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. Because 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 FISTA (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.

中文翻译：

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用于图像恢复的结构化FISTA

在本文中，我们提出了一种有效的数值方案，用于解决一些由于图像恢复而引起的大规模不适定线性逆问题。为了加速计算，利用了两种不同的隐藏结构。首先，系数矩阵近似为少量Kronecker乘积之和。该过程不仅在计算中引入了更高级别的并行性，而且还使得在随后的优化过程中可以使用计算密集型矩阵-矩阵乘法。然后，我们导出相应的Tikhonov正则化最小化模型，并扩展快速迭代收缩阈值算法（FISTA）以解决由此产生的优化问题。由于出现在Kronecker乘积近似中的矩阵都是结构化矩阵（Toeplitz，Hankel等）。），我们可以在每次迭代时进一步利用它们的快速矩阵-矢量乘法算法。因此，所提出的算法称为结构化FISTA（sFISTA）。特别是，我们证明了sFISTA引入的近似误差可以很好地控制，并且sFISTA可以达到与FISTA相同的图像恢复精度。最后，提供了理论上的复杂度分析和一些数值结果来证明sFISTA的效率。