Compressed Sensing Off the Grid

Abstract

This work investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0,1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. Even with this continuous dictionary, it is shown that O(s log s log n) random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well separated. Extensive numerical experiments are performed to illustrate the effectiveness of the proposed method.

Compressed Sensing Of Complex Sinusoids Off The Grid

“…For each PSNR, 100 independent trials are executed. Reconstruction accuracy is measured by the "reconstruction signal-to-noise ratio" (RSNR), which is defined as RSNR = 20 log 10 x 2 x −x 2 (10) wherex is the estimate of x. Average RSNRs of different PSNR levels are shown in Fig.…”

“…As the computational cost is very high, L. Hu then developed a fast and accurate reconstruction algorithm [9], which applies a liner approximation to the true unknown dictionary. G. Tang has investigated compressed sensing off the gird and proposed an atomic norm minimization approach [10]. But it is provided that the frequencies are well separated.…”

High-Resolution Grid-Independent Imaging for Terahertz 2-D Synthetic Aperture Radar With Spatial Under-Sampling

“…This assumes both that the imaging scene consists of 2-D predefined grids and that all the scatterers are located exactly on these grids. Otherwise, those off-grid scatterers would severely affect the imaging performance [10,11].…”

“…However, CS-based reconstruction still belongs to the scope of discrete parameter estimation. Thus, it cannot avoid the off-grid phenomenon, and would lead to the well-known basis mismatch problem [10,11]. Besides, conventional methods for continuous parameter estimation [14,15], such as the estimating signal parameters via rotational invariance techniques (ESPRIT), and the matrix pencil (MP) method, which are free of grid dependence, would become less effective if the number of given samples is small by under-sampling.…”

A recursive way for sparse reconstruction of parametric spaces

“…In general approximation does not hold very well and off-grid problem is introduced. In the literature, the effect of this basis mismatch has been observed and analyzed in several studies [3]- [6].…”