Images obtained using traditional approaches relying on hardware are limited by the level of resolution achievable. However, there is an increasing demand for high image resolutions to enhance medical diagnosis, seismic hazard monitoring, and camera power saving that are limited by what hardware can support without increasing cost. In order to break the resolution bottleneck imposed by hardware, software-based super-resolution techniques that rely on machine learning are being developed that achieve unprecedented performance. However, even the most sophisticated techniques in this category lack solid theoretical guidance about how to choose tuning parameters or about under what conditions exact recovery and stability can be guaranteed. This project will address the theory behind a certain class of techniques, called convex super-resolution techniques for high-dimensional sensing geometries, and seek to determine theoretical limits and performance guarantees for problems such as exact and stable recovery. The successful outcome of this the project will help advance super-resolution techniques in imaging science.
Imaging techniques such as time-reversal suffer from a resolution bottleneck determined by the physical conditions of the imaging modality including limited source wavelengths, small azimuth in the acquisition geometry, and band-limited characteristics of the receiver arrays. The proposed research lies at the intersection of the fields of compressed sensing and inverse problems. Super-resolution is a long-standing demand in inverse problems and imaging science, while the involvement of sparsity is quite recent. Although sparsity is only rigorously shown to be useful to the super-resolution of 1D signals under direct Fourier measurements, the theory has the potential to be extended to high dimensional settings and to partial differential equation measurements. In this project, the Principal Investigator aims to build such a theoretical extension and to demonstrate that the resulting high dimensional inversion scheme can outperform a series of ad-hoc applications of the existing 1D inversion algorithms to each column of the data matrix. More explicitly, the following two types of optimization problems that lead to super-resolution will be analyzed: 1) the L1 norm regularized optimization framework for inverse source problems with physics-driven particle differential equation constraints; and 2) the total variation norm regularized optimization framework for image quantization under a finite bit budget.
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.