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MELISSA BLOCK, host:

From NPR News, this is ALL THINGS CONSIDERED. I'm Melissa Block.

MICHELE NORRIS, host:

And I'm Michele Norris.

And it's time now for All Tech Considered.

(Soundbite of music)

NORRIS: A story now about a bit of mathematical magic. It's a trick that in the future could shorten the amount of time you have to spend inside an MRI scanner. And it's based on something called sparsity. With the right math you can use sparsity to paint a full picture from a little bit of information. And that's good news for all kinds of high-tech devices, as NPR's Art Silverman explains.

ART SILVERMAN: This is a story about syphilis, math and guessing - and that's only part of it. I'm not very good at math, so let's start with the syphilis. And Jordan Ellenberg gets us going.

Professor JORDAN ELLENBERG (Math, University of Wisconsin-Madison): There's this problem that happens in the middle of the century when you're drafting hundreds of thousands of men to go fight in World War II.

SILVERMAN: Ellenberg is an associate professor of math at the University of Wisconsin in Madison. He says during World War II recruiters needed to find out which GIs had syphilis and pronto.

Prof. ELLENBERG: What can you do besides test them all? What you can do is this: You take the men and you put them in groups of 10. You take blood from the 10 men, you mix them, and you do the syphilis test on that.

SILVERMAN: Most of those mixed batches are going to come out negative. That is, it's sparse in the population.

Prof. ELLENBERG: But of course, every time one of those tests comes up positive, then you take those 10 men and test them individually. But as you can imagine, what you've done is drastically reduced the number of tests that you do.

SILVERMAN: The lesson - gather less information, not more. It works with music, too.

(Soundbite of music)

SILVERMAN: Does that tune sound familiar? Probably not. That's because when it was copied, only some of the information was captured. Normally when we make digital copies of sounds we take little sonic snapshots of it thousands a times a second. That's called sampling. But for the example we just heard, the music was sampled at only a few hundred times a second.

(Soundbite of music)

SILVERMAN: So our ears can't decode the original tune. But a computer with the right compressed sensing formula can, by intelligently filling in the blanks between the sparse samples.

(Soundbite of music)

Prof. ELLENBERG: And that's what it's supposed to sound like. That actually -that is not the original, that's the reconstructed version. But if I played you the original, it would sound exactly the same.

SILVERMAN: Jordan Ellenberg insists, that's not magic, it's just math. With "Mary Had a Little Lamb," the sparse information is just the right kind for the computer to know what it has to do to fill in the missing bits.

Prof. ELLENBERG: Whether you can get away with the information you didn't record, that actually depends in a serious way on exactly which part of the information you missed. Like, if you take all the vowels out of words, you can reconstruct the sentence, you can reconstruct the paragraph, you can reconstruct the whole page. But if you were to just say, like, cut off the lower half of the page, you have no chance of reconstructing.

SILVERMAN: And it's important to reconstruct the data correctly when it comes to the lifesaving, practical side of all this.

(Soundbite of MRI)

SILVERMAN: This is the sound of a typical closed MRI, a magnetic resonance imaging machine that twangs the atoms in your body to construct a picture of what's going on inside.

I had to get an MRI done once. It felt like they were putting me in a morgue drawer. The technician told me to stay still for an hour. She added, some people will freak out in there, so there's a little emergency buzzer to push if you panic. They then slipped me in.

(Soundbite of screaming)

SILVERMAN: I felt I was the victim of this scary Dutch film I once saw called "The Vanishing" and I was being buried alive in a box. Within a minute I pushed the emergency buzzer. I was out of there. If they could've gotten by by twanging fewer of my atoms, I might've been able to take it.

Professor EMMANUEL CANDES (Mathematics, Stanford University): Now, the problem with magnetic resonance imaging and one of the reason it's not used much more, it's because the acquisition process is slow. That is, it takes a long time.

SILVERMAN: That's Stanford University math professor Emmanuel Candes. MRI scientists came to him and his colleagues. They wanted the math guys to speed up their MRIs.

Prof. CANDES: And so they submitted to us a bunch of test images with just very few measurements. And they were asking us to try to take our best shot at what these images could be, given that we were given so little information about each image.

SILVERMAN: In effect, the math shortcut mimicked a full MRI scan. Candes and company used their complicated algorithms, this impossible dense world of symbols and manipulations whose terms include Nyquist sampling, discrete cosine transformations, computational harmonic analysis and, of course, the always popular uniform uncertainty principle.

Prof. CANDES: We ran one algorithm and something almost magical occurred, which is that when we ran the algorithm on images they were giving us, even though we were - they were hiding from us most of the information about this image, the algorithm seemed not to care and always returned exactly the image that they wanted us to recover.

SILVERMAN: But they're still in the testing phase. One experimenter predicts that within five years those tests will be done and real world products will be using the technology. And it's all because, as math professor Jordan Ellenberg frames it, machines are being allowed to, well, guess.

Prof. ELLENBERG: Oh, of course it's guessing. That's the great thing about modern mathematics, that we've started to understand that very often guesses are what you should do if you want to get the most accurate answer. People have this prejudice against guessing and I think it's quite important that people let go of that.

SILVERMAN: Agreed. So let's do this, guess how the story will end.

Art Silverman, NPR News. Good guess.

NORRIS: For more examples of sparsity, go to our All Tech blog. That's at NPR.org/alltech.

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