Weighted inhomogeneous regularization for inverse problems with indirect and incomplete measurement data

TL;DR

This work addresses inverse problems with indirect and incomplete measurements by introducing a weighted inhomogeneous regularization framework that unifies a redesigned exponent design with spatial weights. By fusing patch-wise and pixel-wise feature estimation and leveraging directional derivatives of smoothed reconstructions, the method improves the discrimination between edges, oscillations, and smooth regions, while maintaining convexity with $p_j\in[1,2]$ and solving via an ADMM-based algorithm. Numerical experiments on synthetic 2D images and a real sea ice image show that the proposed exponents reduce misclassification and that the additional weights enhance recovery of small features and mixed regions, outperforming standard inhomogeneous regularization in challenging scenarios. The approach offers a practical pathway to more accurate reconstructions in multifeatured inverse problems, with potential extensions to data fusion and variational data assimilation in geophysical applications.

Abstract

Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's characteristics, such as sparsity or smoothness. Inhomogeneous regularization, which incorporates a spatially varying exponent $p$ in the standard $\ell_p$-norm-based framework, has been used to recover signals with spatially varying features. This study introduces weighted inhomogeneous regularization, an extension of the standard approach incorporating a novel exponent design and spatially varying weights. The proposed exponent design mitigates misclassification when distinct characteristics are spatially close, while the weights address challenges in recovering regions with small-scale features that are inadequately captured by traditional $\ell_p$-norm regularization. Numerical experiments, including synthetic image reconstruction and the recovery of sea ice data from incomplete wave measurements, demonstrate the effectiveness of the proposed method.

Figures (17)

• Figure 1: True signals and exponent distributions from the standard inhomogeneous regularization
• Figure 2: Kernel A: triangular, Kernel B: Gaussian, Kernel C: shifted cosine
• Figure 3: True signal and exponent distributions from our proposed inhomogeneous regularization
• Figure 4: 2D synthetic images (A-C) and their scaled images (D-F) by a factor of 1/4 along $x$-axis
• Figure 5: Standard and proposed exponents, and VBJS and proposed weights for Images A-C in Fig \ref{['fig:true_2d_synthetic']}(a)-(c)