Golden Number Sampling Applied to Compressive Sensing

Abstract

In a common compressive sensing (CS) formulation, limited Discrete Fourier Transform samples of a signal allow someone to reconstruct it by using an optimization procedure provided that certain well-known conditions hold. However, the frequencies in the Discrete Fourier Transform correspond to equally spaced samples of the continuous frequency domain, and the other possible frequency distributions are not usually considered in compressive sensing. This paper presents an irregular sampling of the normalized frequencies of the Discrete Fourier Transform which converges to an equidistributed sequence. This is done by taking the sequence of the fractional parts of the successive multiples of the golden number. That sequence was considered in applications in computer graphics and in magnetic resonance imaging [1], [2]. We also show that sub-matrices of the Discrete Fourier Transform with frequencies corresponding to fractional parts of multiples of the golden number produce signal-to-error ratios almost as high as the equally spaced counterpart. In addition, we show that the proposed irregular sampling converges faster to a uniform distribution in the range (0, 1). Thus, it reduces the discrepancy of pairwise distances of consecutive elements in the frequency sampling.