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

TL;DR

This paper reframes optical deconvolution in light microscopy as an inverse problem and introduces the multi-stage residual BCR net (m-rBCR), a physics-informed neural network that leverages Beylkin–Coifman–Rokhlin wavelet representations to approximate the inverse of the forward imaging operator. By decomposing the forward/convolution via a forward operator $K^{T}$ and a pseudo-differential inverse $B=(K^{T}K+\varepsilon I)^{-1}$ and fusing posteriors across scales, m-rBCR achieves high restoration quality with dramatically fewer trainable parameters and shorter runtimes compared to state-of-the-art models. The method demonstrates competitive or superior PSNR/SSIM on four datasets, including simulated BioSR and ImageNet as well as real dSTORM and widefield/confocal data, outperforming several deep-learning baselines and reducing computational burden by up to an order of magnitude. Overall, the work highlights the practical value of embedding physical constraints into neural architectures for microscopy deconvolution and points to avenues for further fusion-strategy enhancements.

Abstract

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.

Figures (9)

• Figure 1: BCR-based decomposition. a) presents the decomposed operator fan_bcr-net_2019 in Eq. \ref{['doubInt']} at one resolution $l$ with Beylkin, Coifman, and Rokhlin (BCR) wavelet theory. b) illustrates a simple multi-level decomposition based on a). All matrices shown here are band matrices with only values on the green lines.
• Figure 2: Single stage residual BCR Net. a) indicates the structure of the NN model for microscopy deconvolution. it consists of the Sequence Convolution Block (SCB) in b) as $K^{T}$ and pseudo-differential operator from sub-units of c) and d). c) illustrates the regularised BCR decomposition through the residual structure.
• Figure 3: Multi-Stage Residual BCR Net (m-rBCR). The m-rBCR model reintegrates the input image from other resolutions as posterior for the pseudo-differential operator $B = \left( K^{T}K + \varepsilon I \right) ^{-1}$.
• Figure 4: Test results on the simulated widefield microscope dataset from BioSR. Our m-rBCR model successfully restored comparable details to the benchmark MIMO-U-Net, demonstrating efficiency with fewer parameters and a shorter runtime.
• Figure 5: Test results on the simulated widefield microscope dataset from Imagenet. Multi-stage residual BCR Net (m-rBCR) restores the most details without introducing additional artifacts due to noise.