# Deep Decomposition Learning for Inverse Imaging Problems

arXiv: https://arxiv.org/abs/1911.11028 · HTML: https://mlsanity.com/paper/1911.11028/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Nov 2019.

## Built on by (most cited first, 25 of 25)

- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [Learning Enriched Features via Selective State Spaces Model for Efficient Image Deblurring](https://mlsanity.com/paper/2403.20106/) (2024)
- [GAN Prior based Null-Space Learning for Consistent Super-Resolution](https://mlsanity.com/paper/2211.13524/) (2022)
- [Unlimited-Size Diffusion Restoration](https://mlsanity.com/paper/2303.00354/) (2023)
- [Mixed Hierarchy Network for Image Restoration](https://mlsanity.com/paper/2302.09554/) (2023)
- [Invertible Diffusion Models for Compressed Sensing](https://mlsanity.com/paper/2403.17006/) (2024)
- [A Conditional Normalizing Flow for Accelerated Multi-Coil MR Imaging](https://mlsanity.com/paper/2306.01630/) (2023)
- [A Regularized Conditional GAN for Posterior Sampling in Image Recovery Problems](https://mlsanity.com/paper/2210.13389/) (2022)
- [Self-Supervised Scalable Deep Compressed Sensing](https://mlsanity.com/paper/2308.13777/) (2023)
- [Deep Physics-Guided Unrolling Generalization for Compressed Sensing](https://mlsanity.com/paper/2307.08950/) (2023)
- [Progressive Content-aware Coded Hyperspectral Compressive Imaging](https://mlsanity.com/paper/2303.09773/) (2023)
- [NPN: Non-Linear Projections of the Null-Space for Imaging Inverse Problems](https://mlsanity.com/paper/2510.01608/) (2025)
- [Projection-Based Correction for Enhancing Deep Inverse Networks](https://mlsanity.com/paper/2505.15777/) (2025)
- [NullFlow: One-Step Generative Reconstruction](https://mlsanity.com/paper/2606.22696/) (2026)
- [A Range-Null Space Decomposition Approach for Fast and Flexible Spectral Compressive Imaging](https://mlsanity.com/paper/2305.09746/) (2023)
- [Exploring Richer and More Accurate Information via Frequency Selection for Image Restoration](https://mlsanity.com/paper/2407.08950/) (2024)
- [SHARE: A Fully Unsupervised Framework for Single Hyperspectral Image Restoration](https://mlsanity.com/paper/2601.13987/) (2026)
- [GSNR: Graph Smooth Null-Space Representation for Inverse Problems](https://mlsanity.com/paper/2602.20328/) (2026)
- [Adaptive Identification of Blurred Regions for Accurate Image Deblurring](https://mlsanity.com/paper/2502.20880/) (2025)
- [HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Training](https://mlsanity.com/paper/2601.01228/) (2026)
- [Data-Consistent Learning of Inverse Problems](https://mlsanity.com/paper/2601.12831/) (2026)
- [Equivariant Learning for Unsupervised Image Dehazing](https://mlsanity.com/paper/2601.13986/) (2026)
- [Integrated Forward-Inverse Network for Lensless Image Reconstruction](https://mlsanity.com/paper/2607.04608/) (2026)
- [SONAR: A Structure-Consistent Neural Operator for Null-Space-Aware Sparse View CT Reconstruction](https://mlsanity.com/paper/2609.13688/) (2026)

## Abstract (arXiv)

Deep learning is emerging as a new paradigm for solving inverse imaging problems. However, the deep learning methods often lack the assurance of traditional physics-based methods due to the lack of physical information considerations in neural network training and deploying. The appropriate supervision and explicit calibration by the information of the physic model can enhance the neural network learning and its practical performance. In this paper, inspired by the geometry that data can be decomposed by two components from the null-space of the forward operator and the range space of its pseudo-inverse, we train neural networks to learn the two components and therefore learn the decomposition, i.e. we explicitly reformulate the neural network layers as learning range-nullspace decomposition functions with reference to the layer inputs, instead of learning unreferenced functions. We empirically show that the proposed framework demonstrates superior performance over recent deep residual learning, unrolled learning and nullspace learning on tasks including compressive sensing medical imaging and natural image super-resolution. Our code is available at https://github.com/edongdongchen/DDN.