Off-the-grid regularisation for Poisson inverse problems

TL;DR

This work develops a grid-less (off-the-grid) regularisation framework for Poisson inverse problems by coupling Total Variation regularisation with a Kullback-Leibler data term under nonnegativity constraints in the space of Radon measures ${\mathcal{M}(\Omega)}$. It provides a rigorous optimality/duality analysis for the Poisson KL-TV model, and introduces a Sliding Frank-Wolfe algorithm complemented by an algorithmic homotopy to automatically select the regularisation parameter ${\lambda}$. Through 1D, 2D, and 3D simulations and a real 3D fluorescence dataset, the authors demonstrate that the Poisson KL-TV setup yields superior spike localisation and amplitude accuracy compared to Gaussian-based BLASSO, with robust performance under Poisson noise. The proposed approach is particularly relevant for high-resolution molecular imaging (e.g., fluorescence microscopy), where photon-counting noise is inherently Poisson and off-the-grid reconstruction can mitigate discretisation biases. The combination of convex analytic insights, efficient SFW-based optimization, and a practical parameter-homotopy strategy offers a scalable, automated tool for sparse, grid-free spike deconvolution in imaging applications.

Abstract

Off-the-grid regularisation has been extensively employed over the last decade in the context of ill-posed inverse problems formulated in the continuous setting of the space of Radon measures $\mathcal{M}(\mathcal{X})$. These approaches enjoy convexity and counteract the discretisation biases as well the numerical instabilities typical of their discrete counterparts. In the framework of sparse reconstruction of discrete point measures (sum of weighted Diracs), a Total Variation regularisation norm in $\mathcal{M}(\mathcal{X})$ is typically combined with an $L^2$ data term modelling additive Gaussian noise. To asses the framework of off-the-grid regularisation in the presence of signal-dependent Poisson noise, we consider in this work a variational model coupling the Total Variation regularisation with a Kullback-Leibler data term under a non-negativity constraint. Analytically, we study the optimality conditions of the composite functional and analyse its dual problem. Then, we consider an homotopy strategy to select an optimal regularisation parameter and use it within a Sliding Frank-Wolfe algorithm. Several numerical experiments on both 1D/2D simulated and real 3D fluorescent microscopy data are reported.

Figures (10)

• Figure 1: Comparison between discrete (on-the-grid) and off-the-grid sparse reconstructions with a Poisson data term. In black: the ground-truth spikes to retrieve. In Fig.\ref{['fig:grid1']}: the acquired blurred and noisy signal $y\in\mathbb{R}^M$ lying on a low-resolution grid of size $M$. In Fig.\ref{['fig:grid2']}: in red, a discrete reconstruction with support on a grid with $M$ pixels. In Fig.\ref{['fig:grid3']}: in red, discrete reconstruction with support on a grid with $N>M$ pixels. In Fig.\ref{['fig:grid4']}: in green, off-the-grid reconstruction.
• Figure 2: Reconstructions obtained using SFW for a 1D sparse deconvolution problem with Poisson noise. Ground truth spikes (black) and reconstructed ones (green) using Algorithm \ref{['alg_sfw']} for some choices of $\lambda$ are shown. When $\lambda\ll 1$, the number and intensities of spikes are overestimated, while for $\lambda\gg 1$
• Figure 3: Homotopy algorithms explore the Pareto frontier iteratively for a strictly decreasing sequence of regularisation parameters $\lambda$. In blue: fine discretisation of the Pareto frontier with grid search. In red: homotopy iterations, corresponding to different values of $\lambda$. In gray: target value for the fidelity.
• Figure 4: 1D comparison between Gaussian (left) and Poisson (right) models. In black: ground truth spikes. In green: reconstructed spikes. For both models, $\lambda=8.82$.
• Figure 5: Mean values over 100 different randomly generated ground truth signals with 6 spikes and their corresponding reconstructions. Shaded area corresponds to standard deviation. Maximum number of iterations of SFW: $2N_{\text{spikes}}$. Tolerance radius for computation of the Jaccard Index is $\delta=0.05$.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Proposition 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• Remark 1