# 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.

• 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
• 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