چكيده به لاتين
In this Thesis, 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 exists a dual certificate that is sufficient for exact recovery.
In the second work, 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 channel estimation, one has some additional information. 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 regular atomic norm minimization and 2-D MUSIC that do not consider prior information.
