Superresolution without Separation

This paper provides a theoretical analysis of diffraction-limited superresolution, demonstrating that arbitrarily close point sources can be resolved in ideal situations. Precisely, we assume that the incoming signal is a linear combination of M shifted copies of a known waveform with unknown shifts and amplitudes, and one only observes a finite collection of evaluations of this signal. We characterize properties of the base waveform such that the exact translations and amplitudes can be recovered from 2M + 1 observations. This recovery is achieved by solving a a weighted version of basis pursuit over a continuous dictionary. Our methods combine classical polynomial interpolation techniques with contemporary tools from compressed sensing.Comment: 23 pages, 8 figure

Parameter selection in sparsity-driven SAR imaging

We consider a recently developed sparsity-driven synthetic aperture radar (SAR) imaging approach which can produce superresolution, feature-enhanced images. However, this regularization-based approach requires the selection of a hyper-parameter in order to generate such high-quality images. In this paper we present a number of techniques for automatically selecting the hyper-parameter involved in this problem. In particular, we propose and develop numerical procedures for the use of Stein’s unbiased risk estimation, generalized cross-validation, and L-curve techniques for automatic parameter choice. We demonstrate and compare the effectiveness of these procedures through experiments based on both simple synthetic scenes, as well as electromagnetically simulated realistic data. Our results suggest that sparsity-driven SAR imaging coupled with the proposed automatic parameter choice procedures offers significant improvements over conventional SAR imaging

Advanced parallel magnetic resonance imaging methods with applications to MR spectroscopic imaging

Parallel magnetic resonance imaging offers a framework for acceleration of conventional MRI encoding using an array of receiver coils with spatially-varying sensitivities. Novel encoding and reconstruction techniques for parallel MRI are investigated in this dissertation. The main goal is to improve the actual reconstruction methods and to develop new approaches for massively parallel MRI systems that take advantage of the higher information content provided by the large number of small receivers. A generalized forward model and inverse reconstruction with regularization for parallel MRI with arbitrary k-space sub-sampling is developed. Regularization methods using the singular value decomposition of the encoding matrix and pre-conditioning of the forward model are proposed to desensitize the solution from data noise and model errors. Variable density k-space sub-sampling is presented to improve the reconstruction with the common uniform sub-sampling. A novel method for massively parallel MRI systems named Superresolution Sensitivity Encoding (SURE-SENSE) is proposed where acceleration is performed by acquiring the low spatial resolution representation of the object being imaged and the stronger sensitivity variation from small receiver coils is used to perform intra-pixel reconstruction. SURE-SENSE compares favorably the performance of standard SENSE reconstruction for low spatial resolution imaging such as spectroscopic imaging. The methods developed in this dissertation are applied to Proton Echo Planar Spectroscopic Imaging (PEPSI) for metabolic imaging in human brain with high spatial and spectral resolution in clinically feasible acquisition times. The contributions presented in this dissertation are expected to provide methods that substantially enhance the utility of parallel MRI for clinical research and to offer a framework for fast MRSI of human brain with high spatial and spectral resolution

Enhancing SDO/HMI images using deep learning

The Helioseismic and Magnetic Imager (HMI) provides continuum images and magnetograms with a cadence better than one per minute. It has been continuously observing the Sun 24 hours a day for the past 7 years. The obvious trade-off between full disk observations and spatial resolution makes HMI not enough to analyze the smallest-scale events in the solar atmosphere. Our aim is to develop a new method to enhance HMI data, simultaneously deconvolving and super-resolving images and magnetograms. The resulting images will mimic observations with a diffraction-limited telescope twice the diameter of HMI. Our method, which we call Enhance, is based on two deep fully convolutional neural networks that input patches of HMI observations and output deconvolved and super-resolved data. The neural networks are trained on synthetic data obtained from simulations of the emergence of solar active regions. We have obtained deconvolved and supper-resolved HMI images. To solve this ill-defined problem with infinite solutions we have used a neural network approach to add prior information from the simulations. We test Enhance against Hinode data that has been degraded to a 28 cm diameter telescope showing very good consistency. The code is open source.Comment: 13 pages, 10 figures. Accepted for publication in Astronomy & Astrophysic

Statistical performance analysis of a fast super-resolution technique using noisy translations

It is well known that the registration process is a key step for super-resolution reconstruction. In this work, we propose to use a piezoelectric system that is easily adaptable on all microscopes and telescopes for controlling accurately their motion (down to nanometers) and therefore acquiring multiple images of the same scene at different controlled positions. Then a fast super-resolution algorithm \cite{eh01} can be used for efficient super-resolution reconstruction. In this case, the optimal use of $r^2$ images for a resolution enhancement factor $r$ is generally not enough to obtain satisfying results due to the random inaccuracy of the positioning system. Thus we propose to take several images around each reference position. We study the error produced by the super-resolution algorithm due to spatial uncertainty as a function of the number of images per position. We obtain a lower bound on the number of images that is necessary to ensure a given error upper bound with probability higher than some desired confidence level.Comment: 15 pages, submitte