Precise Semidefinite Programming Formulation of Atomic Norm Minimization for Recovering d-Dimensional (d≥2d\geq 2) Off-the-Grid Frequencies

Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In particular, atomic norm minimization was proposed in \cite{tang2012csotg} to recover $1$-dimensional spectrally sparse signal. However, in spite of existing research efforts \cite{chi2013compressive}, it was still an open problem how to formulate an equivalent positive semidefinite program for atomic norm minimization in recovering signals with $d$-dimensional ($d\geq 2$) off-the-grid frequencies. In this paper, we settle this problem by proposing equivalent semidefinite programming formulations of atomic norm minimization to recover signals with $d$-dimensional ($d\geq 2$) off-the-grid frequencies.Comment: 4 pages, double-column,1 Figur

Super-resolution Line Spectrum Estimation with Block Priors

We address the problem of super-resolution line spectrum estimation of an undersampled signal with block prior information. The component frequencies of the signal are assumed to take arbitrary continuous values in known frequency blocks. We formulate a general semidefinite program to recover these continuous-valued frequencies using theories of positive trigonometric polynomials. The proposed semidefinite program achieves super-resolution frequency recovery by taking advantage of known structures of frequency blocks. Numerical experiments show great performance enhancements using our method.Comment: 7 pages, double colum

Blind Two-Dimensional Super-Resolution and Its Performance Guarantee

In this work, we study the problem of identifying the parameters of a linear system from its response to multiple unknown input waveforms. We assume that the system response, which is the only given information, is a scaled superposition of time-delayed and frequency-shifted versions of the unknown waveforms. Such kind of problem is severely ill-posed and does not yield a unique solution without introducing further constraints. To fully characterize the linear system, we assume that the unknown waveforms lie in a common known low-dimensional subspace that satisfies certain randomness and concentration properties. Then, we develop a blind two-dimensional (2D) super-resolution framework that applies to a large number of applications such as radar imaging, image restoration, and indoor source localization. In this framework, we show that under a minimum separation condition between the time-frequency shifts, all the unknowns that characterize the linear system can be recovered precisely and with very high probability provided that a lower bound on the total number of the observed samples is satisfied. The proposed framework is based on 2D atomic norm minimization problem which is shown to be reformulated and solved efficiently via semidefinite programming. Simulation results that confirm the theoretical findings of the paper are provided

2D phaseless super-resolution

In phaseless super-resolution, we only have the magnitude information of continuously-parameterized signals in a transform domain, and try to recover the original signals from these magnitude measurements. Optical microscopy is one application where the phaseless super-resolution for 2D signals arise. In this paper, we propose algorithms for performing phaseless super-resolution for 2D or higher-dimensional signals, and investigate their performance guarantees