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Structured FISTA for Image Restoration

Zixuan Chen, James Nagy, Yuanzhe Xi, Bo Yu

TL;DR

This work tackles large-scale ill-posed image restoration problems by exploiting a Kronecker-product approximation A ≈ A_s = ∑_{i=1}^s K_i ⊗ H_i to enable a structured variant of FISTA, termed sFISTA. By solving a matrix minimization problem that leverages the Kronecker structure and the Toeplitz/Hankel patterns of K_i and H_i, the method achieves fast matrix-matrix operations and parallelism while maintaining the same restoration accuracy as traditional FISTA. The authors provide a convergence- and complexity-focused analysis, including an error bound separating the Kronecker-approximation error from the iterative error, and demonstrate substantial empirical speedups (roughly 5×) with minimal accuracy loss across diverse blurred/noisy images. The approach offers scalable, efficient image restoration from large-scale data and opens avenues for extending to nonsmooth regularization and enhanced preconditioning in future work.

Abstract

In this paper, we propose an efficient numerical scheme for solving some large scale ill-posed linear inverse problems arising from image restoration. In order to accelerate the computation, two different hidden structures are exploited. First, the coefficient matrix is approximated as the sum of a small number of Kronecker products. This procedure not only introduces one more level of parallelism into the computation but also enables the usage of computationally intensive matrix-matrix multiplications in the subsequent optimization procedure. We then derive the corresponding Tikhonov regularized minimization model and extend the fast iterative shrinkage-thresholding algorithm (FISTA) to solve the resulting optimization problem. Since the matrices appearing in the Kronecker product approximation are all structured matrices (Toeplitz, Hankel, etc.), we can further exploit their fast matrix-vector multiplication algorithms at each iteration. The proposed algorithm is thus called structured fast iterative shrinkage-thresholding algorithm (sFISTA). In particular, we show that the approximation error introduced by sFISTA is well under control and sFISTA can reach the same image restoration accuracy level as FISTA. Finally, both the theoretical complexity analysis and some numerical results are provided to demonstrate the efficiency of sFISTA.

Structured FISTA for Image Restoration

TL;DR

This work tackles large-scale ill-posed image restoration problems by exploiting a Kronecker-product approximation A ≈ A_s = ∑_{i=1}^s K_i ⊗ H_i to enable a structured variant of FISTA, termed sFISTA. By solving a matrix minimization problem that leverages the Kronecker structure and the Toeplitz/Hankel patterns of K_i and H_i, the method achieves fast matrix-matrix operations and parallelism while maintaining the same restoration accuracy as traditional FISTA. The authors provide a convergence- and complexity-focused analysis, including an error bound separating the Kronecker-approximation error from the iterative error, and demonstrate substantial empirical speedups (roughly 5×) with minimal accuracy loss across diverse blurred/noisy images. The approach offers scalable, efficient image restoration from large-scale data and opens avenues for extending to nonsmooth regularization and enhanced preconditioning in future work.

Abstract

In this paper, we propose an efficient numerical scheme for solving some large scale ill-posed linear inverse problems arising from image restoration. In order to accelerate the computation, two different hidden structures are exploited. First, the coefficient matrix is approximated as the sum of a small number of Kronecker products. This procedure not only introduces one more level of parallelism into the computation but also enables the usage of computationally intensive matrix-matrix multiplications in the subsequent optimization procedure. We then derive the corresponding Tikhonov regularized minimization model and extend the fast iterative shrinkage-thresholding algorithm (FISTA) to solve the resulting optimization problem. Since the matrices appearing in the Kronecker product approximation are all structured matrices (Toeplitz, Hankel, etc.), we can further exploit their fast matrix-vector multiplication algorithms at each iteration. The proposed algorithm is thus called structured fast iterative shrinkage-thresholding algorithm (sFISTA). In particular, we show that the approximation error introduced by sFISTA is well under control and sFISTA can reach the same image restoration accuracy level as FISTA. Finally, both the theoretical complexity analysis and some numerical results are provided to demonstrate the efficiency of sFISTA.

Paper Structure

This paper contains 8 sections, 4 theorems, 38 equations, 5 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Kronecker Decomposition
  3. Structured FISTA
  4. FISTA: A fast iterative shrinkage-thresholding algorithm
  5. Accelerating FISTA by exploiting structures
  6. Complexity Analysis
  7. Numerical Results
  8. Conclusion

Key Result

Theorem 2.1

\newlabelKronecker approximation theoremNagy2004. Assume the blurring matrix $A$ is constructed from a PSF $P$ with center $p_{lq}$ located at $(l,q)$, then for both zero boundary condition and reflective boundary condition, we have where $\bar{P}=W_aPW_b$ with $W_a=\mathrm{diag}\left([\ \sqrt{n-l+1}\ \cdots\ \sqrt{n-1}\ \sqrt{n}\ \sqrt{n-1}\ \cdots\ \sqrt{l}\ ]^T\right)$, $W_b=\mathrm{diag}$$\le

Figures (5)

  • Figure 3.1: Singular values of $P$ and $\bar{P}$ for test image 'hst' in Example \ref{['example1']}.
  • Figure 4.1: Figures for 'hst' extracted from PRblurdefocus. \newlabelfig1
  • Figure 4.2: Figures for 'satellite' extracted from PRblurdefocus
  • Figure 4.3: Figures for 'pattern1' extracted from PRblurshake
  • Figure 4.4: Figures for 'ppower' extracted from PRblurshake

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Example 1
  • Example 2