Greedy techniques for inverse problems

TL;DR

The paper introduces a greedy sampling framework for inverse problems that selects optimal indirect measurements in the operator codomain. It combines kernel-based interpolation with a regularized inversion in a two-step scheme and proposes residual-based and error-based greedy strategies, the latter accompanied by explicit error bounds. Focusing on kernel/interpolation theory, the authors derive stability and error propagation results, particularly using Lebesgue constants and power-function indicators. They validate the approach on solar hard X-ray imaging (STIX) data, showing that high-quality reconstructions can be achieved with a small fraction of measurements. The work offers a principled method for measurement design in ill-posed inverse problems with practical implications for imaging applications.

Abstract

Inverse imaging problems rely on limited and indirect measurements, making reconstruction highly dependent on both regularization and sample locations. We introduce a novel greedy framework for the optimal selection of indirect measurements in the operator codomain, specifically tailored to inverse problems. Our approach employs a two-step scheme combining kernel-based interpolation and extrapolation. Within this framework, greedy schemes can be residual-based, where points are selected according to the current approximation error for a specific target function, or error-based, where points are chosen using a priori error indicators independent of the residual. For the latter, we derive explicit error bounds that quantify the propagation of approximation errors through both interpolation and extrapolation. Numerical applications to solar hard X-ray imaging demonstrate that the proposed greedy sampling strategy achieves high-quality reconstructions using only a few available measurements.

Figures (7)

• Figure 1: The scheme: combination of kernel interpolation and regularization. In the first image, the most important samples are selected (first image, red circles). This gives an interpolant, which is then evaluated at more points (second image, black stars). A regularization is then applied on such data to reconstruct the desired image (third image).
• Figure 2: The full set of Fibonacci nodes used in our experiments is shown in blue in both the left and right panels, whereas the subset of frequencies selected by the error-based greedy strategy is highlighted in red in the right panel.
• Figure 3: Ground truth images of the simulated flare morphologies: single, double, and loop (from left to right, respectively).
• Figure 4: Sampling points selected by the residual-based greedy strategy for the single, double, and loop configuration (top, middle, and bottom row, respectively). From left to right, the panels correspond to the sampling subsets chosen when using the uv_smooth, MEM_GE, and Clean.
• Figure 5: Reconstructed images in the case of the single configuration. Columns correspond to the three reconstruction methods: uv_smooth, MEM_GE, and Clean (from left to right, respectively). Rows show the reconstruction obtained using (top) all sampling points, (middle) the subset selected by the error-based greedy method, and (bottom) the subset selected by the residual-based greedy method.

Theorems & Definitions (6)

• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Remark 3.4