arXiv subjects

Sajad Daei

Publications and source records attributed to Sajad Daei.

40 records · Page 3Linked to original sources

arXiv2018

Sample Complexity of Total Variation Minimization

This work considers the use of Total variation (TV) minimization in the recovery of a given gradient sparse vector from Gaussian linear measurements. It has been shown in recent studies that there exist a sharp phase transition behavior in TV minimization in asymptotic regimes. The phase transition curve specifies the boundary of success and failure of TV minimization for large number of measurements. It is a challenging task to obtain a theoretical bound that reflects this curve. In this work, we present a novel upper-bound that suitably approximates this curve and is asymptotically sharp. Numerical results show that our bound is closer to the empirical TV phase transition curve than the previously known bound obtained by Kabanava.

cs.IT↗

arXiv2017

Two-Dimensional Super-Resolution via Convex Relaxation

In this paper, we address the problem of recovering point sources from two dimensional low-pass measurements, which is known as super-resolution problem. This is the fundamental concern of many applications such as electronic imaging, optics, microscopy, and line spectral estimation. We assume that the point sources are located in the square $[0,1]^2$ with unknown locations and complex amplitudes. The only available information is low-pass Fourier measurements band-limited to integer square $[-f_c,f_c]^2$. The signal is estimated by minimizing Total Variation $(\mathrm{TV})$ norm, which leads to a convex optimization problem. It is shown that if the sources are separated by at least $1.68/f_c$, there exist a dual certificate that is sufficient for exact recovery.

cs.IT↗

arXiv2017

Off-the-grid Two-Dimensional Line Spectral Estimation With Prior Information

In this paper, we provide a method to recover off-the-grid frequencies of a signal in two-dimensional (2-D) line spectral estimation. Most of the literature in this field focuses on the case in which the only information is spectral sparsity in a continuous domain and does not consider prior information. However, in many applications such as radar and sonar, one has extra information about the spectrum of the signal of interest. The common way of accommodating prior information is to use weighted atomic norm minimization. We present a new semidefinite program using the theory of positive trigonometric polynomials that incorporate this prior information into 2-D line spectral estimation. Specifically, we assume prior knowledge of 2-D frequency subbands in which signal frequency components are located. Our approach improves the recovery performance compared with the previous work that does not consider prior information. Through numerical experiments, we find out that the amount of this improvement depends on prior information we have about the locations of the true frequencies.

cs.IT↗

arXiv2017

How to exploit prior information in low-complexity models

Compressed Sensing refers to extracting a low-dimensional structured signal of interest from its incomplete random linear observations. A line of recent work has studied that, with the extra prior information about the signal, one can recover the signal with much fewer observations. For this purpose, the general approach is to solve weighted convex function minimization problem. In such settings, the convex function is chosen to promote the low-dimensional structure and the optimal weights are so chosen to reduce the number of measurements required for the optimization problem. In this paper, we consider a generalized non-uniform model in which the structured signal falls into some partitions, with entries of each partition having a definite probability to be an element of the structure support. Given these probabilities and regarding the recent developments in conic integral geometry, we provide a method to choose the unique optimal weights for any general low-dimensional signal model. This class of low-dimensional signal model includes many popular examples such as $\ell_1$ analysis (entry-wise sparsity in an arbitrary redundant dictionary), $\ell_{1,2}$ norm (block sparsity) and total variation semi-norm (for piece-wise constant signals). We show through precise analysis and simulations that the weighted convex optimization problem significantly improves the regular convex optimization problem as we choose the unique optimal weights.

cs.IT↗