# Linear Combinations of Patches are Unreasonably Effective for Single-Image Denoising

arXiv: https://arxiv.org/abs/2212.00422 · HTML: https://mlsanity.com/paper/2212.00422/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 58 references that are in the map.
- First posted on arXiv 1 Dec 2022.

## Builds on (most cited first, 27 of 27)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Restormer: Efficient Transformer for High-Resolution Image Restoration](https://mlsanity.com/paper/2111.09881/) (2021)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising](https://mlsanity.com/paper/1710.04026/) (2017)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [Noise2Self: Blind Denoising by Self-Supervision](https://mlsanity.com/paper/1901.11365/) (2019)
- [Benchmarking Denoising Algorithms with Real Photographs](https://mlsanity.com/paper/1707.01313/) (2017)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [Multi-level Wavelet-CNN for Image Restoration](https://mlsanity.com/paper/1805.07071/) (2018)
- [Efficient Frequency Domain-based Transformers for High-Quality Image Deblurring](https://mlsanity.com/paper/2211.12250/) (2022)
- [High-Quality Self-Supervised Deep Image Denoising](https://mlsanity.com/paper/1901.10277/) (2019)
- [Non-Local Recurrent Network for Image Restoration](https://mlsanity.com/paper/1806.02919/) (2018)
- [Noisier2Noise: Learning to Denoise from Unpaired Noisy Data](https://mlsanity.com/paper/1910.11908/) (2019)
- [Neural Nearest Neighbors Networks](https://mlsanity.com/paper/1810.12575/) (2018)
- [Practical Blind Image Denoising via Swin-Conv-UNet and Data Synthesis](https://mlsanity.com/paper/2203.13278/) (2022)
- [A Trilateral Weighted Sparse Coding Scheme for Real-World Image Denoising](https://mlsanity.com/paper/1807.04364/) (2018)
- [Non-Local Color Image Denoising with Convolutional Neural Networks](https://mlsanity.com/paper/1611.06757/) (2016)
- [Rethinking Deep Image Prior for Denoising](https://mlsanity.com/paper/2108.12841/) (2021)
- [Noise2Fast: Fast Self-Supervised Single Image Blind Denoising](https://mlsanity.com/paper/2108.10209/) (2021)
- [Adaptive Denoising via GainTuning](https://mlsanity.com/paper/2107.12815/) (2021)
- [Training deep learning based image denoisers from undersampled measurements without ground truth and without image prior](https://mlsanity.com/paper/1806.00961/) (2018)
- [DCT2net: an interpretable shallow CNN for image denoising](https://mlsanity.com/paper/2107.14803/) (2021)
- [Towards a unified view of unsupervised non-local methods for image denoising: the NL-Ridge approach](https://mlsanity.com/paper/2203.00570/) (2022)

## Built on by (most cited first, 4 of 4)

- [Normalization-Equivariant Neural Networks with Application to Image Denoising](https://mlsanity.com/paper/2306.05037/) (2023)
- [Image Denoising Using Green Channel Prior](https://mlsanity.com/paper/2408.05923/) (2024)
- [Mixed Degradation Image Restoration via Local Dynamic Optimization and Conditional Embedding](https://mlsanity.com/paper/2411.16217/) (2024)
- [On normalization-equivariance properties of supervised and unsupervised denoising methods: a survey](https://mlsanity.com/paper/2402.15352/) (2024)

## Abstract (arXiv)

In the past decade, deep neural networks have revolutionized image denoising in achieving significant accuracy improvements by learning on datasets composed of noisy/clean image pairs. However, this strategy is extremely dependent on training data quality, which is a well-established weakness. To alleviate the requirement to learn image priors externally, single-image (a.k.a., self-supervised or zero-shot) methods perform denoising solely based on the analysis of the input noisy image without external dictionary or training dataset. This work investigates the effectiveness of linear combinations of patches for denoising under this constraint. Although conceptually very simple, we show that linear combinations of patches are enough to achieve state-of-the-art performance. The proposed parametric approach relies on quadratic risk approximation via multiple pilot images to guide the estimation of the combination weights. Experiments on images corrupted artificially with Gaussian noise as well as on real-world noisy images demonstrate that our method is on par with the very best single-image denoisers, outperforming the recent neural network based techniques, while being much faster and fully interpretable.