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''.

A state-of-the-art CAT scan of a human head. The scan shows a diagonal slice going through the eyeballs and the nape. Notethe sharp detail of bone structure and the differentiation betweentissues.

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 torealize how miraculous it should seem: a CAT scangives 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. Thedifference 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 singleimage 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 froma series of 1-dimensional samplings of its density. The mathematicaltechnique is called the Radon Transform. We will examine a roughsimulation of part of this procedure that can be carried out by hand, anddemonstrated, if necessary, with poker chips on a checkerboard.

--Tony Phillips
SUNY at Stony Brook

• 1. The CAT scan

   

• 2. A numerical simulation: the analysis

   

• 3. A numerical simulation: the reconstruction

   

• 4. The Poker-chip CAT scan