# Is there any theory of the minimum amount of data for tomographic reconstruction?

I'm doing an experiment on synthetic data and I want to generate enough data but not too much. So I wonder if there is any rule for the minimum number of projection angles and detector count.

For example, in 2D parallel beam case, if the image size is [H, H], I usually set the detector count as int(1.4*H) to cover the image. But I'm not sure about the number of angles.

• In the CT lab I was working at, where we scanned non-live samples (so no fear of radiation damage), we would scan pi*H number of angles in a CBCT geometry. I don't know the particular maths of this, but the reason I was given was "that way we fully sample the Fourier (I suppose they meant Radon) space". Definetly the filtered backproojection-type of algorithms would do a good job on it with this sampling. Iterative algorithms are a different thing, they can do quite a decent job with considerably less data. Jan 28 '20 at 10:20

## 1 Answer

It's a well-known result that the 2d tomography problem is weakly ill-posed (singular values decay as $$O(1/\sqrt{n})$$ even with full data and strongly ill-posed (the singular values decay exponentially) if you don't have complete angular coverage.

See for example:

M. E. Davison. The Ill-Conditioned Nature of the Limited Angle Tomography Problem. SIAM Journal on Applied Mathematics 42(3):428-448, 1983. https://doi.org/10.1137/0143028

With respect to discrete samples and a discretized model for the density in the area being scanned, you'd have to regularize the solution (or use a coarse discretization which effectively regularizes the solution.) You can then analyze the resolution of the regularized solution. However, the resolution matrix isn't a measure of bias in the solution and you can't bound the bias without making assumptions about the smoothness of the solution.

Could you edit your question to add more context? What exactly are you trying to do?

• Actually, the problem is ill-posed even with complete angular coverage (the Radon transform is a compact operator). Jan 24 '20 at 23:37
• Thanks- corrected. Jan 25 '20 at 1:52
• I would have used "mildly" and "severely" instead of "weakly" and "strongly" for the degree of ill-posedness. Don't know if this is a language/translation issue, but I like these terms better as they reduce overloading of "weak" and "strong".
– Dirk
Jan 26 '20 at 13:44
• @Brian For the moment, what I want to do is to choose the reasonable number of angles for the synthetic test. Jan 27 '20 at 20:34
• Anyone who can elaborate more why the problem is ill-posed even with compute angular coverage? Jan 27 '20 at 20:36