In yesterday's article I demonstrated an intuitive explanation of the Fourier transform, which is used in nearly every branch of physics and mathematics. It allows an image to be stored in terms of its properties, such as average brightness and weighting of different regions of the image, and therefore is more useful for data analysis than a bitmapped image would be.

Today I will demonstrate one of the many applications of the Fourier transform. As promised, today I will be demonstrating how a computed tomography imaging machine, which is prevalent in hospitals and medical centers everywhere, automatically generates a Fourier transform of a patient's internal bits. (And I also apologize in advance to any experts in CT reconstruction algorithms who find this explanation too simplistic :) )

Let us begin by considering how a traditional x-ray image is created. This is the imaging process that everyone knows about and thinks about when talking about medical imaging - the patient stands in front of a piece of film, and a big, scary machine bombards them with high energy photons known as x-rays. X-rays that miss the patient hit the film with no change, and are recorded in the final image as black pixels. Where the patient is dense, such as in bones, the x-rays get blocked or scattered away, and the pixels in the film are white. And everywhere else in the patient, the number of x-rays gets reduced as some are absorbed by tissue, but some still hit the film and result in shades of gray, with lighter shades corresponding to denser tissues. This method is easy and affordable, and works very well for imaging bone fractures and certain tissue injuries, but it is limited because the x-rays interact with all materials in their path and not just the organ of interest.

That is where the computed tomography imaging methods become useful. In a CT scan, the patient is placed inside a tube that is surrounded by x-ray emitters and digital detectors. (Actually most CT scanners use a single emitter and a single detector, capable of recording a high resolution image in the x-ray spectrum, and rotate these around the patient at a high speed, but that is a technical detail that isn't important to today's discussion). Within a few seconds the CT scanner takes approximately eighty to a hundred separate x-ray images, covering a range of 180o. It is possible for a radiologist to then look at each of these images separately to analyse the patient from all angles - effectively the doctor has access to a hundred separate x-ray images. However there is a more interesting method of viewing these images.

Let us first consider a single x-ray image, taken with the emitter below the patient and the digital detector above the patient. (We will assume that the x-ray emitter creates parallel beams of x-rays, although some machines create a cone of x-rays and require a more complicated reconstruction method than is presented here). Each beam passes through the patient, getting reduced in intensity as it passes through different types and thicknesses of material (ie tissue, bone, assorted fluids).

When the beam arrives at the detector, its intensity is proportional to the average rate of absorption or scattering of the materials that it has passed through. If we think about a cross-section of the human being like a grayscale photograph, as shown above, then the detector is recording the average grayscale number along a vertical strip of this image. This is just a subset of the Fourier transform of the image!

If we average the values of each pixel in our detector, then we have the average value of the entire image - which in the notation that we used yesterday is F(0,0). If we average the right side of the detector and subtract the average of the left side of the detector, we have F(1,0). We can continue in this way to measure the values of F(k,0) for all values of k up to the resolution of the detector. We have effectively measured one strip of the Fourier transform of the cross-section of our patient.

Now we take a second image from our original CT scan. In this case we will consider the image taken when the x-ray emitter is on the patient's right side, and the detector is on their left side. In this case, each x-ray beam will measure the average grayscale value along a horizontal strip of the image, or the average absorption rate of the materials along this same horizontal strip.  

Using our Fourier transform notation once again, the average value of all the pixels in the detector gives us F(0,0). The average of the bottom half subtracted from the average of the top half gives us F(0,1). We can continue with this analysis until we have all values of F(0,k), with the maximum value of k corresponding to the resolution of the detector. This gives us another strip from the Fourier transform of the cross-section of the patient.

The next x-ray that we consider is only slightly more complicated. In this case, the x-ray beams travel from the lower right of the cross-section, and are detected in the upper left. This gives us the element of the Fourier transform that we referred to as F(1,1), in which the average of the bottom left quadrant and the top right quadrant of the original image were subtracted from the average of the other two quadrants. In fact this x-ray can be used in the same manner as the previous two x-rays to give us all values of F(k,k).
This process is continued for each of the x-ray images that were collected by the CT scanner, and eventually we are able to measure more strips from the Fourier transform of the cross-section. The first x-ray gave us a horizontal strip, the second gave us a vertical strip, and the third one gives us a diagonal strip.

By applying this same process to the remaining x-ray images that were collected by the CT scanner, we eventually have a good approximation for the entire Fourier transform of the cross-section of the patient,

and then by applying the usual mathematical methods of inverting a Fourier transform, we arrive at a grayscale image such as the one below,

which displays a detailed cross-section of the patients organs and bones, without having to cut into the patient or insert cameras anywhere. The doctor and other healthcare professionals can look at this image, and knowing which shades of gray correspond to which materials can determine and diagnose a large number of ailments and disorders. 

And all of this highly detailed non-invasive imaging and medical examination owes its existence to the power and simplicity of the Fourier Transform!