Deterministic Sparse FFT Algorithms

The discrete Fourier transform (DFT) is a well-known transform with many applications in various fields. By fast Fourier transform (FFT) algorithms, the DFT of a vector can be efficiently computed. Using these algorithms, one can reconstruct a complex vector x of length N from its discrete Fourier transform applying O(N log N) arithmetical operations. In order to improve the complexity of FFT algorithms, one needs additional a priori assumptions on the vector x. In this thesis, the focus is on vectors with small support or sparse vectors for which several new deterministic algorithms are proposed that have a lower complexity than regular FFT algorithms. We develop sublinear time algorithms for the reconstruction of complex vectors or matrices with small support from Fourier data as well as an algorithm for the reconstruction of real nonnegative vectors. The algorithms are analyzed and evaluated in numerical experiments. Furthermore, we generalize the algorithm for real nonnegative vectors with small support and propose an approach to the reconstruction of sparse vectors with real nonnegative entries

Deterministic Construction of Binary, Bipolar and Ternary Compressed Sensing Matrices

In this paper we establish the connection between the Orthogonal Optical Codes (OOC) and binary compressed sensing matrices. We also introduce deterministic bipolar $m\times n$ RIP fulfilling $\pm 1$ matrices of order $k$ such that $m\leq\mathcal{O}\big(k (\log_2 n)^{\frac{\log_2 k}{\ln \log_2 k}}\big)$. The columns of these matrices are binary BCH code vectors where the zeros are replaced by -1. Since the RIP is established by means of coherence, the simple greedy algorithms such as Matching Pursuit are able to recover the sparse solution from the noiseless samples. Due to the cyclic property of the BCH codes, we show that the FFT algorithm can be employed in the reconstruction methods to considerably reduce the computational complexity. In addition, we combine the binary and bipolar matrices to form ternary sensing matrices ($\{0,1,-1\}$ elements) that satisfy the RIP condition.Comment: The paper is accepted for publication in IEEE Transaction on Information Theor