How to obtain projections from sinogram in ART reconstruction technique?

I'm kind new in the Computed Tomography field and I'm trying to understand and implement ART technique. Said it, I started to read the book The Mathematics of Medical Imaging - A Beginners Guide by Timothy G. Feeman. In the chapter 9, author describes the ART method. Well, If I understood fine (correct me If I'm wrong, please), in ART we are just trying to solve this system: Ax=p. One approach to solve it is to use Kaczmarz method (I read it and it is ok to understand it).

Now, let's understand each variable in the equation above: x is the image matrix that I want to found. A is the projection matrix and p is the sinogram. Is this reasoning right? If yes, how I obtain A from p? Can anyone show me an example with some numbers? (Ex: let A = [...], p = [....], etc.) A very simple example for the sake of simplicity would be fine.

Thanks in advance, guys!

• Welcome to SciComp.SE! Your interpretation is correct, but you don't find $A$ from $p$; $A$ is determined by the scanning methodology and geometry -- i.e., you have to fix this a priori depending on the specific tomography setup you wish to model. – Christian Clason Oct 9 '16 at 20:45
• Thanks for the response, Christian. Two questions: (1) What the CT machine produces and its available for me is the sinogram, right? (Parameter p above). (2) How to effectivaly obtain A? I read somewhere it was related to the pixels that were affected from each projection, but no idea how to mount its A matrix. Thanks!!! – Flávio Schuindt Oct 9 '16 at 20:58
• (1) That's right. (2) Mathematically, CT amounts to inverting the Radon transform of the image, so you should first read up on that (and its discretization; there are some explicit examples in Feeman's book). See also math.stackexchange.com/questions/1017965/… – Christian Clason Oct 9 '16 at 21:12
• @ChristianClason Just read it here and if I understood, I could do it: Let p be the sinogram (I have it because its provided by the CT machine), x be the image that I want to find (so x is a unknown variable) and A be the radon projection matrix. To obtain A I should do this: A = linsolve(reshape(image,1,[]), reshape(projection,1,[])); right? But the question here that is quite confused to me yet is: What I want in this whole history is find the image x. And to find A I`m still dependent in a equation in the image. So how to solve it? – Flávio Schuindt Oct 11 '16 at 3:22
• No, a single pair of image and projection is not nearly enough to determine the projection matrix (just count the degrees of freedom: if you have an $n\times m$ image and an $n\times p$ sinogram, the projection matrix is $(nm)\times(np)$. Setting up the matrix really is a separate step that needs concrete information about the actual measurement process you're trying to model. (Of course, for academic purposes, you are free to invent your own; to start with, you could use stackoverflow.com/questions/12166562/…) – Christian Clason Oct 11 '16 at 6:44