from just a few samples. In fact, they could recover high-resolution pictures even when an MRI is not given enough time to complete a scan. To quote from Wired, “That was the beginning of compressed sensing, or CS, the paradigm-busting field in mathematics that’s reshaping the way people work with large data sets.”19

Despite being only a few years old, compressed sensing algorithms are already in use in some form in several hospitals in the country. For example, compressed sensing has been used clinically for over 2 years at Lucile Packard Children’s Hospital at Stanford. This new method produces sharp images from brief scans. The potential for this method is such that both General Electric and Phillips Corporation have medical imaging products in the pipeline that will incorporate compressed sensing.

However, what research into compressed sensing discovered is not just a faster way of getting MR images. It revealed a protocol for acquiring information, all kinds of information, in the most efficient way. This research addresses a colossal paradox in contemporary science, in that many protocols acquire massive amounts of data and then discard much of it, without much or any loss of information, through a subsequent compression stage, which is usually necessary for storage, transmission, or processing purposes. Digital cameras, for example, collect huge amounts of information and then compress the images so that they fit on a memory card or can be sent over a network. But this is a gigantic waste. Why bother collecting megabytes of data when we know very well that we will throw away 95 percent of it? Is it possible to acquire a signal in already compressed form? That is, Can we directly measure the part of the signal that carries significant information and not the part of the signal that will end up being thrown away? The surprise is that mathematical scientists provide an affirmative answer. It was unexpected and counterintuitive, because common sense says that a good look at the full signal is necessary in order to decide which bits one should keep or measure and which bits can be ignored or discarded. This view, although intuitive, is wrong. A very rich mathematical theory has emerged showing when such compressed acquisition protocols are expected to work.

This mathematical discovery is already changing the way engineers think about signal acquisition in areas ranging from analog-to-digital conversion, to digital optics, and seismology. In communication and electronic intelligence, for instance, analog-to-digital conversion is key to transducing information from complex radiofrequency environments into the digital domain for analysis and exploitation. In particular, adversarial communications can hop from frequency to frequency. When the frequency range is large, no analog-to-digital converter (ADC) is fast enough to scan the

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19 Jordan Ellenberg, 2010, Fill in the blanks: Using math to turn lo-res datasets into hi-res samples. Wired, March.



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