# Generalized Compressed Sensing for Image Reconstruction with Diffusion Probabilistic Models

arXiv: https://arxiv.org/abs/2405.17456 · HTML: https://mlsanity.com/paper/2405.17456/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 28 of its 84 references that are in the map.
- First posted on arXiv 22 May 2024.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 28 of 28)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [Density Modeling of Images using a Generalized Normalization Transformation](https://mlsanity.com/paper/1511.06281/) (2015)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Robust and interpretable blind image denoising via bias-free convolutional neural networks](https://mlsanity.com/paper/1906.05478/) (2019)
- [DeepRED: Deep Image Prior Powered by RED](https://mlsanity.com/paper/1903.10176/) (2019)
- [Solving Linear Inverse Problems Using the Prior Implicit in a Denoiser](https://mlsanity.com/paper/2007.13640/) (2020)
- [An Online Plug-and-Play Algorithm for Regularized Image Reconstruction](https://mlsanity.com/paper/1809.04693/) (2018)
- [Provable Probabilistic Imaging using Score-Based Generative Priors](https://mlsanity.com/paper/2310.10835/) (2023)
- [Deep Compressed Sensing](https://mlsanity.com/paper/1905.06723/) (2019)
- [Active MR k-space Sampling with Reinforcement Learning](https://mlsanity.com/paper/2007.10469/) (2020)
- [Learning a Compressed Sensing Measurement Matrix via Gradient Unrolling](https://mlsanity.com/paper/1806.10175/) (2018)
- [FourierNets enable the design of highly non-local optical encoders for computational imaging](https://mlsanity.com/paper/2104.10611/) (2021)

## Built on by (most cited first, 1 of 1)

- [Learning Normalized Energy Models for Linear Inverse Problems](https://mlsanity.com/paper/2605.15487/) (2026)

## Abstract (arXiv)

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.