[Paper Review] Random mesh projectors for inverse problems

Sidharth Gupta, Konik Kothari+2 more|arXiv (Cornell University)|May 29, 2018

Seismic Imaging and Inversion Techniques65 references4 citations

TL;DR

This paper proposes a two-stage learning-based method for solving severely ill-posed inverse problems in imaging, especially when ground truth data is scarce. Instead of directly learning the inverse mapping, it trains neural networks to estimate projections of the unknown image into random piecewise-constant subspaces (Delaunay-based), then combines these projections via a deconvolution-like reconstruction, achieving superior robustness to noise and unseen corruptions compared to direct inversion.

ABSTRACT

We propose a new learning-based approach to solve ill-posed inverse problems in imaging. We address the case where ground truth training samples are rare and the problem is severely ill-posed - both because of the underlying physics and because we can only get few measurements. This setting is common in geophysical imaging and remote sensing. We show that in this case the common approach to directly learn the mapping from the measured data to the reconstruction becomes unstable. Instead, we propose to first learn an ensemble of simpler mappings from the data to projections of the unknown image into random piecewise-constant subspaces. We then combine the projections to form a final reconstruction by solving a deconvolution-like problem. We show experimentally that the proposed method is more robust to measurement noise and corruptions not seen during training than a directly learned inverse.

Motivation & Objective

• Address the challenge of solving severely ill-posed inverse problems in imaging when ground truth training data is extremely limited.
• Overcome the instability of direct deep learning inversion under low signal-to-noise ratio and measurement corruption.
• Develop a method that generalizes across diverse training datasets and remains robust to unseen corruptions.
• Enable accurate reconstruction of geometric features even when measurements are sparse or corrupted in ways not seen during training.
• Provide a stable alternative to direct inversion by decomposing the problem into simpler, low-dimensional projection learning tasks.

Proposed method

• Use random Delaunay triangulations to define piecewise-constant subspaces for projecting the unknown image into lower-dimensional, structured representations.
• Train convolutional neural networks to map noisy or corrupted measurements to projections in these random subspaces, reducing the Lipschitz constant and improving generalization.
• Reconstruct the final image by solving a deconvolution-like inverse problem that combines estimates from multiple random subspaces.
• Formulate the combined reconstruction as a linear system where, in expectation over random subspaces, the problem reduces to a deconvolution.
• Use a two-stage framework: first estimate multiple low-dimensional projections, then fuse them into a coherent full-resolution image.
• Leverage the fact that local averaging (piecewise-constant projection) is a more stable learning task than pixel-wise reconstruction.

Experimental results

Research questions

• RQ1 Can learning projections into random low-dimensional subspaces improve generalization in inverse problems with scarce training data?
• RQ2 How does the proposed method compare to direct inversion in terms of robustness to measurement noise and unseen corruptions?
• RQ3 Does the method maintain geometric accuracy when reconstructions are tested on corruptions not present in the training data?
• RQ4 To what extent does the method generalize across different training datasets, especially when real ground truth is unavailable?
• RQ5 Can the reformulated inverse problem be interpreted as a deconvolution in expectation over random subspaces?

Key findings

• The proposed method achieves higher average SNR (14.49 dB) than direct inversion (13.51 dB) when trained and tested on 10 dB noisy measurements.
• Under input SNR mismatch, the method maintains reconstruction quality (12.88 dB) while direct inversion degrades significantly (10.34 dB).
• For erasure corruptions with probability 1/12, the method achieves 12.08 dB average SNR, while direct inversion fails completely (9.03 dB).
• The method correctly reconstructs geometric features such as cavity outlines even when direct inversion hallucinates sharp but incorrect structures.
• Reconstructions remain consistent across different training datasets (LSUN, CelebA, synthetic shapes), demonstrating robustness to dataset overfitting.
• Visual comparisons show that the method preserves correct geometry in checkerboard and x-ray phantoms, while direct inversion fails to recover salient features.