# Deep Unfolding of a Proximal Interior Point Method for Image Restoration

arXiv: https://arxiv.org/abs/1812.04276 · HTML: https://mlsanity.com/paper/1812.04276/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2018.

## Built on by (most cited first, 24 of 24)

- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Graph Signal Restoration Using Nested Deep Algorithm Unrolling](https://mlsanity.com/paper/2106.15910/) (2021)
- [ADMM-DAD net: a deep unfolding network for analysis compressed sensing](https://mlsanity.com/paper/2110.06986/) (2021)
- [Proximal Algorithm Unrolling: Flexible and Efficient Reconstruction Networks for Single-Pixel Imaging](https://mlsanity.com/paper/2505.23180/) (2025)
- [On Adversarial Robustness of Deep Image Deblurring](https://mlsanity.com/paper/2210.02502/) (2022)
- [Deep Unfolding of the DBFB Algorithm with Application to ROI CT Imaging with Limited Angular Density](https://mlsanity.com/paper/2209.13264/) (2022)
- [Regularization via deep generative models: an analysis point of view](https://mlsanity.com/paper/2101.08661/) (2021)
- [Unrolled Variational Bayesian Algorithm for Image Blind Deconvolution](https://mlsanity.com/paper/2110.07202/) (2021)
- [DECONET: an Unfolding Network for Analysis-based Compressed Sensing with Generalization Error Bounds](https://mlsanity.com/paper/2205.07050/) (2022)
- [Monotonically Convergent Regularization by Denoising](https://mlsanity.com/paper/2202.04961/) (2022)
- [Learning Variational Models with Unrolling and Bilevel Optimization](https://mlsanity.com/paper/2209.12651/) (2022)
- [Model-Guided Multi-Contrast Deep Unfolding Network for MRI Super-resolution Reconstruction](https://mlsanity.com/paper/2209.07030/) (2022)
- [Reconstruct Anything Model: a lightweight general model for computational imaging](https://mlsanity.com/paper/2503.08915/) (2025)
- [Photothermal-SR-Net: A Customized Deep Unfolding Neural Network for Photothermal Super Resolution Imaging](https://mlsanity.com/paper/2104.10563/) (2021)
- [Alternative design of DeepPDNet in the context of image restoration](https://mlsanity.com/paper/2202.09810/) (2022)
- [UNEM: UNrolled Generalized EM for Transductive Few-Shot Learning](https://mlsanity.com/paper/2412.16739/) (2024)
- [Stability Bounds for the Unfolded Forward-Backward Algorithm](https://mlsanity.com/paper/2412.17888/) (2024)
- [Unrolled Graph Neural Networks for Constrained Optimization](https://mlsanity.com/paper/2509.17156/) (2025)
- [Variational models for signal processing with Graph Neural Networks](https://mlsanity.com/paper/2103.16337/) (2021)
- [Learning-Based Symbol Level Precoding: A Memory-Efficient Unsupervised Learning Approach](https://mlsanity.com/paper/2111.08110/) (2021)
- [Explainable bilevel optimization: an application to the Helsinki deblur challenge](https://mlsanity.com/paper/2210.10050/) (2022)
- [CQnet: convex-geometric interpretation and constraining neural-network trajectories](https://mlsanity.com/paper/2302.10895/) (2023)
- [Proximal Vision Transformer: Enhancing Feature Representation through Two-Stage Manifold Geometry](https://mlsanity.com/paper/2508.17081/) (2025)
- [Unrolled Neural Networks for Constrained Optimization](https://mlsanity.com/paper/2601.17274/) (2026)

## Abstract (arXiv)

Variational methods are widely applied to ill-posed inverse problems for they have the ability to embed prior knowledge about the solution. However, the level of performance of these methods significantly depends on a set of parameters, which can be estimated through computationally expensive and time-consuming methods. In contrast, deep learning offers very generic and efficient architectures, at the expense of explainability, since it is often used as a black-box, without any fine control over its output. Deep unfolding provides a convenient approach to combine variational-based and deep learning approaches. Starting from a variational formulation for image restoration, we develop iRestNet, a neural network architecture obtained by unfolding a proximal interior point algorithm. Hard constraints, encoding desirable properties for the restored image, are incorporated into the network thanks to a logarithmic barrier, while the barrier parameter, the stepsize, and the penalization weight are learned by the network. We derive explicit expressions for the gradient of the proximity operator for various choices of constraints, which allows training iRestNet with gradient descent and backpropagation. In addition, we provide theoretical results regarding the stability of the network for a common inverse problem example. Numerical experiments on image deblurring problems show that the proposed approach compares favorably with both state-of-the-art variational and machine learning methods in terms of image quality.