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CT Scans are so common these days that practically everyone has heard of them and has either had one or knows someone who has. The high resolution images the machine produces are truly remarkable and allow physicians to spot tumors, hemorrhages, and bone trauma, among other maladies. But I suspect most people don’t know what “CT” stands for (computed tomography), or, if they do, they don’t know what it means.
Old-fashioned X-ray images provide useful, but coarse, information about our insides. The problem is that the rays pass through the body onto the film in such a way that each point on the image is the aggregate of the points in the body lying above it. Thus, more dense areas of the body appear lighter on the output image. Sometimes this is good enough–a large light area might correspond to a growth of some kind (a tumor, perhaps). X-ray images are also great for spotting broken bones.
But if your physician wants more detailed information, a CT scan might be warranted. Tomography is the imaging of an object by cross-sections. The basic procedure for medical tomography is to shoot X-rays through a thin cross-section of the body. The machine then computes the amount of energy that comes out the other side along each straight line. Mathematically, this means that if f(x,y) is the density of the body at the point (x,y) in the cross-section, and if L is the line the X-ray moves along, then the machine is gathering the various line integrals (here, z is the arc length parameter)
This is what an X-ray does, too, but instead of a computer gathering the integral data it is a piece of film catching the intensity of the X-ray that reaches it. What’s more, the lines L make some angle with the horizontal; by varying the angle, we get a collection of such data for each one. The end result is a function Rf, called the Radon transform of the density function f. It is a function of two variables, the distance s of L from the origin and the angle α that L makes with the horizontal.
So what? Well, the remarkable thing is that the Radon transform can be inverted; that is, if we know the function Rf, we can recover the function f ! This falls into the general area of inverse problems. The computer attached to the CT machine does this inversion, and the resulting images are then assembled together to give a remarkably accurate representation of the interior of the body.
Wow! Math can save lives.