Gazzola Silvia, Scott Sebastian James, Spence Alastair
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
J Imaging. 2021 Oct 18;7(10):216. doi: 10.3390/jimaging7100216.
Many successful variational regularization methods employed to solve linear inverse problems in imaging applications (such as image deblurring, image inpainting, and computed tomography) aim at enhancing edges in the solution, and often involve non-smooth regularization terms (e.g., total variation). Such regularization methods can be treated as iteratively reweighted least squares problems (IRLS), which are usually solved by the repeated application of a Krylov projection method. This approach gives rise to an inner-outer iterative scheme where the outer iterations update the weights and the inner iterations solve a least squares problem with fixed weights. Recently, flexible or generalized Krylov solvers, which avoid inner-outer iterations by incorporating iteration-dependent weights within a single approximation subspace for the solution, have been devised to efficiently handle IRLS problems. Indeed, substantial computational savings are generally possible by avoiding the repeated application of a traditional Krylov solver. This paper aims to extend the available flexible Krylov algorithms in order to handle a variety of edge-enhancing regularization terms, with computationally convenient adaptive regularization parameter choice. In order to tackle both square and rectangular linear systems, flexible Krylov methods based on the so-called flexible Golub-Kahan decomposition are considered. Some theoretical results are presented (including a convergence proof) and numerical comparisons with other edge-enhancing solvers show that the new methods compute solutions of similar or better quality, with increased speedup.
许多用于解决成像应用（如图像去模糊、图像修复和计算机断层扫描）中线性逆问题的成功变分正则化方法旨在增强解中的边缘，并且通常涉及非光滑正则化项（例如总变差）。此类正则化方法可被视为迭代加权最小二乘问题（IRLS），通常通过反复应用克里洛夫子空间投影方法来求解。这种方法产生了一种内外迭代方案，其中外层迭代更新权重，内层迭代求解具有固定权重的最小二乘问题。最近，为了有效处理IRLS问题，已经设计出了灵活的或广义的克里洛夫子空间求解器，它们通过在解的单个近似子空间中纳入依赖于迭代的权重来避免内外迭代。实际上，通过避免传统克里洛夫子空间求解器的反复应用，通常可以大幅节省计算量。本文旨在扩展现有的灵活克里洛夫子空间算法，以便处理各种边缘增强正则化项，并在计算上方便地选择自适应正则化参数。为了处理方阵和长方阵线性系统，考虑了基于所谓灵活的Golub-Kahan分解的灵活克里洛夫子空间方法。给出了一些理论结果（包括收敛性证明），并且与其他边缘增强求解器的数值比较表明，新方法能够计算出质量相似或更好的解，且加速比有所提高。
