加权总变分正则化用于具有显著空空间的反问题
Weighted total variation regularization for inverse problems with significant null spaces
AI总结：
本文提出加权总变分正则化方法，用于解决具有显著空空间的反问题，有效恢复远离边界的源位置和大小。
摘要：
我们考虑具有大空空间的反问题，这些反问题在重要的应用中出现，例如逆ECG和EEG过程。标准正则化方法通常产生在或接近前向算子空空间的正交补集中的解。这往往导致不充分的结果，其中内部源被错误地解释为接近数据采集位置——例如，在EEG和ECG记录中接近或在身体表面。为了缓解这一问题，我们之前提出了用于Tikhonov和稀疏正则化的加权方案。在这里，我们将这种方法扩展到总变分（TV）正则化，这特别适合于识别具有近似恒定值的空间扩展区域。我们介绍了一种加权TV正则化方法，提供了支持性分析，并通过数值实验展示了其性能。与标准TV正则化不同，加权版本能够恢复远离边界的大型分段常数源的位置和大小，尽管不能精确恢复其形状。此外，我们还探讨了一种混合加权稀疏性和TV正则化的 approaches，该方法能更好地捕捉小和大源，尽管与单独的加权TV方法相比，重建结果稍显模糊。
英文摘要：
We consider inverse problems with large null spaces, which arise in important applications such as in inverse ECG and EEG procedures. Standard regularization methods typically produce solutions in or near the orthogonal complement of the forward operator's null space. This often leads to inadequate results, where internal sources are mistakenly interpreted as being near the data acquisition sites -- e.g., near or at the body surface in connection with EEG and ECG recordings. To mitigate this, we previously proposed weighting schemes for Tikhonov and sparsity regularization. Here, we extend this approach to total variation (TV) regularization, which is particularly suited for identifying spatially extended regions with approximately constant values. We introduce a weighted TV-regularization method, provide supporting analysis, and demonstrate its performance through numerical experiments. Unlike standard TV regularization, the weighted version successfully recovers the location and size of large, piecewise constant sources away from the boundary, though not their exact shape. Additionally, we explore a hybrid weighted-sparsity and TV regularization approach, which better captures both small and large sources, albeit with somewhat more blurred reconstructions than the weighted TV method alone.
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