full range, and surveys of high-speed ADC technologies show that they are advancing at a very slow rate. However, compressed sensing ideas show that such signals can be acquired at a much lower rate, and this has led to the development of novel ADC architectures aiming at the reliable acquisition of signals that are in principle far outside the range of current data converters. In the area of digital optics, several systems have been designed. Guided by compressed sensing research, engineers have more design freedom in three dimensions: (1) they can consider high-resolution imaging with far fewer sensors than were once thought necessary, dramatically reducing the cost of such devices; (2) they can consider designs that speed up signal acquisition time in microscopy by orders of magnitude, opening up new applications; and (3) they can sense the environment with greatly reduced power consumption, extending sensor life. Remarkably, a significant fraction of this work takes place in industry, and a number of companies are already engineering faster, cheaper, and more-efficient sensors based on these recently developed mathematical ideas.

Not only is compressed sensing one of the most applicable theories coming out of the mathematical sciences in the last decade, but it is also very sophisticated mathematically. Compressed sensing uses techniques of probability theory, combinatorics, geometry, harmonic analysis, and optimization to shed new light on fundamental questions in approximation theory: How many measurements are needed to recover an object of interest? How is recovery possible from a minimal number of measurements? Are there tractable algorithms to retrieve information from condensed measurements? Compressed sensing research involves the development of mathematical theories, the development of numerical algorithms and computational tools, and the implementation of these ideas into novel hardware. Thus, progress in the field involves a broad spectrum of scientists and engineers, and core and applied mathematicians, statisticians, computer scientists, circuit designers, optical engineers, radiologists, and others regularly gather to attend scientific conferences together. This produces a healthy cycle in which theoretical ideas find new applications and where applications renew theoretical mathematical research by offering new problems and suggesting new directions.

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