Thread: Tomography problem: is the result of a CT-scan unique?

Hello,
I hope someone can help me with my problem. I was reading something about how a CT-scan works, and how an image is made.

After the scanning is done, the result is the amount of radiation that passed all the internal tissue (when he object is for example a human body) in it's path, and this is done 360 degrees around the object.

Now, through tomography a 2D image is made. The way I see it (I don't know if this is right), the image is made by filling in the pixels where the intersection of the object is made of. The colour of the pixel gives the amount of radiation that is absorbed.

Now here is my problem. How do you know that the final image that is created by the algorithm is unique? Aren't there more possibilities if you only know the total radioation that is absorbed? or are there just so many pixels that there can only be one solution?

I hope someone can help me, or explain to me on what part I'm wrong (or do I think to easily?)

thanks!

This is a matter (as algebra teachers tell us) of "solving for simultaneous equations", and its solution requires having more equations than unknowns.

With CT scans, the "equations" involve the strength of the signal that the detector receives after the radiation beam passes through various elements ("pixels" as you call them) and the unknowns are the absorptions of the elements to the radiation.

So, in this super-simplified example, consider this cross-section with these densities of its elements.

Code:

7 3 4 4
5 8 1 3
7 8 4 2
1 2 3 1

The CT scan software doesn't know these values yet, so it calls them

Code:

a b c d
e f g h
i j k l
m n o p

If we shoot a beam down the leftmost column, it's strength would be reduced by 20 (that is, 7+5+7+1), and the software would create the equation a+e+i+m=20. Scanning from left to right also produces b+f+j+n=21, c+g+k+o=12, and d+h+l+p=10. Scanning across the rows generates the equations a+b+c+d=18, e+f+g+h=17, i+j+k+l=21, and m+n+o+p=7. But this gives us only 8 equations for 16 unknowns ... not enough equations for a [unique] solution.

So let's "scan" the cross section from other angles. Diagonally, we get m=1, i+n=9, etc, etc, c+h=7, and d=4. And from another angle, we get a=7, b+e=8, etc, etc, l+o=5, and p=1. In all, we get 22 equations to solve for 16 unknowns —this allows us to solve for the unknowns (which I won't do here).

This article in the Journal of Nuclear Medicine and Technology is a much better description of what I gave here. Please note that the figures show a graphical method of "combining" the data from the scans to create a solution which is not the same method as solving for simultaneous equations. (For example, in the graphical method, too few scans results in reconstructed parts of the image that don't actually exist, but too few scans in solving for simultaneous equations would result in several [obviously non-unique] solutions.)

So, to answer your question, a CT scanner will produce a unique solution because it's programmed to scan finely enough and from enough angles to solve for a certain number of elements. That's why it take so long to conduct a scan.

That was very helpful, thank you so much for taking the time to explain!