The mathematical CAT scan


1. The CAT Scan

The medical site Imaginis.net has a page on How does CT work and a very good Brief history of CT. The University of Colorado Physics2000 project has an excellent CAT Scans page with wonderful applets, especially the ``Rib Cage Projection''.

CAT scan of human head
A state-of-the-art CAT scan of a human head. The scan shows a diagonal slice going through the eyeballs and the nape. Note the sharp detail of bone structure and the differentiation between tissues.
The CAT in ``CAT scan'' stands for Computed Axial Tomography (the ``Axial'' is often dropped, leaving ``CT scan''). The procedure, some 25 years old (see the Imaginis.net history), has become a very widely used tool in diagnosis and in medical research.

The procedure has become so common in fact that it is difficult to realize how miraculous it should seem: a CAT scan gives an image of what an actual slice would look like (through the patient's head, for example, as in our illustration), but without any actual slicing.

The technique relies on the different X-ray penetrability of different tissues, just like conventional X-ray imaging. The difference is that here, instead of a single 2-dimensional X-ray picture, a series of 1-dimensional X-ray images, taken from different angles, is recombined mathematically (hence ``computed'') into a single image of a 2-dimensional slice. ``Tomography'' means literally ``writing the slice.''

In this column we will examine the mathematics that permits reconstructing a 2-dimensional cross-section of an object from a series of 1-dimensional samplings of its density. The mathematical technique is called the Radon Transform. We will examine a rough simulation of part of this procedure that can be carried out by hand, and demonstrated, if necessary, with poker chips on a checkerboard.

--Tony Phillips
SUNY at Stony Brook

© copyright 1999, American Mathematical Society.