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.

翻译：我们研究了使用分裂Bregman算法实现各向同性与各向异性正则化的$\ell_1$正则化图像去模糊问题的求解。针对大规模问题，我们通过为重组矩阵$\mathcal{R}(A)$构建近似截断奇异值分解，获得克罗内克积近似以替代系统矩阵$A$。为获得$\mathcal{R}(A)$的近似分解，我们提出了扩展Golub Kahan双对角化算法，该算法通过将Krylov子空间扩展至超出给定近似秩，或采用自动停止准则确定合适的近似秩。所得展开式与使用相同项数的截断及随机奇异值分解进行了对比。为进一步扩大可处理问题的规模，我们采用单精度确定近似分解，同时以标准双精度执行正则化的所有步骤。数值实验表明，将近似单精度克罗内克积展开应用于矩阵$A$，结合分裂Bregman算法实现的各向同性或各向异性正则化，能有效求解图像去模糊问题。随着问题规模增大，结果显示主要计算成本在于确定克罗内克积近似，而非正则化算法本身。此外，在估计近似奇异值分解方面，扩展Golub Kahan双对角化算法相比随机奇异值分解展现出显著竞争力。