A Toy Model For Tomography
In Chapter 12 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss tomography. The algorithms for solving the tomography problem involve calculus, Fourier analysis, and convolutions. I love all that mathematics, but some people don’t (😮). Is there a way to introduce tomography to students who don’t have a mathematical background?
Let’s start with a simple object divided into six pixels. Our goal is to determine the value of some property for each pixel. If this were positron emission tomography, this property would be the concentration of a radioactive substance. If this were a CT scan, this property would be the x-ray absorption. How we interpret the property doesn’t matter; we’re just going to assign a number to each pixel.
The Froward Problem
Suppose this is your object.
We assume what you can measure is the sum of the pixels along one direction: a projection.
You can take projections from different orientations; sum the pixels in that direction.
A tomography machine in the hospital measures projections.
The Inverse Problem
So far we have examined the forward problem: determine the projections from the object. Next, consider the inverse problem: determine the image from the projections. Here’s another example.
How do you obtain an image of the object from multiple projections? That is the fundamental problem of tomography. In other words, how do you figure out what number to put into each pixel so that it gives the projections shown above? Stop reading and try to guess the image. When you’re done, continue reading.
Perhaps you found this image.
Good job, but your friends working from the same projections may have found different images.
Check for yourself; they all have the same three projections. If you permit negative values, you can find even more images.
The values don’t have to be integers.
The number of solutions is infinite. What can we do to pick the correct image? Use more projections!
You’ll find only one image consistent with all six projections: the first one I listed (try it yourself). If you have enough projections, you can find a unique image. You’ve solved the problem of tomography.
If you add noise to the projections, you may have no solution. In that case, you would need to use something like the least squares method to estimate the image. But that’s another story.
If you have only six pixels (and no noise) you can compute the image using trial and error, and some logic, sort of like Sudoku. A medical image, however, might have tens of thousands of pixels! What do you do then? That’s exactly what Russ and I discuss in Chapter 12 of Intermediate Physics for Medicine and Biology.
I will leave you with a final example to solve on your own. Enjoy!
Originally published at http://hobbieroth.blogspot.com.