Computed Tomography

Computed tomography (the CT or CAT scan) has become an absolutely fundamental tool in the modern diagnosis of disease. Unlike the X-Ray scan, which presents a two dimensional view of the general tissue density in a scanned region, a CT scan allows doctors to see a detailed 3D scan of a patient’s body, essentially peering into the body without exploratory surgery. Take a look at the image below to see the dramatic difference between a normal 2D X-ray (left) and a CT scan (right).

Principles behind Computed Tomography

CT scans work by taking multiple X-rays and using the “overlapping” information to combine them into a composite 3D image. From a single X-Ray scan, you really only get the equivalent of a “shadow” of the insides.

Defining the problem

The first mathematical tool we need to discuss is the “Radon transform.” The Radon transform is defined as “the integral transform which takes a function ‘f’ defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line.” 

 The Radon transform of an object is basically the raw data that the CT scanner collects during the scan, which is later transformed into a reconstruction of the slice of the patient later on.

In more mathematical terminology, the Radon transform is defined as:

Where the bottom line is the parameterization of the line integral in terms of a line at angle α.

CT-scanners basically “collect” the Radon transforms of slices of patients’ bodies and then reconstruct those transforms back into the slices that generated them.

Approaches

There exist both iterative and “exact” solutions to the Radon transform which we will go through, although only a handful are really used in practice.

First, let’s take the most basic approach. Imagine a slice of your patient as a two dimensional grid, where each pixel of the image can be represented by Cartesian coordinates (x, y). The intensity of an X-ray passing through the slice of your patient is basically the sum of the intensities of all the pixels that the ray passes through. So really, the radon transform represents a huge system of simultaneous linear equations. Luckily, these equations are extremely sparse, since simple geometry tells us that a ray will pass through about √n pixels on average, where n is the total number of pixels in the image.

In a perfect world, there exist exact solutions to these massive systems of equations which will yield exactly the image of the slice of the patient. In practice, iterative methods and approximations are used to solve such huge systems, since it’s computationally impractical to directly solve them. This method of solution is known as the “Simultaneous Algebraic Reconstruction Technique.”

Other methods take a sort of enhanced “guess and check” process, where an initial image of the slice is guessed randomly and transformed into what the CT would see if that guess was coghurrect. That transformation is then compared to what the CT actually saw. The guess is then modified slightly, and the process is repeated until the predicted transformation matches the observed data within a certain error bound.