Weighted total variation regularization for inverse problems with significant null spaces

TL;DR

This work develops a weighted total variation framework to address inverse problems with large null spaces, notably in ECG/EEG contexts, by defining weights via forward operators and Green's function derivatives. It provides 1D and 2D analyses showing that the weighted TV can accurately locate and size extended sources away from boundaries, with an extended TV variant incorporating boundary effects to mitigate depth bias. A regularized formulation yields near-true jump reconstructions under suitable conditions, and a hybrid weighted TV–sparsity model offers improved recovery for mixed small and large sources at the cost of somewhat blurred shapes. Numerical experiments corroborate the theoretical insights and highlight practical considerations for weight selection and potential extensions to learning-based weight design.

Abstract

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.

Figures (12)

• Figure 1: True sources and inverse recoveries computed with standard total variation regularization. No noise was added to the synthetic boundary data $d=K f_{\textnormal{true}}$ and $\alpha = 10^{-6}$.
• Figure 2: A domain $\Omega$, with boundary $\partial \Omega$, and the support (in gray) of two prototypical sources.
• Figure 3: A visualization of how the exact solution $\rho \bar{H}_{x^*}+\tau$ (in blue color) and the inverse solution $f_\alpha$ (in red color) presented in Theorem \ref{['thm:variational_rigorous_proof']} are related. The jump is correctly located, but slightly smaller and the center line is slightly shifted depending on the inner-product $(K\bar{H}_{x^*}, K1)$.
• Figure 4: A rectangular domain $\Omega$ and the support $R \subset \Omega$, with boundary $\Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4$, of the true source $f^*=\chi_R$. The blue dots represent some arbitrary $y \in \Gamma_1$ and $y' \in \Gamma_3$.
• Figure 5: Total variation weights $\left\|K\partial_{y_i}G(\cdot;y) \right\|_{L^p(E)}$ for $i= 1, 2$ and different values of $p$, cf. \ref{['eq:alfred2p']}.

Theorems & Definitions (24)

• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Theorem 4.3
• proof
• Theorem 4.4
• Remark 5.1
• Theorem 5.2
• proof