Generalized Compressed Sensing for Image Reconstruction with Diffusion Probabilistic Models

TL;DR

This paper introduces a novel method for selecting optimal linear measurements for image reconstruction by leveraging the implicit prior of diffusion models, outperforming traditional PCA, ICA, and compressed sensing in accuracy.

Contribution

The authors propose a general approach to optimize linear measurements for images using neural network-based priors, capturing richer statistical structures than traditional methods.

Findings

Optimized measurements differ from PCA, ICA, and CS.

Results in lower mean squared reconstruction error.

Measurement distributions are notably skewed.

Abstract

We examine the problem of selecting a small set of linear measurements for reconstructing high-dimensional signals. Well-established methods for optimizing such measurements include principal component analysis (PCA), independent component analysis (ICA) and compressed sensing (CS) based on random projections, all of which rely on axis- or subspace-aligned statistical characterization of the signal source. However, many naturally occurring signals, including photographic images, contain richer statistical structure. To exploit such structure, we introduce a general method for obtaining an optimized set of linear measurements for efficient image reconstruction, where the signal statistics are expressed by the prior implicit in a neural network trained to perform denoising (known as a "diffusion model"). We demonstrate that the optimal measurements derived for two natural image datasets differ from those of PCA, ICA, or CS, and result in substantially lower mean squared reconstruction error. Interestingly, the marginal distributions of the measurement values are asymmetrical (skewed), substantially more so than those of previous methods. We also find that optimizing with respect to perceptual loss, as quantified by structural similarity (SSIM), leads to measurements different from those obtained when optimizing for MSE. Our results highlight the importance of incorporating the specific statistical regularities of natural signals when designing effective linear measurements.