Understanding the maths of Computed Tomography (CT) scans
Noseman is having a headache and as an old-school hypochondriac he goes to see his doctor. His doctor is quite worried and makes an appointment with a radiologist for Noseman to get a CT scan.
Because Noseman always wants to know how things work he asks the radiologist about the inner workings of a CT scanner.
The basic idea is that X-rays are fired from one side of the scanner to the other. Because different sorts of tissue (like bones, brain cells, cartilage etc.) block different amounts of the X-rays the intensity measured on the other side varies accordingly.
The problem is of course that a single picture cannot give the full details of what is inside the body because it is a combination of different sorts of tissue in the way of the respective X-rays. The solution is to rotate the scanner and combine the different slices.
How, you ask? Good old linear algebra to the rescue!
We start with the initial position and fire X-rays with an intensity of 30 (just a number for illustrative purposes) through the body:
As can be seen in the picture the upper ray goes through areas 1, 2 and 3 and let』s say that the intensity value of 12 is measured on the other side of the scanner:
The rest of the formula is found accordingly:
We then rotate the scanner for the first time…
…which gives the following formula:
And a second rotation…
…yields the following formula:
Now we are combining all three systems of equations:
or written out in full:
Here is the data of the matrixfor you to download:ct-scan.txt).
We now have 9 equations with 9 unknown variables… which should easily be solvable by R, which can also depict the solution as a gray-scaled image… the actual CT-scan!
A <- read.csv("data/ct-scan.txt") b <- c(18, 21, 18, 18, 21, 9, 18, 14, 16) v <- solve(A, b) matrix(v, ncol = 3, byrow = TRUE) ## [,1] [,2] [,3] ## [1,] 9 9 0 ## [2,] 9 5 7 ## [3,] 9 9 0 image(matrix(v, ncol = 3), col = gray(4:0 / 4))
The radiologist looks at the picture… and has good news for Noseman: everything is how it should be! Noseman is relieved and his headache is much better now…
Real CT scans make use of the same basic principles (of course with a lot of additional engineering and maths magic 😉 )
Here are real images of CT scans of a human brain…
… which can be combined into a 3D-animation:
Isn』t it fascinating how a little bit of maths can save lives!
原文 : Learning Machines