How to define the transpose of system matrix

[Hznnn]

ART Equation x^(k+1) = x^k + lambda * AT(b - A(x))/ATA

For low-dose CT reconstruction, how do I define a full-view fan-beam projection process A and AT without using kaczmarz function.

[ozanoktem]

Honestly, I don't understand your question. In ODL, Kaczmarz iterates (and any other iterative and/or optimisation based scheme) do not require information about how A (and A*) are digitised.

Regarding the projection operator A (and its transpose/adjoint A*), in ODL you typically use a helper function to construct A. This helper function needs a specification of the tomographic acquisition geometry.

[Hznnn]

How do I construct the projection operator A (and its transpose/adjoint A*) by using helper function.

[adler-j]

For constructing the operator, look at the examples here: https://github.com/odlgroup/odl/tree/master/examples/tomo

For finding the adjoint, simply call operator.adjoint on the constructed operator.

[Hznnn]

Are adjoint and transpose equal？ In fan beam projection, how to define a geometry with projection number 720 and detector unit number 860？

[ozanoktem]

Are adjoint and transpose equal？

For reconstruction, it is the adjoint you need. In most cases the adjoint coincides with the transpose.

[ozanoktem]

In fan beam projection, how to define a geometry with projection number 720 and detector unit number 860？

I suggest you check the documentation: https://odlgroup.github.io/odl/generated/odl.tomo.geometry.conebeam.FanBeamGeometry.html

[adler-j]

Quick answer is

geometry = odl.tomo.cone_beam_geometry(space, src_radius, det_radius, num_angles=720, det_shape=860)