New Data Dependent Mathematical Methods to Achieve Super-Resolution
Abstract
Super-resolution has become a catch-all term in applied mathematics, image processing, and other fields. Generally, one wants finer information about an image than seems possible to discern from a handful of available snapshots of that image. If we distinguish between spectral and spatial super-resolution, then spectral super-resolution can be subsumed in the classical theory of spectral estimation that includes topics such as maximum entropy. In fact, spectral super-resolution is concerned with recovering fine details or high frequencies from given coarse or low-frequency information. As we noted in the proposal, our focus is on that aspect of spectral super-resolution dealing with total variation (TV) minimization. As such, our spectral super-resolution problem for spectra in the circle group is a generalization of the basis pursuit algorithm for under-sampled discrete Fourier transform (DFT) data. Consequently, TV minimization can be considered in terms of a continuous theory of compressed sensing. This does not make solving the problems or doing effective computation any easier. The major goal of this grant as formulated in our proposal is to capture an image optimally from given spectral information about the image on a small set of frequencies. This is an ill-posed realistic predicament. To further complicate matters, even in this spectral case, progress for some specialists open problems is not necessarily the ultimate goal of others, especially with regards to noise and modeling. Our proposal to do super-resolution brought to the table a new way of interpreting old, deep ideas of Beurling in the context of recent significant progress of Cand{`e}s and Fernandez-Granda. As background for {it spatial super-resolution}, we think in terms of spatial interpolation that arises in image processing, e.g., with in-painting problems, where we have significant experience with our colleague Professor Wojciech Czaja. Such problems naturally require non-uniform sampling technology as well as reconciling multiple measurements, where one must consider registration, warping, and the like. We have made basic contributions with non-uniform sampling, especially as regards balayage and Fourier frames, going back to Beurling and Henry Landau. As background for {it spectral super-resolution} or {it TV minimization}, we think in terms of spectral extrapolation that arises in optical or diffraction problems or, e.g., with MRI, where we have contributed in the setting of finite frame theory and computation (see the chapter [BNPW] with Nava-Tudela, Powell, and Y. Wang uploaded in the section, Publications for this Proposal). Generally, spectral super-resolution techniques are employed to capture information beyond the inherent resolution limit of a given measurement system, when one has access to low resolution data. These problems abound in microscopy, astronomy, and many facets of medical imaging, such as MRI. In other applications, these spectral techniques are used to extrapolate missing information, in cases where obtaining new samples is expensive or impossible. The fields of radar, neuroscience, or geophysics provide a host of examples. Our proposal was formulated listing several broad problem areas and a number of specific problems. We suggested techniques and ideas for addressing all of the problems. Generally, the problems fell into one of the following categories: 1. Reconstruction of singular continuous measures in image processing; 2. Spectral super-resolution and noise reduction; 3. Spatial super-resolution: 4. Deterministic sampling masks and effective computation.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 14, 2019
- Source ID
- W911NF1710014
Entities
People
- John Benedetto
Organizations
- Army Contracting Command
- United States Army
- University of Maryland