AI 中文总结
研究将图拉普拉斯算子作为正则化器用于图像恢复，提出从当前重建迭代更新拉普拉斯算子的三种方案，包括标准、误差方程及混合方案，证明其收敛性，数值实验显示在重建质量和细节恢复上有提升。
AI 中文摘要
我们研究图拉普拉斯算子作为广义蒂霍诺夫框架中线性不适定问题的正则化器。拉普拉斯算子从当前重建中迭代更新，以便将关于解的结构信息逐步更清晰地输入到正则化项中。我们引入三种方案：一种从每个新迭代重建拉普拉斯算子的标准方案；一种基于迭代蒂霍诺夫正则化的误差方程方案，从重建误差估计而非图像本身构建拉普拉斯算子；以及一种结合两者的混合方案。我们在噪声水平趋于零时，在先验参数和停止规则下建立了所有三种方案对噪声数据的收敛性。二维计算机断层扫描和图像去模糊的数值实验表明，在重建质量上有一致的提升，并且能更清晰地恢复精细细节。
英文摘要
We study the graph Laplacian operator as a regularizer in a generalized Tikhonov framework for linear ill-posed problems. The Laplacian is updated iteratively from the current reconstruction, so that progressively sharper structural information about the solution is fed into the regularization term. We introduce three schemes: a standard one that rebuilds the Laplacian from each new iterate; an error-equation scheme that, following the error-based formulation of iterated Tikhonov regularization, builds the Laplacian from an estimate of the reconstruction error rather than of the image itself; and a mixed scheme combining the two. We establish convergence of all three schemes for noisy data under a priori parameter and stopping rules as the noise level tends to zero. Numerical experiments in two-dimensional computed tomography and image deblurring show consistent gains in reconstruction quality and sharper recovery of fine details.
