We present a new inner-outer iterative algorithm for edge enhancement in imaging problems. At each outer iteration, we formulate a Tikhonov-regularized problem where the penalization is expressed in the 2-norm and involves a regularization operator designed to improve edge resolution as the outer iterations progress, through an adaptive process. An efficient hybrid regularization method is used to project the Tikhonov-regularized problem onto approximation subspaces of increasing dimensions (inner iterations), while conveniently choosing the regularization parameter (by applying well-known techniques, such as the discrepancy principle or the ${\mathcal L}$-curve criterion, to the projected problem). This procedure results in an automated algorithm for edge recovery that does not involve regularization parameter tuning by the user, nor repeated calls to sophisticated optimization algorithms, and is therefore particularly attractive from a computational point of view. A key to the success of the new algorithm is the design of the regularization operator through the use of an adaptive diagonal weighting matrix that effectively enforces smoothness only where needed. We demonstrate the value of our approach on applications in X-ray CT image reconstruction and in image deblurring, and show that it can be computationally much more attractive than other well-known strategies for edge preservation, while providing solutions of greater or equal quality.

中文翻译：

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一种图像恢复重建中边缘保留的内外迭代方法

我们提出了一种新的内外迭代算法，用于成像问题中的边缘增强。在每次外部迭代中，我们制定了一个 Tikhonov 正则化问题，其中惩罚以 2 范数表示，并涉及一个正则化算子，旨在通过自适应过程随着外部迭代的进展提高边缘分辨率。一种有效的混合正则化方法用于将 Tikhonov 正则化问题投影到维度增加（内部迭代）的近似子空间上，同时方便地选择正则化参数（通过应用众所周知的技术，例如差异原理或 ${\mathcal L}$-curve 准则，到预计的问题）。此过程产生了一种自动边缘恢复算法，不涉及用户调整正则化参数，也不会重复调用复杂的优化算法，因此从计算的角度来看特别有吸引力。新算法成功的关键是通过使用自适应对角加权矩阵设计正则化算子，该矩阵仅在需要时有效地强制执行平滑度。我们展示了我们的方法在 X 射线 CT 图像重建和图像去模糊中的应用价值，并表明它在计算上比其他众所周知的边缘保留策略更具吸引力，同时提供更高或同等质量的解决方案。新算法成功的关键是通过使用自适应对角加权矩阵设计正则化算子，该矩阵仅在需要时有效地强制执行平滑度。我们展示了我们的方法在 X 射线 CT 图像重建和图像去模糊中的应用价值，并表明它在计算上比其他众所周知的边缘保留策略更具吸引力，同时提供更高或同等质量的解决方案。新算法成功的关键是通过使用自适应对角加权矩阵设计正则化算子，该矩阵仅在需要时有效地强制执行平滑度。我们展示了我们的方法在 X 射线 CT 图像重建和图像去模糊中的应用价值，并表明它在计算上比其他众所周知的边缘保留策略更具吸引力，同时提供更高或同等质量的解决方案。