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

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.

💡 Research Summary

Microscopy image deconvolution is fundamentally an inverse problem: the recorded image is the convolution of the true object with the point spread function (PSF) plus noise. Classical explicit methods such as Richardson‑Lucy rely on an accurate PSF model, but in practice PSFs are often unknown, spatially varying, or corrupted by measurement noise, leading to sub‑optimal restoration. Recent deep‑learning approaches (U‑Net, CARE, DDPM, MIMO‑U‑Net, etc.) achieve impressive visual results by learning a direct mapping from degraded to high‑quality images, yet they ignore the underlying physics, resulting in very large models and limited interpretability.

The authors address these shortcomings by formulating deconvolution as a physics‑informed inverse problem and by leveraging the non‑standard wavelet compression scheme introduced by Beylkin, Coifman, and Rokhlin (BCR). BCR provides an O(N) representation of integral and pseudo‑differential operators through multi‑level scaling and wavelet coefficients, yielding banded matrices that can be efficiently approximated with local convolutions. By mapping these banded matrices onto neural‑network layers, the forward operator A is represented as a trainable convolutional block K_T, while the pseudo‑differential inverse (A⁻¹) is approximated by B = (K_TᵀK_T + εI)⁻¹.

Because microscopy data are heavily contaminated with non‑linear noise and artifacts, a plain BCR‑NN architecture is unstable. To improve robustness, the authors embed residual dense blocks (RDB) within the BCR structure, creating a single‑stage residual BCR network (s‑rBCR). They then extend this to a multi‑stage residual BCR (m‑rBCR) architecture that re‑injects the intermediate reconstructions as posterior information at each resolution level, fusing features across scales. This multi‑stage design mitigates the information loss caused by truncating the BCR decomposition at a finite level L₀ and enables the network to learn a regularized inverse mapping without excessive parameters.

Four datasets are used for evaluation: two simulated sets (ImageNet‑based and BioSR) generated with realistic PSFs and noise, and two real microscopy collections (dSTORM super‑resolution and wide‑field fluorescence). m‑rBCR is compared against a broad spectrum of baselines, including explicit deconvolution (Richardson‑Lucy), classic CNNs (U‑Net, RCAN, ESRGAN), denoising‑oriented networks (DnCNN, Noise2Noise, CARE), diffusion models (DDPM), and the recent MIMO‑U‑Net.

Quantitatively, m‑rBCR achieves the highest PSNR and SSIM on both real datasets and the BioSR simulation, outperforming the best competing deep models while using roughly 30 × fewer parameters than MIMO‑U‑Net and up to 210 × fewer than ESRGAN. On the ImageNet simulation it ranks second, just behind MIMO‑U‑Net, but it runs about three times faster than MIMO‑U‑Net and up to 300 × faster than DDPM. Qualitatively, m‑rBCR restores fine sub‑cellular structures with clear edges and effective noise suppression, demonstrating that embedding physical constraints yields both efficiency and fidelity.

The study highlights several important insights: (1) BCR’s banded‑matrix representation naturally aligns with convolutional neural networks, enabling a compact physics‑aware backbone; (2) residual dense connections provide the non‑linear robustness needed for real microscopy noise; (3) multi‑stage posterior integration compensates for truncation errors inherent in any finite BCR decomposition. Limitations include sensitivity to hyper‑parameters such as the bandwidth nb and truncation level L₀, and the current focus on 2‑D data; extending to full 3‑D volumes, time‑lapse imaging, or more complex noise models remains future work.

In conclusion, the paper presents a novel multi‑stage residual BCR network that fuses wavelet‑based operator compression with deep residual learning, delivering a lightweight, fast, and high‑quality solution to microscopy deconvolution. The approach opens avenues for real‑time, physics‑guided image restoration in biomedical imaging and suggests that further integration of operator theory with deep learning could benefit a wide range of inverse problems.