Reconstructing high-quality images with sharp edges requires the use of edge-preserving constraints in the regularized form of the inverse problem. The use of the $\ell_q$-norm on the gradient of the image is a common such constraint. For implementation purposes, the $\ell_q$-norm term is typically replaced with a sequence of $\ell_2$-norm weighted gradient terms with the weights determined from the current solution estimate. While (hybrid) Krylov subspace methods can be employed on this sequence, it would require generating a new Krylov subspace for every new two-norm regularized problem. The majorization-minimization Krylov subspace method (MM-GKS) addresses this disadvantage by combining norm reweighting with generalized Krylov subspaces (GKS). After projecting the problem using a small dimensional subspace - one that expands each iteration - the regularization parameter is selected. Basis expansion repeats until a sufficiently accurate solution is found. Unfortunately, for large-scale problems that require many expansion steps to converge, storage and the cost of repeated orthogonalizations presents overwhelming memory and computational requirements. In this paper we present a new method, recycled MM-GKS (RMM-GKS), that keeps the memory requirements bounded through recycling the solution subspace. Specifically, our method alternates between enlarging and compressing the GKS subspace, recycling directions that are deemed most important via one of our tailored compression routines. We further generalize the RMM-GKS approach to handle experiments where the data is either not all available simultaneously, or needs to be treated as such because of the extreme memory requirements. Numerical examples from dynamic photoacoustic tomography and streaming X-ray computerized tomography (CT) imaging are used to illustrate the effectiveness of the described methods.

翻译：为了重建具有清晰边缘的高质量图像，需要在逆问题的正则化形式中采用保边约束。一种常见的约束是对图像梯度施加$\ell_q$范数。在实际实现中，通常将$\ell_q$范数项替换为一系列加权$\ell_2$范数梯度项，其中权重由当前解的估计值确定。虽然（混合）Krylov子空间方法可应用于该序列，但每个新的二范数正则化问题都需要重新生成Krylov子空间。最小化-最大化Krylov子空间方法（MM-GKS）通过将范数重新加权与广义Krylov子空间（GKS）相结合克服了这一缺陷。该方法利用低维子空间（该子空间在每次迭代中扩展）投影问题后，选择正则化参数。基扩展过程将持续进行，直至获得足够精确的解。但对于需要大量扩展步骤才能收敛的大规模问题，存储开销和重复正交化的计算代价将带来沉重的内存与计算负担。本文提出一种新方法——循环式MM-GKS（RMM-GKS），通过循环利用解子空间将内存需求控制在有限范围内。具体而言，该方法交替进行GKS子空间的扩展与压缩，并利用我们设计的专用压缩算法保留被认为最重要的方向。我们进一步将RMM-GKS方法推广至处理数据无法同时获取、或因极端内存需求需分批处理的情况。通过动态光声层析成像和流式X射线计算机断层成像（CT）的数值实验，验证了所述方法的有效性。