Home
Research
Papers
Talks
Teaching
Group
Service
News

Inverse Problems and Super-Resolution Imaging

The goal of image inversion is to invert the source locations of the wavefields that produced the image acquired at the sensor suite, where the image broadly takes the form of a time series, a space series, a space-time series, a 2-D image, and so on. Typically, one is interested in super-resolution, such that the desired resolution for source locations exceeds the temporal or spatial resolution of the image itself. Conventional methods assume a separable linear model, wherein a sparse parametric modal representation for the source signal is posited and estimates of linear parameters (complex amplitudes of modes) and nonlinear mode parameters (frequency, wavenumber, delay, and/or Doppler) are extracted.

Compressive Sensing (CS) suggests that if a signal can be represented in some basis with a few nonzero coefficients (i.e. admits a sparse representation), then it can be reconstructed from a much smaller number of samples than the ambient dimension. This opens up exciting possibilities in signal processing to consider sparsity as a new prior in algorithm design to increase resolution and reduce complexity. However, the success of CS relies on the assumption that the signal is sparse in an a priori known basis, yet, in spectrum analysis or parameter estimation problems in radar and sonar, there is an inevitable mismatch between the basis assumed in CS and the true basis imposed by the physics of scattering. The performance of CS degenerates considerably in the presence of this basis mismatch, no matter how fine the grid is in the assumed basis.

To address this problem, we develop novel optimization-based algorithms for super-resolution imaging, based on structured matrix completion and atomic norm minimization. The versatility of the optimization framework allows tackling imaging modalities that are out of reach of traditional methods, particularly when the measurements are susceptible to missing data, outliers, calibration errors, poor models of the physical field, and interference from multiple sources. The proposed algorithms leverage physically-meaningful constraints of the propagation field as well as sparsity constraints of the source scene, admit provably accurate estimates under appropriate mathematical models, and provide important insights on the fundamental limits. Built on the theory, our group further designed tailored super-resolution algorithms for single-molecule fluorescence microscopy imaging, which produce high-resolution image reconstructions at a reduced computational time by an order-of-magnitude than the state-of-the-art.

Overview

Super-Resolution Off the Grid

Bilinear Inverse Problems and Blind Super-Resolution

Compressed Sensing and Basis Mismatch

Single-Molecule Localization Microscopy Imaging

Source Localization in Sensing Systems