The magic lies, of course, in the mathematical sciences.
The magic lies, of course, in the mathematical sciences. Even though there may be millions of scenes that would reproduce the million pictures you took with your kaleidoscopic camera, it is highly likely that there will be only one sparse scene that does. Therefore, if you know the scene you photographed is information-sparse (e.g., it contains a heart and a kidney and nothing else) and measurement noise is controlled, you can reconstruct it perfectly. L1 minimization happens to be a good technique for zeroing in on that one sparse solution. Compressed sensing actually built on, and helped make coherent, ideas that had been applied or developed in particular scientific contexts, such as geophysical imaging and theoretical computer science, and even in mathematics itself (e.g., geometric functional analysis). Lots of other reconstruction algorithms are possible, and a hot area for current research is to find the ones that work best when the scene is not quite so sparse.
As with wavelets, seeing is believing. Compressed sensing has the potential to cut down imaging time with an MRI from 2 minutes to 40 seconds. Other researchers have used compressed sensing in wireless sensor networks that monitor a patient’s heartbeat without tethering him or her to an electrocardiograph. The sensors strap to the patient’s limbs and transmit their measurements to a remote receiver. Because a heartbeat is information-sparse (it’s flat most of the time, with a few spikes whose size and timing are the most important information), it can be reconstructed perfectly from the sensors’ sporadic measurements.
Compressed sensing is already changing the way that scientists and engineers think about signal acquisition in areas ranging from analog-to-digital conversion to digital optics and seismology. For instance, the country’s intelligence services have struggled with the problem of eavesdropping on enemy transmissions that hop from one frequency to another. When the frequency range is large, no analog-to-digital converter is fast enough to scan the full range in a reasonable time. However, compressed sensing ideas demonstrate that such signals can be acquired quickly enough to allow such scanning, and this has led to new analog-to-digital converter architectures.
Ironically, the one place where you aren’t likely to find compressed sensing used, now or ever, is digital photography. The reason is that optical sensors are so cheap; they can be packed by the millions onto a computer chip. Even though this may be a waste of sensors, it costs essentially nothing. However, as soon as you start acquiring data at other wavelengths (such as radio or infrared) or in other forms (as in MRI scans), the savings in cost and time offered by compressed sensing take on much greater importance. Thus compressed sensing is likely to continue to be fertile ground for dialogue between mathematicians and all kinds of scientists and engineers.