Texture tomography with high angular resolution utilizing sparsity

We demonstrate a novel approach to the reconstruction of scanning probe x-ray diffraction tomography data with anisotropic poly crystalline samples. The method involves reconstructing a voxel map containing an orientation distribution function in each voxel of an extended 3D sample. This method differs from existing approaches by not relying on a peak-finding and is therefore applicable to sample systems consisting of small and highly mosaic crystalline domains that are not handled well by existing methods. Samples of interest include bio-minerals and a range of small-graines microstructures common in engineering metals. By choosing a particular kind of basis functions, we can effectively utilize non-negativity in orientation-space for samples with sparse texture. This enables us to achieve stable solutions at high angular resolutions where the problem would otherwise be under determined. We demonstrate the new approach using data from a shot peened martensite sample where we are able to map the twinning micro structure in the interior of a bulk sample without resolving the individual lattice domains. We also demonstrate the approach on a piece of gastropods shell with a mosaic micro structure.

💡 Research Summary

The authors present a novel reconstruction approach for scanning‑probe X‑ray diffraction tomography (XRD‑CT) that directly recovers an orientation distribution function (ODF) in each voxel of a three‑dimensional sample. Unlike conventional texture tomography, which first identifies discrete diffraction peaks and then builds pole figures, this method—named ODF‑based Tensor Tomography (ODF‑TT)—bypasses peak‑finding entirely. By expanding the ODF on a grid of symmetric Gaussian basis functions and imposing two physical priors—non‑negativity in orientation space and sparsity of the texture—the inverse problem becomes a convex optimization that remains well‑posed even at high angular resolution where the number of unknowns far exceeds the number of measured data points.

Mathematically, the measured diffraction intensity is modeled as a linear combination of the unknown ODF coefficients weighted by the experimental geometry (rotation angles, detector positions). The reconstruction minimizes a least‑squares data‑misfit term subject to (i) a non‑negativity constraint, ensuring that the ODF never takes unphysical negative values, and (ii) an L1‑type sparsity regularizer, reflecting the empirical observation that most bulk materials exhibit only a few dominant crystallographic orientations despite a potentially huge orientation space. This combination suppresses the high‑frequency artefacts that would otherwise arise from the severe under‑determination (hundreds of millions of unknown coefficients versus tens of millions of data points).

Two very different specimens were used to validate the method. The first is a shot‑peened martensitic steel, whose microstructure consists of nanoscale ferrite laths formed by transformation twinning. Traditional 3D‑XRD or s3D‑XRD cannot resolve such fine twins because the diffraction rings are highly smeared. ODF‑TT successfully maps the dominant orientation, a secondary orientation, and the misorientation axis and magnitude in each voxel. Quantitative texture indices reveal lower texture near the peened surface, and the reconstructed ODFs show that only about half of the total intensity is captured by the primary peak, confirming a highly dispersed texture. The second specimen is a gastropod shell composed mainly of aragonite. Here the ODF‑TT reconstruction yields a smoothly varying c‑axis direction that aligns with the shell’s columnar wall, and it clearly distinguishes two concentric layers that differ by a rotation about the c‑axis. When compared with pole‑figure‑based tensor tomography (PF‑TT) on the same data, ODF‑TT demonstrates far fewer “missing‑wedge” artefacts; PF‑TT shows abrupt, erratic orientation changes, especially when only a zero‑tilt dataset is used, whereas ODF‑TT remains stable thanks to the sparsity and non‑negativity constraints.

The discussion places ODF‑TT in the broader landscape of 3D‑XRD techniques. While grain‑mapping methods aim to resolve individual grains or strain fields and require low misorientation or a small number of grains, ODF‑TT targets the voxel‑averaged texture of highly deformed, fine‑grained materials. Its reliance on a single rotation axis simplifies experimental geometry and reduces acquisition time. However, the high dimensionality of the ODF makes visualization and quantitative analysis non‑trivial, and the method’s performance may degrade for textures that are not sparse, requiring careful tuning of regularization parameters. Future work is suggested on automated parameter selection, incorporation of multiple rotation axes, and coupling ODF‑TT with strain tomography to provide a more complete picture of bulk microstructures.

In summary, the paper demonstrates that by exploiting physical priors—non‑negativity and sparsity—ODF‑TT can reconstruct high‑resolution, three‑dimensional texture maps for materials with small grains, high mosaic spread, or complex twinning, where existing X‑ray diffraction tomography methods fail. This opens new possibilities for non‑destructive bulk texture analysis in metallurgy, biomineralization, and other fields where bulk crystallographic orientation is a key property.