Professor Stewart's Hoard of Mathematical Treasures (18 page)

How to transform him.

The raw material for the Radon transform is a ‘function’ f defined on all points x of the plane. This means that f defines some rule, which, for any given choice of x, leads to a specific number f(x). Examples are things like ‘form the square of x’, in which case
f
(
x
) =
x
²
, and so on. The transform turns f into a related function F defined on lines in the plane. The value
F
(
L
) of
F
at some line
L
can be thought of as the average of
f
(x) as x runs along the line.

That’s not terribly intuitive (except to professionals), so I’ll restate it in terms of something that, in this computer age, may be more familiar. Consider a ‘black and white’ picture such as the photo of Radon on the opposite page. We can can associate a number with each shade of grey in the picture. So, if 0 = white and 1 = black, then
would be whatever grey you get by mixing equal amounts of black with white, and so on. These numbers determine a ‘grey scale’: the bigger the number, the darker the shade of grey. So points in Radon’s collar are at 0, most of his face is around 0.25 or so, his jacket is 0.5 or higher, and some of the shadows are close to 1.

We can associate a function f with the photo. To do so, let x be any point in the photo, and let f(x) be the number for the shade of grey at that point. For instance, f(point in collar) = 0, f(point in face) = 0.25, and so on. This function is defined at all points in the plane (within the edges of the photo). We can also reconstruct the photo from the function - in fact, that’s how the image is stored in a computer, give or take a few technicalities.

To define the Radon transform F, take any line in the plane - say the line marked
L
in the right-hand picture. Let
F
(
L
) be the average grey-scale value of the photo along line
L
. Here
L
cuts across Radon’s face, and the average is (say) 0.38. So
F
(
L
) = 0.38. The line M has a lot more dark grey along it, so maybe
F
(
M
) = 0.72. You have to do this for every possible line, not just these two: there’s a formula for the answer in terms of an integral.

Starting with a function and working out its Radon transform is straightforward, though a bit messy. It is less clear that, given the Radon transform, you can work out the function. Radon’s
key discovery is that this is possible, and he gave another formula for that calculation. It implies that, if all we know is the average grey-scale value along every line across Radon’s photo, then we can work out what Radon looks like.

What does any of this have to do with CAT scans?

Suppose a doctor could take a ‘slice’ of your body, along a plane, and make a grey-scale image of the tissues that the slice cuts. Dense organs would show up as dark grey, less dense ones as light grey, and so on. It would be just like a plane slice through a sort of ‘three-dimensional X-ray’ image. And it would tell the doctor exactly where your bodily tissues are, relative to that slice.

Unfortunately, no X-ray machine exists that can take that sort of picture
directly.
But what you can do is pass an X-ray beam - essentially, a straight line - through the body, and measure how strong the radiation is when it comes out at the far side. This strength is related to the average density of tissue - the average grey-scale value of the hypothetical slice - observed along that line. The greater the average tissue density, the weaker the emerging rays are. So, if you shone such a beam along every possible line in the slice plane, you would be able to work out the Radon transform of the grey-scale function for that slice. Then Radon’s formula would tell you the grey-scale function itself, and that would be a direct representation of the image created by the plane slice. That is, what that slice of you looks like in real space. So it’s a way to see inside solid objects.

In practice you can’t measure the Radon transform along every line, but you can measure it along enough lines to reconstruct a useful approximation to the image. (Many of the tweaks are to do with this loss of precision.) And this, give or take a few million dollars’ worth of technicalities,
²³
is what a CAT scanner does. You lie inside a machine that takes X-ray images from a series of closely spaced angles in a plane that slices
through your body. A computer uses tweaked versions of Radon’s formula, or related methods, to work out the corresponding cross-sectional image. The scanner does one more thing: it shifts you along a millimetre or so, and repeats the same process on a parallel slice. And then another, and then another ... building up a full three-dimensional image of your body.

Slices through a human head, made by a CAT scanner.

PET scans use similar technology, and are often performed using the same machines, but with positrons in place of X-rays. The patient is given a dose of a mildly radioactive version of a common body sugar, usually one called fluorodeoxyglucose. The sugar concentrates at different levels in different tissues. As the radioactive element decays, it emits positrons, and the more sugar there is in any location, the more positrons that region emits. The scanner picks up the positrons and measures how much activity there is along any given line. The rest is much as before.

If you ever need a medical scan, it could be worth bearing in mind that what makes it possible is some equations doodled by a mathematical physicist, and a formula discovered nearly a
hundred years ago by a pure mathematician interested in a technical question about integral transforms.

Mathematics is written for mathematicians.
Nicolaus Copernicus

 

Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game; we cannot ascertain whether the game is fair.
Tobias Dantzig

 

With me, everything turns into mathematics.
René Descartes

 

Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone-cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well placed internal charges.
Howard W. Eves

 

Nature’s great book is written in mathematical symbols.
Galileo Galilei

 

Mathematics is the queen of the sciences. Carl Friedrich Gauss

 

Mathematics is a language.
Josiah Willard Gibbs

 

Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
Richard W. Hamming

 

Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
Godfrey Harold Hardy

 

One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to
know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics.
Leon Henkin

 

Mathematics is a game played according to certain simple rules with meaningless marks on paper.
David Hilbert

 

Mathematics is the science of what is clear by itself.
Carl Gustav Jacob Jacobi

 

Mathematics is the science which uses easy words for hard ideas.
Edward Kasner and James Newman

 

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.
Johannes Kepler

 

In mathematics you don’t understand things. You just get used to them.
John von Neumann

 

Mathematics is the science which draws necessary conclusions.
Benjamin Peirce

 

Mathematics is the art of giving the same name to different things.
Henri Poincaré

 

We often hear that mathematics consists mainly of ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences’?
Gian-Carlo Rota

 

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Bertrand Russell

 

Mathematics is the science of significant form.
Lynn Arthur Steen

Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze.
James Joseph Sylvester

 

Mathematics transfigures the fortuitous concourse of atoms into the tracery of the finger of God. Herbert Westren Turnbull

 

In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs.
Stanislaw Ulam

 

God exists since mathematics is consistent, and the Devil exists since we cannot prove it.
Andre Weil

 

Mathematics as a science commenced when first someone, probably a Greek, proved propositions about ‘any’ things or about ‘some’ things, without specifications of definite particular things.
Alfred North Whitehead

 

Philosophy is a game with objectives and no rules. Mathematics is a game with rules and no objectives.
Anonymous

This story is told by the Cambridge analyst John Edensor Littlewood in his lovely little book A Mathematician’s Miscellany:

Littlewood says that he asked the Cambridge philosopher Ludwig Wittgenstein whether this was a profound philosophical joke, and he said it was.

It was early afternoon in Geronimo’s Pizzeria, and business was slow. Angelina, one of the servers, was amusing herself by piling pizza delivery boxes on top of each other on the edge of a table. It all looked rather precarious, and Luigi said as much.

‘I’m trying to see how far out I can make the pile go without the boxes actually falling off,’ Angelina explained. ‘I’ve discovered that with just three boxes, I can almost get the top one outside the line of the table.’