发表机构
Norwegian University of Life Sciences（挪威生命科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究正向算子零空间大时逆问题的加权全变差正则化，通过格林函数进行灵敏度分析得出空间变化权重，介绍并分析两个模型，证明相关条件和结果，数值实验验证其优势。
AI 中文摘要
我们研究正向算子具有大零空间的逆问题的加权全变差（TV）正则化，在此设置中标准（未加权）TV会产生如空间偏差等系统重建伪影。为解决此问题，我们考虑通过相关格林函数从正向算子表达式的灵敏度分析得出的空间变化权重。这些权重重新平衡TV惩罚，从而补偿正向算子的非均匀灵敏度。我们引入并分析两个相关模型：一个各向同性加权TV泛函，其中权重反映最坏情况方向灵敏度，以及一个方向加权细化，其中加权取决于解的实际跳跃方向。除了标准的蒂霍诺夫公式，我们还研究相应的基追踪问题。我们推导最优性条件并在合适的结构假设下证明精确恢复结果。此外，我们研究可恢复性如何取决于解的几何形状及其到观测边界的距离。逆源问题和相关应用的数值实验表明，所提出的加权公式显著减少了未加权TV重建中存在的伪影，提高了定位和大小恢复。这些结果突出了空间加权在具有大零空间的逆问题的TV正则化中的关键作用。
英文摘要
We study weighted total variation (TV) regularization for inverse problems in which the forward operator has a large null space, a setting in which standard (unweighted) TV is known to produce systematic reconstruction artifacts such as spatial bias. To address this issue, we consider spatially varying weights derived from a sensitivity analysis of the forward operator expression via an associated Green's function. These weights rebalance the TV penalty and thereby compensate for the inhomogeneous sensitivity of the forward operator. We introduce and analyze two related models: an isotropic weighted TV functional, where the weight reflects worst-case directional sensitivity, and a directionally weighted refinement in which the weighting depends on the actual jump direction of the solution. In addition to the standard Tikhonov formulation, we also study corresponding basis-pursuit problems. We derive optimality conditions and prove exact recovery results under suitable structural assumptions. Furthermore, we investigate how recoverability depends on the geometry of the solution and its distance to the observation boundary. Numerical experiments for inverse source problems and related applications demonstrate that the proposed weighted formulations significantly reduce artifacts present in unweighted TV reconstructions, yielding improved localization and size recovery. These results highlight the crucial role of spatial weighting in TV regularization for inverse problems with large null spaces.
