"In the past decade tremendous research efforts focused on signals with specific features, especially sparse and low rank signals. Researchers showed that these signals can be recovered from much smaller number of samples than the Nyquist rate. These efforts were promising for several applications in which the nature of the data is known to be sparse or low rank, but the available samples are much fewer than what is required by the traditional signal processing algorithms to grant an exact recovery. Our objective in the first part of this thesis is to develop new algorithms for low rank data recovery from few observed samples and for robust low rank and sparse data separation using the Robust Principal Component Analysis (RPCA). Most current approaches in this class of algorithms are based on using the computationally expensive Singular Value Decomposition (SVD) in each iteration to minimize the nuclear norm. In particular, we first develop new algorithms for low rank matrix completion that are more robust to noise and converge faster than the previous algorithms. Furthermore, we generalize our recovery function to the multi-dimensional tensor domain to target the applications that deal with multi-dimensional data. Based on this generalized function, we propose a new tensor completion algorithm to recover multi-dimensional tensors from few observed samples. We also used the same generalized functions for robust tensor recovery to reconstruct the sparse and low rank tensors from the tensor that is formed by the superposition of those parts. The experimental results for this application showed that our algorithms provide comparable performance, or even outperforms, state-of-the-art matrix completion, tensor completion and robust tensor recovery algorithms; but at the same time our algorithms converge faster. The main objective of the second part of the thesis develops new algorithms for example based single image super-resolution. In this type of applications, we observe a low-resolution image and using some external "example" high-resolution - low-resolution images pairs, we recover the underlying high-resolution image. The previous efforts in this field either assumed that there is a one-to-one mapping between low-resolution and high-resolution image patches or they assumed that the high-resolution patches span the lower dimensional space. In this thesis, we propose a new algorithm that parts away from these assumptions. Our algorithm uses a subspace similarity measure to find the closes high-resolution patch to each low-resolution patch. The experimental results showed that DMCSS achieves clear visual improvements and an average of 1dB improvement in PSNR over state-of-the-art algorithms in this field. Under this thesis, we are currently pursuing other low rank and image super-resolution applications to improve the performance of our current algorithms and to find other algorithms that can run faster and perform even better."--Pages ii-iii.
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