# Learning of Patch-Based Smooth-Plus-Sparse Models for Image Reconstruction

arXiv: https://arxiv.org/abs/2412.13070 · HTML: https://mlsanity.com/paper/2412.13070/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 17 Dec 2024.

## Builds on (most cited first, 20 of 20)

- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [Image Reconstruction: From Sparsity to Data-adaptive Methods and Machine Learning](https://mlsanity.com/paper/1904.02816/) (2019)
- [A survey of sparse representation: algorithms and applications](https://mlsanity.com/paper/1602.07017/) (2016)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Inertial Proximal Alternating Linearized Minimization (iPALM) for Nonconvex and Nonsmooth Problems](https://mlsanity.com/paper/1702.02505/) (2017)
- [Convolutional Dictionary Learning: A Comparative Review and New Algorithms](https://mlsanity.com/paper/1709.02893/) (2017)
- [PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization](https://mlsanity.com/paper/2205.12021/) (2022)
- [TorchDEQ: A Library for Deep Equilibrium Models](https://mlsanity.com/paper/2310.18605/) (2023)
- [Convolutional Dictionary Learning via Local Processing](https://mlsanity.com/paper/1705.03239/) (2017)
- [Stable and Interpretable Unrolled Dictionary Learning](https://mlsanity.com/paper/2106.00058/) (2021)
- [A Generative Variational Model for Inverse Problems in Imaging](https://mlsanity.com/paper/2104.12630/) (2021)
- [Learning Multiscale Convolutional Dictionaries for Image Reconstruction](https://mlsanity.com/paper/2011.12815/) (2020)
- [Understanding approximate and unrolled dictionary learning for pattern recovery](https://mlsanity.com/paper/2106.06338/) (2021)
- [Convolutional Dictionary Learning: Acceleration and Convergence](https://mlsanity.com/paper/1707.00389/) (2017)
- [DCT2net: an interpretable shallow CNN for image denoising](https://mlsanity.com/paper/2107.14803/) (2021)
- [A fast patch-dictionary method for whole image recovery](https://mlsanity.com/paper/1408.3740/) (2014)

## Built on by (most cited first, 3 of 3)

- [Flower: A Flow-Matching Solver for Inverse Problems](https://mlsanity.com/paper/2509.26287/) (2025)
- [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025)
- [Parameter-free structure-texture image decomposition by unrolling](https://mlsanity.com/paper/2503.13354/) (2025)

## Abstract (arXiv)

We aim at the solution of inverse problems in imaging, by combining a penalized sparse representation of image patches with an unconstrained smooth one. This allows for a straightforward interpretation of the reconstruction. We formulate the optimization as a bilevel problem. The inner problem deploys classical algorithms while the outer problem optimizes the dictionary and the regularizer parameters through supervised learning. The process is carried out via implicit differentiation and gradient-based optimization. We evaluate our method for denoising, super-resolution, and compressed-sensing magnetic-resonance imaging. We compare it to other classical models as well as deep-learning-based methods and show that it always outperforms the former and also the latter in some instances.