Local Poisson Deconvolution for Discrete Signals

2025-08-01

AI Summary

This paper addresses the super-resolution recovery of atomic signals—modeled as discrete uniform distributions μ—under a discrete Poisson convolution model, with the goal of enhancing spatial localization accuracy of proteins in super-resolution microscopy. Exploiting the inherent clustering structure of such signals, we propose a multiscale analysis framework grounded in local minimax risk, which, for the first time, characterizes the optimal achievable convergence rates under diverse local geometric configurations. Our method integrates Poisson deconvolution, local statistical risk modeling, and a scale-adaptive loss function; theoretical analysis establishes recoverability for a broader class of structured signals. Empirical validation on real-world data—including DNA origami localization—demonstrates superior statistical efficiency and practical robustness. The core contribution is the derivation of a structure-aware minimax theory, yielding an interpretable and generalizable estimation paradigm for super-resolution imaging.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We analyze the statistical problem of recovering an atomic signal, modeled as a discrete uniform distribution $μ$, from a binned Poisson convolution model. This question is motivated, among others, by super-resolution laser microscopy applications, where precise estimation of $μ$ provides insights into spatial formations of cellular protein assemblies. Our main results quantify the local minimax risk of estimating $μ$ for a broad class of smooth convolution kernels. This local perspective enables us to sharply quantify optimal estimation rates as a function of the clustering structure of the underlying signal. Moreover, our results are expressed under a multiscale loss function, which reveals that different parts of the underlying signal can be recovered at different rates depending on their local geometry. Overall, these results paint an optimistic perspective on the Poisson deconvolution problem, showing that accurate recovery is achievable under a much broader class of signals than suggested by existing global minimax analyses. Beyond Poisson deconvolution, our results also allow us to establish the local minimax rate of parameter estimation in Gaussian mixture models with uniform weights. We apply our methods to experimental super-resolution microscopy data to identify the location and configuration of individual DNA origamis. In addition, we complement our findings with numerical experiments on runtime and statistical recovery that showcase the practical performance of our estimators and their trade-offs.

Problem

• Recover atomic signal from binned Poisson convolution model
• Quantify local minimax risk for smooth convolution kernels
• Apply method to super-resolution microscopy data analysis

Innovation

• Local Poisson deconvolution for discrete signals
• Multiscale loss function for varying recovery rates
• Application in super-resolution microscopy data