Generalized Compressed Sensing for Image Reconstruction with Diffusion Probabilistic Models
Abstract: We examine the problem of selecting a small set of linear measurements for reconstructing high-dimensional signals in the context of probabilistic diffusion models. 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.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: 1. In response to the comments, we have introduced improvements throughout the main text; we also changed the title to slightly to "generalized compressed sensing".
2. Section 3.4, Figure 4G, and Appendix E.6: We show the performance of a greedy version of the algorithm, in which the OLMs are sequentially orthogonalized (analogous to PCA). Performance is only slightly worse than the nonsequential (joint) algorithm.
3. Section 3.4, Figure 4H: We provide a simple analysis and demonstration that reconstruction from OLM measurements is reasonably robust to noise, and can be made more so by training on noisy measurements.
4. Appendix D: We provide additional experiments showing the effects of hyperparameter choices (step size, injected noise, stopping criterion, and number of samples) on OLM reconstruction performance.
Assigned Action Editor: ~Arash_Mehrjou1
Submission Number: 4112
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