Compressive Sensing in Signal Processing: Algorithms and Transform Domain Formulations

Abstract

Compressive sensing has emerged as an area that opens new perspectives in signal acquisition and processing. It appears as an alternative to the traditional sampling theory, endeavoring to reduce the required number of samples for successful signal reconstruction. In practice, compressive sensing aims to provide saving in sensing resources, transmission, and storage capacities and to facilitate signal processing in the circumstances when certain data are unavailable. To that end, compressive sensing relies on the mathematical algorithms solving the problem of data reconstruction from a greatly reduced number of measurements by exploring the properties of sparsity and incoherence. Therefore, this concept includes the optimization procedures aiming to provide the sparsest solution in a suitable representation domain. This work, therefore, offers a survey of the compressive sensing idea and prerequisites, together with the commonly used reconstruction methods. Moreover, the compressive sensing problem formulation is considered in signal processing applications assuming some of the commonly used transformation domains, namely, the Fourier transform domain, the polynomial Fourier transform domain, Hermite transform domain, and combined time-frequency domain.

On some common compressive sensing recovery algorithms and applications

“…and 1 − ℑ is the inverse transform matrix. Different transform domains can be used: discrete Fourier transform domain -DFT [3], [6], [12], discrete cosine transform domain -DCT [3], [6], [12], [46], wavelet domain, Hermite transform domain [48]- [55], time-frequency domain [56], etc. Apart from the sparsity, another important property is incoherence, which enables successful signal reconstruction from small set of acquired samples.…”

“…They provide high reconstruction accuracy, but they are computationally demanding. The commonly used and less computationally demanding are greedy algorithms -Matching Pursuit and Orthogonal Matching Pursuit [1]- [6], [8], [12]. Also, recently proposed threshold based algorithms provide high reconstruction accuracy with low computational complexity: (e.g.…”

“…Iterative Hard Thresholding -IHT, Iterative Soft Thresholding -IST) [1], [6], [13], Automated Threshold Based Iterative Solution [12], [57], Adaptive Gradient-Based Algorithm, [3], [12], [26], [27], etc.…”

“…They are much faster than algorithms based on convex relaxation. An iteration can be described in terms of threshold function as [1], [6], [11], [13]:…”

“…IHT algorithm sets all but the K largest components, in terms of signal X magnitudes, to zero. The hard thresholding function HK is defined as [6], [11], [13]:…”