Compressive Sensing:

Overview

•  Compressive sensing is a recent, growing field that has gained substantial attention in signal processing, statistics, computer science, and other scientific disciplines.
•  Compressive sensing is a mathematical theory with characteristics that disregard the physical-continuous time aspects of the signal and focuses instead on measuring or projecting finite dimensional vectors in RN to lower dimensional ones in RM and a framework that measures the signal by linearly projecting it to a known basis. The signal recovery process is also more involved and is typically achieved using convex-optimization-based recovery methods.
• Most signal processing has moved from the analog to the digital domain, creating sensing systems that are more robust, flexible, and cost-effective than their analog counterparts by exploiting the property of the Nyquist–Shannon theorem on sampling that asserts that signals can be recovered perfectly from a set of uniformly spaced samples, taken at a rate of twice the highest frequency present in the signal of interest.
• However, in many important real-life applications, the resulting Nyquist rate is so high that it is not possible to build a device that can acquire in this rate; and, despite the rapid growth of computational power, the acquisition and processing of signals continues to be a great challenge in both academic and commercial sectors.
• Therefore, practical solutions to resolve these computational and storage challenges of working with high-dimensional data often rely on compression, which focuses on finding the most concise representation that is still able to achieve an acceptable distortion.
• A popular approach for compressive sensing is sparse representation with a methodology that finds a basis that will provide a sparse and compressible representation of the signal. By storing only the values and locations of the nonzero coefficients, sparse representation results in a compressed representation of the signal. When a signal has a sparse representation in a known basis, it is possible to vastly reduce the number of samples that are required that is below the Nyquist rate to still be able to perfectly recover the signal (under appropriate conditions).
•  Compressive sensing has made important contributions to several fields, particularly those that use real-world images like medical imaging. For example, scanning sessions of MRI images can be significantly accelerated by measuring fewer Fourier coefficients and reconstructing the under-sampled MRI image, while still preserving its diagnostic quality.
•  Oher applications for compressed sensing in academic and commercial sectors include building efficient systems for sub-Nyquist sampling and filtering, compression of networked data, and compressive imaging architectures.