基于矩阵分解与列采样的快速对齐去噪算法

A Fast Alignment-Based Denoising Algorithm Based on Matrix Decomposition and Column Sampling

摘要: 针对原始RASL算法在图像对齐去噪任务中需求解核范数正则化问题、每次迭代均需计算大矩阵奇异值分解(SVD)而导致计算复杂度过高、难以适用于大规模数据的不足，本文提出一种基于矩阵分解和列采样的快速对齐去噪算法。首先，将低秩矩阵分解为两个子矩阵的乘积，并使用这两个子矩阵的Frobenius范数约束替代原有的核范数约束，从而避免直接求解SVD，显著提升了计算效率和去噪性能。其次，在求解过程中引入列采样策略，进一步降低内存开销与子问题的计算复杂度。在四个典型数据集上的数值实验表明，本文算法在保持优异对齐去噪效果的同时，计算效率相比原RASL算法获得显著提升。

Abstract: To address the shortcomings of the original RASL algorithm in image alignment and denoising tasks—namely, the need to resolve kernel norm regularization issues, the requirement to compute singular value decomposition (SVD) of large matrices at every iteration (resulting in excessive computational complexity), and its difficulty in handling large-scale data—this paper proposes a fast alignment and denoising algorithm based on matrix decomposition and column sampling. First, we decompose a low-rank matrix into the product of two submatrices and replace the original kernel norm constraint with Frobenius norm constraints on these submatrices. This avoids the need to directly compute SVD, significantly improving computational efficiency and denoising performance. Second, we introduce a column sampling strategy during the solution process, further reducing memory overhead and the computational complexity of subproblems. Numerical experiments on four typical datasets demonstrate that our algorithm achieves significantly improved computational efficiency compared to the original RASL algorithm while maintaining excellent alignment and denoising performance.