An Introduction To Compressive Sampling

Abstract

onventional approaches to sampling signals or images follow Shannon's celebrated theorem: the sampling rate must be at least twice the maximum frequency present in the signal (the so-called Nyquist rate). In fact, this principle underlies nearly all signal acquisition protocols used in consumer audio and visual electronics, medical imaging devices, radio receivers, and so on. (For some signals, such as images that are not naturally bandlimited, the sampling rate is dictated not by the Shannon theorem but by the desired temporal or spatial resolution. However, it is common in such systems to use an antialiasing low-pass filter to bandlimit the signal before sampling, and so the Shannon theorem plays an implicit role.) In the field of data conversion, for example, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation: the signal is uniformly sampled at or above the Nyquist rate.

Compressed Sensing of Excitatory Postsynaptic Potential Bio-Signals

“…Once temporal redundancy is eliminated, a compressed sensing technique was applied to eliminate statistical redundancy in spare EPSPs [10]. Assuming that the signal to be compressed is x ∈ ℝ ே×ଵ and the compressed signal is y ∈ ℝ ெ×ଵ , then it can be expressed as follows:…”

Fully corrective boosting with arbitrary loss and regularization

“…1 norm regularization may create sparse answers and better approximations in relevant cases. 1 norm regularization methods have recently gained much attention in compressed sensing [23] and machine learning due to the induced sparsity and being easy-to-optimize as a surrogate of the non-convex 0 pseudo-norm [24,25]. For boosting algorithms, F (·) takes the form…”

Compressive Direction Finding Based on Amplitude Comparison

“…Compressive Sensing (CS) [5][6][7] theory asserts that one can recover certain signals from far fewer samples or measurements than traditional methods use, such as natural images or communications signals, have a representation in terms of a sparsity inducing basis (or sparsity basis for short) where most of the coefficients are zero or small and only a few are large. For example, smooth signals and piecewise smooth signals are sparse in a Fourier and wavelet basis, respectively.…”