Statistical Reconstruction For Anisotropic X-ray Dark-Field Tomography

TL;DR

This work addresses the challenge of reconstructing anisotropic X-ray dark-field signals in AXDT with correct noise modeling. It introduces numerically stable implementations of the statistical reconstruction framework and a simplified yet effective model $m3$ that preserves noise assumptions while enhancing efficiency. The authors provide convergence analysis via Lipschitz bounds and demonstrate superior noise performance over linear approaches on datasets including crossed sticks and brain tissue. The results suggest that advanced statistical reconstructions can achieve high-quality microstructure imaging with reduced computational burden, enabling broader adoption of AXDT in material testing and biomedical diagnostics.

Abstract

Anisotropic X-ray Dark-Field Tomography (AXDT) is a novel imaging technology that enables the extraction of fiber structures on the micrometer scale, far smaller than standard X-ray Computed Tomography (CT) setups. Directional and structural information is relevant in medical diagnostics and material testing. Compared to existing solutions, AXDT could prove a viable alternative. Reconstruction methods in AXDT have so far been driven by practicality. Improved methods could make AXDT more accessible. We contribute numerically stable implementations and validation of advanced statistical reconstruction methods that incorporate the statistical noise behavior of the imaging system. We further provide a new statistical reconstruction formulation that retains the advanced noise assumptions of the imaging setup while being efficient and easy to optimize. Finally, we provide a detailed analysis of the optimization behavior for all models regarding AXDT. Our experiments show that statistical reconstruction outperforms the previously used model, and particularly the noise performance is superior. While the previously proposed statistical method is effective, it is computationally expensive, and our newly proposed formulation proves highly efficient with identical performance. Our theoretical analysis opens the possibility to new and more advanced reconstruction algorithms, which in turn enable future research in AXDT.

Figures (5)

• Figure 1: Schematic overview of grating-based interferometry setup used in Anisotropic X-ray Dark-Field Tomography. The setup consists of an X-ray source, the three gratings (source, phase and absorption), and a sample stage mounted on an Euler cradle. This setup enables acquiring non-standard acquisition viewpoints needed in AXDT. Figure by sharma2016SixDimensionalXray is licensed under https://creativecommons.org/licenses/by/4.0/
• Figure 2: Convergence plot for reconstruction of the crossed sticks sample. Top: plot of loss over number of iterations. Bottom: Loss over time plots.
• Figure 3: Visualization of slice 68 of the spherical harmonic coefficients $\eta_l^m$ of the crossed wooden sticks, emphasizing the improved noise behavior of the statistical reconstruction methods. The top row visualizes $\eta_0^0$ and the bottom one $\eta_2^1$. From left to right, coefficients of the models m0, m2, m3 respectively. For each row, the images are windowed to the same interval.
• Figure 4: Fiber visualization of two slices from the crossed wooden sticks sample for each of the models overlaid on top of the attenuation X-ray CT reconstruction. The coloring shows the in-plane orientation according to the color wheel. Shown slices are xy-slices through each of the wooden sticks. From left to right, the visualization with the different models m1, m2 and m3 is shown. Fibers are only shown where the scattering strength is above $0.00035$. Red arrows in visualization of model m1 indicate noticeable differences. We show visualization of the iterations 240.0 for both m1 and m3 and 1800.0 for m3.
• Figure 5: Fiber visualization of the center slice from the human brain sample for each of the models overlaid on top of the attenuation X-ray CT reconstruction. The coloring shows the in-plane orientation according to the color wheel. From left to right, the visualization with the different models m1, m2 and m3 is shown. The bottom row shows the crop indicated by the rectangle in the visualization of model m1.