Computed Tomography Reconstruction

June 26, 2019
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!

Fourier Transforms

June 25, 2019
I have spent the last few weeks pondering what I should write about for my first article back. Should I go into something exotic in astrophysics such as magnetic monopoles or topological defects? Should I go into complicated mathematical research and paradoxes? Should I go into the bizarre world of quantum mechanics and its foundations? Nothing seemed quite right.

So I decided to write about Fourier transforms. That's right, after thinking about some of the strangest concepts in physics and ma...
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News & Updates

April 30, 2019
Today's article is going to be very different from my usual writing, in that it does not contain scientific or mathematics news, but rather news of a more personal nature. Ordinarily I would keep my private life private, but enough of my loyal readers have been asking about my recent absence that I feel it is worth giving a few updates.

For those who have not heard, on April 13th I was rushed to hospital for emergency surgery. I had been having abdominal pains for a few days before, but early ...
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Photographing A Black Hole

April 11, 2019
This article was not completed, for reasons that will be addressed in the next entry. 
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Canada's Lunar Gateway

March 1, 2019
And Canada is officially back in the space race, with the announcement today of two billion dollars over twenty-four years going to aerospace companies that will help to build the Lunar Gateway project.

For those who have not followed this project, Lunar Gateway is a plan by NASA to place a manned space station in polar orbit around the Moon, and to use it as a staging area for future missions to Mars or beyond. It will take two decades to complete, but when finished will provide both a labor...
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Opportunity Ends

February 15, 2019
Nearly fifteen years into what was meant to be a three month mission, NASA's Opportunity rover has finally ended its mission.

The Opportunity rover stopped communicating with Earth when a severe dust storm on the surface of Mars covered the rover last June. NASA engineers and technicians have sent over a thousand messages and commands to the rover since then in an attempt to recover it, but all of them have failed. On Tuesday they made one final attempt at communication, and when that was n...
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A Distant FlyBy

January 4, 2019
The new year is only a few days old, and already we have exciting news from the astronomy and astrophysics community with the announcement that the New Horizons probe has just completed the most distant flyby in history.

Many of you will remember the New Horizons probe for its flyby of the planet Pluto and its moons a while back in 2015, with the stunning photos of the icy planet still being studied and analyzed to this day. The data sent back by New Horizons from Pluto and Charon have reveale...
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Happy New Year!!

January 1, 2019
May you all have a happy and healthy 2019, full of enjoyment and prosperity. And may we all still be together again when the year ends and 2020 begins.

Happy New Year!!

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2018 Year In Review

December 31, 2018
Greetings all! I hope that you each had a relaxing and enjoyable holiday, in whichever form you celebrate it, and are refreshed and re-energized for the coming year!

It is also that time of year again when everyone looks back on the achievements of the year, and this year has been a good one for the scientific community. And so without further ado (or a don't), here we go...

This year has been particularly interesting for the astronomy community. Back in February we had the announcement that a ...
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Merry Christmas!

December 25, 2018

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About Me

Dr. Chris Bird I am a theoretical physicist & mathematician, with training in electronics, programming, robotics, and a number of other related fields.


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