Eight years ago, Australian-born mathematician Terence Tao launched a completely new and highly sophisticated mathematical theory which may be set to bring about enormous savings in the health sector as well as in the oil industry. Tao is considered one of the world’s most eminent mathematicians.
Compressed sensing enables sampling to be carried out without having to look at the raw data first. Physicists at the University of Oslo, Norway are referring to the method as one of this century’s most significant mathematical discoveries. “The idea is to solve a task by involving as few measurements as possible. Whenever data capture is expensive, investment in this new mathematical approach may soon prove cost effective,” says Professor Anders Malthe-Sørenssen at the Department of Physics.
Colleague Andreas Solbrå is the first person at the University of Oslo to have put his theory to use. “Every measurement provides much more information than you think, provided you are smart about the sampling,” he says,
Faster hospital examinations
The method may be used to reduce the number of measuring points used for MR examinations to one-sixth of the present level. It also means that the MR imaging process may become six times faster than it is today. By being “smart” about selecting the sampling points for the MR scan, the method has been successfully tested at the American Lucile Packard Children’s Hospital in Stanford.
MR scanners are very expensive machines, with a price tag ion the range £1 to 2 million. The duration of an MR examination varies between 10 and 60 minutes, depending on what the doctors are looking for. They expose patients to doses of radiation which equal ten years’ natural background radiation.
This means that hospitals may examine six times as many patients without having to buy more scanners and increase their staffing levels. And there is reason to believe that the new mathematical method can reduce the level of radiation from CT scans by five-sixths.
The method is highly calculation-intensive: in order to compensate for the sparsity of measurements, more time must be spent on the calculations. “Today, doctors are able to analyse the medical images straight away. Compressed sensing will introduce a need for computer calculations before the results are ready,” adds Solbrå. “People who are tasked with implementing the compressed sensing method in medical diagnostics must be highly skilled in maths as well as calculation theory.”
The method has been tested on MR for quite a while, without resulting in its widespread adoption, which may be due to the complexity of the reconstruction. Nevertheless, one manufacturer of MR scanners has already implemented the method; however, the new imaging technique has been approved only for research purposes.
Even if compressed sensing may be set to revolutionize medical diagnostics, the method is not well known. “Most hospitals will probably not have heard about it yet”, says Solbrå. “In Norway very few research scientists are familiar with the method, but many will be able to benefit from the new method once they have figured out what it involves.”
Cheaper oil exploration
The new mathematical method may also bring about major savings for the oil industry. Petroleum geologists depend on seismic measurements which involve sending sound waves into the ground. The sound waves are reflected off the various sediment layers and are then caught by microphones on the surface.
“Seismic sampling is very costly. Compressed sensing enables us to collect fewer data points while ending up with equally good information,” says Malthe-Sørenssen.
Compressed sensing enables you to calculate all the things you dont measure
To illustrate the idea about performing as few measurements as possible, consider this classic party brain teaser: there are 12 coins. They look identical, but one of them is counterfeit, and is either heavier or lighter than the other coins. You need to establish which one is the fake by using a pair of scales. Most people can identify the counterfeit coin in four or five weighings. Incredibly, it is also possible to solve the task in only three weighings.
It is precisely this principle of sparse sampling which underpins the idea conceived by the Australian mathematician.