Direct extraction of spatial correlation functions from limited x-ray tomography data for microstructural quantification

Accurately quantifying the microstructure of a heterogeneous material is crucial to establishing quantitative structure-property relations for material optimization and design. There is a preponderance of previous work focused on structural quantification based on 2D images and reconstructed 3D volumes obtained via different imaging techniques. Here, we introduce novel procedures that allow one to extract key structural information in the form of spatial correlation functions from limited x-ray tomography data. In the case where only a very small number of x-ray tomographic radiographs (projections) are available, we derive a formalism based on the Fourier slice theorem to compute angularly averaged correlation functions directly from the radiographs. When a larger number of projections are available, we develop a procedure to extract full vector-based correlation functions. The key component of this procedure is the computation of a “probability map,” which provides the probability of an arbitrary point in the material system belonging to a specific phase, via inverse superposition of the scaled attenuation intensities available in the tomography projections. The correlation functions of interest are then computed based on their corresponding probability interpretations from the probability map. The utilities of both of our procedures are demonstrated by obtaining lower-order correlation functions (including both the standard two-point correlation functions and non-standard surface functions) for a tin-clay composite material from both parallel-beam (synchrotron) and cone-beam (lab-scale) x-ray tomography projection data sets. Our procedure directly transforms the key morphological information contained in limited x-ray tomography projections to a more efficient, understandable, and usable form.