Mathematical Analysis of Super-Resolution via Nonconvex Optimization and Machine Learning

New York University, New York NY

Investigators

Abstract

Diffraction imposes a fundamental limit on the resolution of optical systems. Consequently, in fields such as microscopy, astronomy, and medical imaging, it is often challenging to discern cellular structures, far-away stars, or tumours from the available measurements. The issue also arises in electronic imaging, where shot noise constrains the minimum pixel size, and in other applications, including signal processing, spectroscopy, radar, and seismology. The goal of super-resolution is to meet this challenge, uncovering fine-scale structure from coarse-scale data. In this project the investigators will analyze super-resolution techniques, design new methodology based on the resulting insights, and apply the methodology to fluorescence microscopy, which has become an essential imaging tool in biology. The proposed integrated program of educational and research activities will impact workforce development by training students at the intersection of data science, signal processing, and machine learning. This will contribute to address the rising demand for data scientists and engineers in industry and academia. Modern super-resolution techniques based on nonconvex optimization provide model flexibility, computational efficiency, and yield good empirical results. However, theoretical analysis showing under what conditions these techniques are guaranteed to work, or may fail, is lacking. In addition, recent works show that learning-based methods based on neural networks can be trained to perform super-resolution effectively and efficiently. Calibrating these models requires minimizing a highly nonconvex cost function. The proposed research activities will advance the theoretical underpinnings of nonconvex optimization for super-resolution and related problems such as line-spectra estimation and blind deconvolution. The project will focus on the super-resolution of point sources, which may represent fluorescent particles in microscopy, astral bodies in astronomy, spectral lines in signal processing, or neuron action potentials in neuroscience. The investigators will perform a mathematical analysis of the geometric landscapes that arise when using nonconvex cost functions to fit point sources, and also study the properties of learning-based approaches when deployed on point-source signal models. To complement their theoretical investigations, they will apply these techniques to real fluorescence-microscopy data. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.