# Solving the inverse problem of microscopy deconvolution with a residual Beylkin-Coifman-Rokhlin neural network

arXiv: https://arxiv.org/abs/2407.03239 · HTML: https://mlsanity.com/paper/2407.03239/

## Key facts

- Cited by 2 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 18 of its 51 references that are in the map.
- First posted on arXiv 3 Jul 2024.

## Placed in

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

## Builds on (most cited first, 18 of 18)

- [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)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Image Super-Resolution Using Very Deep Residual Channel Attention Networks](https://mlsanity.com/paper/1807.02758/) (2018)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [UNETR: Transformers for 3D Medical Image Segmentation](https://mlsanity.com/paper/2103.10504/) (2021)
- [Multi-Stage Progressive Image Restoration](https://mlsanity.com/paper/2102.02808/) (2021)
- [Residual Dense Network for Image Super-Resolution](https://mlsanity.com/paper/1802.08797/) (2018)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Rethinking Coarse-to-Fine Approach in Single Image Deblurring](https://mlsanity.com/paper/2108.05054/) (2021)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Sparsity of solutions for variational inverse problems with finite-dimensional data](https://mlsanity.com/paper/1809.05045/) (2018)
- [BCR-Net: a neural network based on the nonstandard wavelet form](https://mlsanity.com/paper/1810.08754/) (2018)
- [Solving Traveltime Tomography with Deep Learning](https://mlsanity.com/paper/1911.11636/) (2019)
- [Solving Optical Tomography with Deep Learning](https://mlsanity.com/paper/1910.04756/) (2019)
- [Blind Deconvolution Microscopy Using Cycle Consistent CNN with Explicit PSF Layer](https://mlsanity.com/paper/1904.02910/) (2019)

## Built on by (most cited first, 2 of 2)

- [Lightweight CycleGAN Models for Cross-Modality Image Transformation and Experimental Quality Assessment in Fluorescence Microscopy](https://mlsanity.com/paper/2510.15579/) (2025)
- [Cryo-SWAN: the Multi-Scale Wavelet-decomposition-inspired Autoencoder Network for molecular density representation of molecular volumes](https://mlsanity.com/paper/2603.03342/) (2026)

## Abstract (arXiv)

Optic deconvolution in light microscopy (LM) refers to recovering the object details from images, revealing the ground truth of samples. Traditional explicit methods in LM rely on the point spread function (PSF) during image acquisition. Yet, these approaches often fall short due to inaccurate PSF models and noise artifacts, hampering the overall restoration quality. In this paper, we approached the optic deconvolution as an inverse problem. Motivated by the nonstandard-form compression scheme introduced by Beylkin, Coifman, and Rokhlin (BCR), we proposed an innovative physics-informed neural network Multi-Stage Residual-BCR Net (m-rBCR) to approximate the optic deconvolution. We validated the m-rBCR model on four microscopy datasets - two simulated microscopy datasets from ImageNet and BioSR, real dSTORM microscopy images, and real widefield microscopy images. In contrast to the explicit deconvolution methods (e.g. Richardson-Lucy) and other state-of-the-art NN models (U-Net, DDPM, CARE, DnCNN, ESRGAN, RCAN, Noise2Noise, MPRNet, and MIMO-U-Net), the m-rBCR model demonstrates superior performance to other candidates by PSNR and SSIM in two real microscopy datasets and the simulated BioSR dataset. In the simulated ImageNet dataset, m-rBCR ranks the second-best place (right after MIMO-U-Net). With the backbone from the optical physics, m-rBCR exploits the trainable parameters with better performances (from ~30 times fewer than the benchmark MIMO-U-Net to ~210 times than ESRGAN). This enables m-rBCR to achieve a shorter runtime (from ~3 times faster than MIMO-U-Net to ~300 times faster than DDPM). To summarize, by leveraging physics constraints our model reduced potentially redundant parameters significantly in expertise-oriented NN candidates and achieved high efficiency with superior performance.