Algorithmic unfolding for image reconstruction and localization problems in fluorescence microscopy

TL;DR

The paper tackles ill-posed image reconstruction and molecule localization in fluorescence microscopy by learning hyperparameters within a bilevel, algorithmic-unrolling framework. It unrolls an accelerated projected gradient descent on a sparse, nonnegative model with Gaussian or Poisson data fidelity, enabling gradient-based learning of both regularization and algorithmic parameters using task-specific losses for reconstruction and localization. The method is validated on simulated SMLM and ISBI datasets as well as fluctuation-based (SOFI-like) deconvolution, demonstrating improved localization metrics and competitive image quality, while preserving the physics-based forward model. This approach offers a physics-informed, data-driven pathway to optimize microscopy reconstructions without abandoning established forward models, and it opens avenues for integrating more advanced regularizers and Plug-and-Play strategies.

Abstract

We propose an unfolded accelerated projected-gradient descent procedure to estimate model and algorithmic parameters for image super-resolution and molecule localization problems in image microscopy. The variational lower-level constraint enforces sparsity of the solution and encodes different noise statistics (Gaussian, Poisson), while the upper-level cost assesses optimality w.r.t.~the task considered. In more detail, a standard $\ell_2$ cost is considered for image reconstruction (e.g., deconvolution/super-resolution, semi-blind deconvolution) problems, while a smoothed $\ell_1$ is employed to assess localization precision in some exemplary fluorescence microscopy problems exploiting single-molecule activation. Several numerical experiments are reported to validate the proposed approach on synthetic and realistic ISBI data.

Figures (9)

• Figure 1: Binarization in [0,1] with $c = 0.5$, $\epsilon = 10^{-4}$ for three different values of $\delta$: 0.1 (blue), 0.25 (red), 0.49 (yellow).
• Figure 2: (a) Radius of correct detection for $\tilde{\delta} = 0$. (b) Radius of correct detection for $\tilde{\delta} = 2$. (c) Radius of correct detection for $\tilde{\delta} = 4$. (d) Example in the case of $\tilde{\delta} = 0$: the white pixel is the true molecule, the yellow pixel is counted as a TP, the red pixel is FP as it is outside of the range, while the orange pixel is still counted as a FP even if it is in range because the yellow one is closer to the true molecule.
• Figure 3: (a) Ground Truth $g_t$. (b) Acquired $f_t$. (c) Optimal reconstruction $u(\hat{\theta})$ obtained by optimizing $\mathcal{L}_2$\ref{['eq:LossL2']}. (d) Optimal reconstruction $B_{\hat{\delta},c,\varepsilon}(u^{(K)}(\hat{\theta}))$ obtained by optimizing $\mathcal{L}_1$\ref{['eq:LossL1']}.
• Figure 4: Columns from left to right: low-resolution (widefield) image, reconstruction computed by our approach, DeepSTORM Nehme_18 reconstruction. Datasets, from top to bottom: Gaussian, Poisson noise with low background and Poisson noise with high background. (a) represents the ground truth stack.
• Figure 5: Left: decrease of the loss functions $\mathcal{L}_1$ through the outer iterations (blue) and value of the binarization parameter $\delta$ (orange). Right: decrease of the inner energy functional (blue) and increase of Jaccard index (orange), through the inner iterations, computed as the mean across the 416 test samples, without applying the binarization.