![]() Rather than first sampling in high rate and then compressing the sampled data, we would like to directly sense the data in a compressed form. ![]() Compressed sensing takes this concept a step further it reduces complexity and the computational cost of the acquisition stage. ![]() Sparse approximation has paved the way towards many standard transform-coding schemes that exploit sparsity for compression, including JPEG, JPEG2000, MPEG, and MP3 standards. By storing only the values and locations of the nonzero coefficients, we get a compressed representation of the signal. Sparse representation means that a signal of length N can be represented with only S ≪ N nonzero coefficients. Consequently, practical solutions addressing these computational and storage challenges of working with high-dimensional data often rely on compression, which aims at finding the most concise representation that is able to achieve an acceptable distortion-one kind of popular approach for signal compression relies on finding a basis that provides a sparse, and thus compressible, representation of the signal. Despite the rapid growth of computational power, the acquisition and processing of signals in many fields continue to pose a great challenge. However, in many important real-life applications, the resulting Nyquist rate is so high that it is not viable, or even physically impossible, to build such a device that can acquire in this rate. By exploiting this property, much of the 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. The classic Nyquist–Shannon theorem on sampling continuous-time band-limited signals 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. ![]() Elad, in Handbook of Numerical Analysis, 2018 2.1 Compressed SensingĬompressed sensing (CS) is a recent, growing field that has attracted substantial attention in signal processing, statistics, computer science and other scientific disciplines.
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