Description
In computerized tomography (CT) the measurement process can be modeled using the Radon transform, which maps the unknown material density to the corresponding absorption loss.
In Nano-CT the scale is so small that even movements of the measuring apparatus lead to unwanted rigid movement of the scanned object. Ignoring this and just building the linear radon operator $A^{\gamma_{0}}:X\to Y$ with the assumed motion $\gamma_{0}$ leads to artifacts in the reconstruction. The true motion $\gamma^{\dagger}$ is not known, i.e. we can not construct the correct linear Radon operator $A^{\gamma^{\dagger}}:X\to Y$ and have to put the motion $\gamma\in\Gamma$ as to be determent parameters $A:X \times \Gamma\to Y$.
Furthermore we are interested X-ray phase contrast. This can also be modeled using the Radon transform, but the phase information can not be measured and needs to be computed first. This phase retrieval leads to artifacts of its own, partially due to fluctuations in the illumination. In conclusion, the input data is misaligned and has background artifacts. Both need to be addressed for sufficient image quality.
The currently used re-projection alignment algorithm uses re-projected filtered back projections and image registration to reconstruct shift and object. We improve on the algorithm by using a thresholded version of normalized cross correlation for the image registration and imposing additional constraints, specifically a non-negativity constraint on the object, smoothness on movement and taking the uncertainty in low frequencies due background data artifacts into account.
We illustrate the algorithm on measurements of nano-porous glass. The data was recorded at the Göttinger Instrument for Nano-Imaging with X-rays (GINIX) operated by the Salditt group (University of Göttingen) located at the P10 beamline at the PETRA III storage ring at DESY in Hamburg.
