Stability of Data-Dependent Ridge-Regularization for Inverse Problems

TL;DR

This work addresses robust reconstruction in inverse problems by introducing pixel-based ridge regularizers with data-dependent, spatially varying strength. By combining set-valued stability analysis, implicit-function-type results, and a Bayesian MAP interpretation, the authors establish existence, stability, and convergence guarantees for the data-to-reconstruction map under both convex and data-dependent regimes. They propose a practical training framework that learns a mask-based data-dependent regularizer from a modest amount of data and demonstrate its effectiveness across denoising, MRI, CT, and microstructure superresolution, particularly when instance-specific data are scarce. The approach offers structure-preserving reconstructions with competitive performance and enables potential uncertainty quantification, highlighting a path toward reliable data-driven regularization in medical imaging and materials science.

Abstract

Theoretical guarantees for the robust solution of inverse problems have important implications for applications. To achieve both guarantees and high reconstruction quality, we propose learning a pixel-based ridge regularizer with a data-dependent and spatially varying regularization strength. For this architecture, we establish the existence of solutions to the associated variational problem and the stability of its solution operator. Further, we prove that the reconstruction forms a maximum-a-posteriori approach. Simulations for biomedical imaging and material sciences demonstrate that the approach yields high-quality reconstructions even if only a small instance-specific training set is available.

Figures (12)

• Figure 1: Visualization of our scheme. First, we use a ridge regularizer $\mathcal{R}$ to compute an initial reconstruction ${\mathbf{x}}_{\mathrm{est}}$ from the data ${\mathbf{y}}$ based on \ref{['eq:var_prob']}. Then, we generate the masks $\bm \Lambda_c ({\mathbf{y}})$ based on ${\mathbf{x}}_{\mathrm{est}}$, and use ${\mathbf{x}}_{\mathrm{est}}$ as initialization for the second minimization.
• Figure 2: Pixel-wise cost $\mathcal{R}({\mathbf{x}}) = \sum_c \boldsymbol \psi_c ({\mathbf{W}}_c {\mathbf{x}})$ for ${\mathbf{x}} \in \{ {\mathbf{x}}_{\mathrm{gt}}, {\mathbf{y}}, {\mathbf{x}}_{\mathrm{est}} \}$ and data-dependent cost $\mathcal{R}_{{\mathbf{y}}}({\mathbf{y}}) = \sum_c \langle \boldsymbol \Lambda_c ({\mathbf{y}} ) , \boldsymbol \psi_c ({\mathbf{W}}_c {\mathbf{y}}) \rangle$. Here, black corresponds to higher values. For the last image $\boldsymbol \Lambda ({\mathbf{y}} ) = \sum_c \boldsymbol \Lambda_c ({\mathbf{y}} )$ black corresponds to smaller values.
• Figure 4: 4-fold single-coil MRI on PD data set. The white box marks the zoomed area. Top: full image; middle: zoomed-in part; bottom: error.
• Figure 6: Average masks $\boldsymbol \Lambda ({\mathbf{y}} ) = \sum_c \boldsymbol \Lambda_c ({\mathbf{y}} )$ for CRR and WCRR in the full view (left) and limited-angle CT (right) setting. Black corresponds to smaller values.
• Figure 9: Superresolution of material microstructures. The white box marks the zoomed area. Top: full image; bottom: zoomed-in part.

Theorems & Definitions (29)

• Theorem 1: Implicit function theorem BolPauSil2024
• Remark 1
• Theorem 2: Implicit functions Ioffe2017
• Lemma 1
• proof
• Theorem 3: Existence
• proof
• Remark 2
• Theorem 4
• Remark 3: Convex $\mathcal{R}$