Mathematical theory of blind super-resolution: multidimensional cases
Creators: name: Suliman, Mohamed Abdalla Elhag
Publisher: name: Imperial College London
Year: 2021

description: Super-resolution is a fundamental problem in signal processing that concerns with extracting fine-scale data from low-resolution information. In this thesis, we focus on developing atomic norm-based algorithms for blind super-resolution. Due to the extra degrees of freedom in blind super-resolution problems resulting from the additional sources of blurring via unknown linear transformations, such problems are known to be ill-posed and very challenging to solve. To address blind super-resolution problems, we usually need to introduce additional structural assumptions on our signal model to resolve the ill-posedness issue. We start by considering the problem of identifying the unknown continuous parameters of a linear system from its response to multiple unknown input waveforms. For this problem, we assume that the system response is a scaled superposition of time-delayed and frequency-shifted versions of the unknown waveforms. Such mathematical formulation is a generic model that appears in a wide range of signal processing, wireless communication, and machine learning applications, such as radar imaging, image restoration, target localization, multi-user communication systems, medical imaging, and astronomy. We develop a general mathematical framework for blind two-dimensional (2D) super-resolution that estimates the continuous 2D shifts (i.e., time and frequency shifts) as well as other unknown parameters that characterize the linear system in the noise-free case. In this framework, we show that under a minimum separation condition between the time-frequency shifts, all the unknowns can be precisely recovered with very high probability provided that a lower bound on the total number of the observed samples is satisfied and that the unknown waveforms lie in a common known low-dimensional subspace that satisfies certain random assumptions. We refer to this problem as super-resolution as estimating the 2D continuous shifts breaks the natural resolution limit of any standard estimation algorithm that is based on discretizing the domain and we call it blind since the input waveforms are assumed to be unknown. Our mathematical framework is based on the 2D atomic norm minimization problem which is reformulated and solved efficiently via semidefinite programming. Following that, we demonstrate how our blind 2D super-resolution framework can be easily extended to higher dimensions by considering the problem of blind 3D super-resolution to recover the unknown continuous 3D shifts as well as unknown amplitudes in a mixture of unknown waveforms using the received signal only. We show that by solving a convex atomic norm minimization problem, and with the fact that the number of observed samples obeys certain complexity bound, an exact recovery for the unknowns holds provided that the 3D shifts are sufficiently separated. Finally, we develop a new mathematical framework for denoising in blind 2D super-resolution upon using the atomic norm. The framework denoises the noisy measurements of a signal that consists of a weighted sum of an unknown number of time-delayed and frequency-shifted unknown waveforms and then estimates its unknown parameters. We show that by solving a regularized least-squares atomic norm minimization problem, we can recover the noise-free signal with very high accuracy under certain assumptions. Moreover, we derive the theoretical mean-squared error of the estimator, and we investigate its relation to the noise level and other system parameters. The theoretical findings for each of the frameworks mentioned above are verified using extensive experiments.