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Recover Cell Tensor: Diffusion-Equivalent Tensor Completion for Fluorescence Microscopy Imaging

Chenwei Wang, Zhaoke Huang, Zelin Li, Wenqi Zhu

TL;DR

This work recasts 3D fluorescence microscopy reconstruction under sparse, noisy, and nonlinear degradation as a robust tensor completion problem. By combining Tucker decomposition with ADMM optimization, it yields a low-rank latent representation that can be completed and denoised from incomplete observations, while modeling sparse noise. The authors reveal a theoretical equivalence between this optimization-driven approach and a conditional score-based diffusion framework guided by structural priors, enabling principled generative reconstruction without explicit score learning. Empirical results on SR-CACO-2 and three in vivo cellular datasets show state-of-the-art quantitative gains (PSNR/SSIM) and superior qualitative fidelity, with demonstrated robustness to sampling density and noise and insights into tensor rank distributions. The approach offers a practical, unsupervised pathway to high-fidelity, temporally consistent 3D cellular reconstructions essential for analyzing dynamic cellular processes such as mitosis, with potential broad impact on live-cell imaging and downstream biological analysis.

Abstract

Fluorescence microscopy (FM) imaging is a fundamental technique for observing live cell division, one of the most essential processes in the cycle of life and death. Observing 3D live cells requires scanning through the cell volume while minimizing lethal phototoxicity. That limits acquisition time and results in sparsely sampled volumes with anisotropic resolution and high noise. Existing image restoration methods, primarily based on inverse problem modeling, assume known and stable degradation processes and struggle under such conditions, especially in the absence of high-quality reference volumes. In this paper, from a new perspective, we propose a novel tensor completion framework tailored to the nature of FM imaging, which inherently involves nonlinear signal degradation and incomplete observations. Specifically, FM imaging with equidistant Z-axis sampling is essentially a tensor completion task under a uniformly random sampling condition. On one hand, we derive the theoretical lower bound for exact cell tensor completion, validating the feasibility of accurately recovering 3D cell tensor. On the other hand, we reformulate the tensor completion problem as a mathematically equivalent score-based generative model. By incorporating structural consistency priors, the generative trajectory is effectively guided toward denoised and geometrically coherent reconstructions. Our method demonstrates state-of-the-art performance on SR-CACO-2 and three real \textit{in vivo} cellular datasets, showing substantial improvements in both signal-to-noise ratio and structural fidelity.

Recover Cell Tensor: Diffusion-Equivalent Tensor Completion for Fluorescence Microscopy Imaging

TL;DR

This work recasts 3D fluorescence microscopy reconstruction under sparse, noisy, and nonlinear degradation as a robust tensor completion problem. By combining Tucker decomposition with ADMM optimization, it yields a low-rank latent representation that can be completed and denoised from incomplete observations, while modeling sparse noise. The authors reveal a theoretical equivalence between this optimization-driven approach and a conditional score-based diffusion framework guided by structural priors, enabling principled generative reconstruction without explicit score learning. Empirical results on SR-CACO-2 and three in vivo cellular datasets show state-of-the-art quantitative gains (PSNR/SSIM) and superior qualitative fidelity, with demonstrated robustness to sampling density and noise and insights into tensor rank distributions. The approach offers a practical, unsupervised pathway to high-fidelity, temporally consistent 3D cellular reconstructions essential for analyzing dynamic cellular processes such as mitosis, with potential broad impact on live-cell imaging and downstream biological analysis.

Abstract

Fluorescence microscopy (FM) imaging is a fundamental technique for observing live cell division, one of the most essential processes in the cycle of life and death. Observing 3D live cells requires scanning through the cell volume while minimizing lethal phototoxicity. That limits acquisition time and results in sparsely sampled volumes with anisotropic resolution and high noise. Existing image restoration methods, primarily based on inverse problem modeling, assume known and stable degradation processes and struggle under such conditions, especially in the absence of high-quality reference volumes. In this paper, from a new perspective, we propose a novel tensor completion framework tailored to the nature of FM imaging, which inherently involves nonlinear signal degradation and incomplete observations. Specifically, FM imaging with equidistant Z-axis sampling is essentially a tensor completion task under a uniformly random sampling condition. On one hand, we derive the theoretical lower bound for exact cell tensor completion, validating the feasibility of accurately recovering 3D cell tensor. On the other hand, we reformulate the tensor completion problem as a mathematically equivalent score-based generative model. By incorporating structural consistency priors, the generative trajectory is effectively guided toward denoised and geometrically coherent reconstructions. Our method demonstrates state-of-the-art performance on SR-CACO-2 and three real \textit{in vivo} cellular datasets, showing substantial improvements in both signal-to-noise ratio and structural fidelity.
Paper Structure (34 sections, 4 theorems, 45 equations, 13 figures, 4 tables, 1 algorithm)

This paper contains 34 sections, 4 theorems, 45 equations, 13 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulations
  3. 3D FM Live Cell Imaging
  4. Tensor Completion Model for 3D FM Cell Recovery
  5. Lower Bound for Exact Cell Recovery
  6. Proposed Solution
  7. Tucker Decomposition-based Solution
  8. Equivalence with Conditional Diffusion Framework
  9. Experiments
  10. Main Results
  11. Quantitative Comparison with State-of-the-Art Methods
  12. Qualitative Comparison with State-of-the-Art Methods
  13. Discussion
  14. Rank Distribution of FM Cell
  15. Performance Under Different Sampling Densities and Noise
  16. ...and 19 more sections

Key Result

Theorem 1

Let $\Omega$ be a uniformly sampled subset of $[I_1]\times\dots\times[I_k]$ and $\hat{\mathcal{T}}$ be the solution to (1) with $\delta_j = \sqrt{\lambda_* r_*/I_j}$. The reason of the sampling of FM imaging serving as a uniformly sampling is shown in Appendix appenxproofRequire1. There exists a con and where $I = \max_j I_j$, $r_*$ denotes the effective rank of the tensor, and $\lambda_*$ is nor

Figures (13)

  • Figure 1: Illustration of the FM imaging process and comparison of reconstructed results across different methods. The schematic (left) shows the fluorescence imaging pipeline, where excitation light passes through optical components and interacts with the specimen before being detected. Raw observations are corrupted by noise and optical degradation. Reconstructed images using inverse-problem-based methods (e.g., IPG, CycleGAN) often suffer from hallucinated structures and residual noise.
  • Figure 2: Overview of the proposed framework. We first model 3D fluorescence microscopy (FM) restoration as a tensor completion task, capturing the nonlinear degradation and partial observation inherent in FM imaging. Then, we reformulate this task into a mathematically equivalent score-based generative process, revealing a principled connection to conditional diffusion modeling. Finally, we introduce structural consistency priors to guide the generative trajectory, enabling accurate and denoised 3D cell volume recovery.
  • Figure 3: 3D/XY-Slice/Zoom-in qualitative denoising comparison of different methods on live 3D fluorescence microscopy volumes at the 2700s time point. Appendix \ref{['appendixexperiments']} show the performance comparison at the 4950s time point.
  • Figure 4: YZ (Yellow) and XZ (Blue) slice qualitative super-resolution comparison of different methods on live 3D fluorescence microscopy volumes at the 2700s time point.
  • Figure 5: 3D/XY-slice/zoom-in qualitative denoising comparison of different methods on live 3D fluorescence microscopy volumes at the 4950s time point.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Theorem 1: Exact Recovery Lower Bound under Eq. \ref{['eq:incoherent_nuclear_min1']}
  • Theorem 2: Exact Recovery Lower Bound under Eq. \ref{['eqtensorc1']}
  • Lemma 1: sufficient condition
  • Theorem 3: Global Convergence