the reordered matrix . To obtain the approximate decomposition for we propose the enlarged Golub Kahan Bidiagonalization algorithm that proceeds by enlarging the Krylov subspace beyond either a given rank for the desired approximation, or uses an automatic stopping test that provides a suitable rank for the approximation. The resultant expansion is contrasted with the use of the truncated and the randomized singular value decompositions with the same number of terms. To further extend the scale of problem that can be considered we implement the determination of the approximation using single precision, while performing all steps for the regularization in standard double precision. The reported numerical tests demonstrate the effectiveness of applying the approximate single precision Kronecker product expansion for , combined with either isotropic or anisotropic regularization implemented using the split Bregman algorithm, for the solution of image deblurring problems. As the size of the problem increases, our results demonstrate that the major costs are associated with determining the Kronecker product approximation, rather than with the cost of the regularization algorithm. Moreover, the enlarged Golub Kahan Bidiagonalization algorithm competes favorably with the randomized singular value decomposition for estimating the approximate singular value decomposition.
Paper Structure (28 sections, 1 theorem, 37 equations, 6 figures, 8 tables, 5 algorithms)

This paper contains 28 sections, 1 theorem, 37 equations, 6 figures, 8 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. The Kronecker Product Approximation
  5. Rearrangement R(A) of A
  6. Enlarged-GKB
  7. RSVD
  8. Isotropic and Anisotropic Regularization
  9. The Split Bregman Algorithm
  10. Anisotropic TV
  11. Isotropic TV
  12. CGLS Algorithm
  13. Computational costs
  14. Computational Cost of the KP sum approximation
  15. The EGKB Algorithm
  16. ...and 13 more sections

Key Result

Theorem 1

golub2013matrix \newlabeleq:Kronecker Product SVD0 Let $A \in \mathbb{R}^{M \times N}$, with $M=m_1m_2$ and $N=n_1 n_2$, be blocked as in eq:block A for KP, and the SVD of $\mathcal{R}(A)$ be given by then where $\widetilde{\sigma}_i$ are the singular values of $\mathcal{R}(A)$, $\widetilde{U}_i = \textbf{array}(\widetilde{\mathbf u}_i, m_1, n_1)$ and $\widetilde{V}_i =\textbf{array}(\widetilde

Figures (6)

  • Figure 1: The non separable speckle PSFs used in \ref{['ex:Blur']} and obtained from gazzola2019ir, with a zoom from the original axes of the PSFs. The sizes of \ref{['Fig:Mild Blur', 'Fig:Medium Blur']} are $7 \times 7$ pixels and $15 \times 15$ pixels, respectively.
  • Figure 2: \ref{['Fig:prodc Mild Blur']} and \ref{['Fig:prodc Medium Blur']} show the monotonic decrease of $\nu_j$, as calculated at step $8$ of \ref{['alg:EGKB']} using the geometric means of $\zeta_j$, until $j=5$ and $j=8$ for \ref{['alg:EGKB']} applied to $\mathcal{R}(\texttt{single}(A_{\text{Mild}}))$ and $\mathcal{R}(\texttt{single}(A_{\text{Medium}}))$, respectively. \ref{['Fig:TSVD EGKB RSVD of R Mild']} and \ref{['Fig:TSVD EGKB RSVD of R Medium']} show the approximations of the singular values for $\mathcal{R}(A_{\text{Mild}})$ and $\mathcal{R}(A_{\text{Medium}})$ in \ref{['ex:Blur']} using the TSVD, EGKB(SP), EGKB, RSVD(SP), and RSVD with the chosen parameters given in \ref{['Tab:Blur']}. The black vertical lines, blue crosses, blue circles, dark red horizontal lines, and dark red squares indicate the estimates of the singular values obtained using TSVD, EGKB(SP), EGKB, RSVD(SP), and RSVD, respectively. All presented results with EGKB(SP) are obtained with full reorthogonalization. \newlabelFig:Blur reults0
  • Figure 3: A true image of $100 \times 100$ pixels \ref{['Fig:True pattern2']} is blurred using a mild non separable speckle PSF of size $100 \times 100$\ref{['Fig:Mild Blur']} from gazzola2019ir and $7\%$ Gaussian noise, which has an approximate SNR$=23$ for the problem in \ref{['ex:Pattern2']}. SB solutions obtained with $A$ and the $\tilde{A}_k$ at convergence for \ref{['Fig:blurred Patteren2']} using $\lambda=0.1150$ and $\beta=0.0066$. Here the approximations to $\tilde{A}_k$, denoted with (SP), are obtained for $R(\texttt{single}(A))$.
  • Figure 4: The corresponding $\text{RC}_{SB}$ and RE in the outer iterations $\ell$ in the SB algorithm with $A$ directly and with $\tilde{A}_k$ estimated by EGKB(SP) and EGKB for the problem in \ref{['ex:Pattern2']}. The matrix $A$ is of size $10000\times 10000$. The horizontal line in each plot indicates the convergence tolerance $\tau_{SB}$.
  • Figure 5: A true image of $128 \times 128$ pixels \ref{['Fig:True satellite']} is blurred using a medium non separable speckle PSF of size $128 \times 128$\ref{['Fig:Medium Blur']} from gazzola2019ir and $4\%$ Gaussian noise, which has an approximate SNR$=28$ for the problem in \ref{['ex:Pattern2']}. SB solutions obtained with $A$ and the $\tilde{A}_k$ at convergence for \ref{['Fig:blurred satellite']} using $\lambda=0.0110$ and $\beta=0.0024$.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • Example 5.4