will be near zero, reflecting the absence of features at that particular scale. If the model in the photograph doesn’t have a blemish on a particular part of her skin, you won’t need the wavelet that would capture such a blemish. Thus you can compress the image by ignoring all of the wavelets with small weighting coefficients and keeping only the others. Instead of storing 10 million pixels, you may only need to store 100,000 or a million coefficients. The picture reconstructed from those coefficients will be indistinguishable from the original to the human eye.
After all, how can we know which 1 percent of information is the most relevant until we have acquired it all?
Curiously, wavelets were discovered and rediscovered more than a dozen times in the 20th century—for example, by physicists trying to localize waves in time and frequency and by geologists trying to interpret Earth movements from seismograms. In 1984, it was discovered that all of these disparate, ad hoc techniques for decomposing a signal into its most informative pieces were really the same. This is typical of the role of the mathematical sciences in science and engineering: Because they are independent of a particular scientific context, the mathematical sciences can bridge disciplines.
Once the mathematical foundation was laid, stronger versions of wavelets were developed and an explosion of applications occurred. Some computer images could be compressed more effectively. Fingerprints could be digitized. The process could also be reversed: Animated movie characters could be built up out of wavelets. A company called Pixar turned wavelets (plus some pretty good story ideas) into a whole series of blockbuster movies (see Figure 1).
In 2004, the central premise of the wavelet revolution was turned on its head with some simple questions: Why do we even bother acquiring 10 million pixels of information if, as is commonly the case, we are going to discard 90 percent or 99 percent of it with a compression algorithm? Why don’t we acquire only the most relevant 1 percent of the information to start with? This realization helped to start a second revolution, called compressed sensing.
Answering these questions might appear almost impossible. After all, how can we know which 1 percent of information is the most relevant until we have acquired it all? A key insight came from the interesting application of how to reconstruct a magnetic resonance image (MRI) from insufficient data. MRI scanners are too slow to allow them to capture dynamic images (videos) at a decent resolution, and they are not ideal for imaging patients such as children, who are unable to hold still and might not be good candidates for sedation. These challenges led to the discovery that MRI test images could, under certain conditions, be reconstructed perfectly—not approximately, but perfectly—from a too-short scan by a mathematical method called L1 (read as “ellone”) minimization. Essentially, random measurements of the image are taken, with each measurement being a randomly weighted average of many randomly selected pixels. Imagine replacing your camera lens with a kaleidoscope. If you do this again and again, a million times, you can get a better image than you can from a camera that takes a 10-megapixel photo through a perfect lens.