In iterative we have a matrix that is corresponding to the operator of projection. This operator contain the contribution of each pixel in each projection. It is known before reconstruction. How is it estimated? Could you give me all details?
Each element of the system matrix (that how the matrix is called) generally gives a probability of the emission from a voxel to be detected in a detector pair. There are several ways how you can estimate the probability. The most common is Length-Of-Intersection - the probability is proportional to the length of intersection of the voxel with the Line Of Response (line that connects two detector surfaces). This approach can be generalized to Surface-Of-Intersection (which is not that common) or to the Volume-Of-Intersection. The latter is actually more accurate. One can also account for solid angle effects (non-uniform probability of detection along Line-Of-Response or Tube-Of-Response), which further improves the system matrix simulation. Apart of these methods you can also measure the system matrix using a point source.
Edited: Have recognized that the question was generally on the tomography. What I wrote refers to PET, however, with certain level of re-interpretation it can be also applied to other modalities.
Many thanks for your response.
I understand this is in case we have a real simulation or we use platform as Gate platform?
In case of CT, if we want to estimate the system matrix of some projections (using (discrete) radon transform) of an discrete image, which formula we have to use?
Thank you in advance.
I would say it's a general approach, both for real and simulated data. All of this models will work, but will lead to different levels of reconstructed resolution (at least in PET it's like this).
I believe there is no formula that you can simply use, however, you can try Siddon's ray tracing algorithm for the system matrix computation. Here is a link to the original work (www.ncbi.nlm.nih.gov/pubmed/4048266). There are plenty of modifications of the algorithm which allow to improve it's performance.
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