Minimax Rates for Discrete Signal Recovery with Applications to Photonic Imaging

Music City Center 

We analyze the statistical problem of recovering a discrete signal, modeled as a k-atomic uniform distribution μ, from a binned Poisson convolution model. This question is motivated from super-resolution microscopy where precise estimation of μ provides insights into spatial configurations, such as protein colocalization in cellular imaging. Our main result quantifies the minimax risk of estimating μ under the Wasserstein distance for Gaussian and compactly supported, smooth convolution kernels. Specifically, we show that the global minimax risk scales with t^{-1/2k} for t→∞, where t denotes the illumination time of the probe, and that this rate is achieved by the method of moments and the maximum likelihood estimator. To address practical settings where atoms of μ may be partially separated, we also analyze a regime with structured clusters and show faster adaptive rates for both estimators and locally minimax optimality. As an application we use our methods on experimental STED microscopy data to locate single DNA origami. In addition, we complement our findings with numerical experiments that showcase the practical performance of both estimators and their trade-offs.

Gaussian Mixture Models

Method of Moments

Maximum Likelihood Estimation

Microscopy

Polynomial Root Stability

Chebyshev Systems 

IMS