Reconstructing grain-shape statistics from electron back-scatter diffraction microscopy

Reconstructing the three-dimensional (3D) size and shape distribution of randomly oriented grains using only images of cross sections remains an important challenge. Even for ellipsoids, a solution is only possible when they are solids of revolution, and may still be numerically unstable. Here we show that crystallographic orientation data, for example from electron back-scatter diffraction (EBSD), provides enough additional information to obtain moments of the 3D grain distribution, provided grain shapes can be assumed to align with crystal axes. We show that this moment method can give an average 3D grain size and shape (with error estimate) which is rigorous for ellipsoids and a good approximation for cuboidal grains, indicating that it may be a useful technique for polycrystalline materials in general. High throughput image analysis and EBSD now make the necessary sample sizes practical. We illustrate by applying the method to a basaltic rock specimen.

Figure 1

(a) A collection of randomly oriented polydisperse ellipsoids. (b) A plane section through the collection produces a set of ellipses (black) from which we wish to reconstruct statistical information about the 3D structure.

Figure 2

Top: Random distribution of randomly oriented monodisperse ellipsoids in space. Each ellipsoid has major axes of length $(A_0,B_0,C_0)=(9,3,1)$. Bottom: The same distribution cut by a plane, showing (in black) the elliptical cross sections produced.

Figure 3

Reconstructed major axis lengths for monodisperse ellipsoids with major axes $(A_0,B_0,C_0)=(9,3,1)$. The vertical bars in the left panel show the range (mean plus or minus one standard deviation) for ten repeats. Right-hand panel shows the fractional error in the reconstructed axis lengths, in the sense of the greatest distance of the standard deviation bar from the true value, divided by the true value. We see that the standard deviation of predictions falls as the number of ellipses ${\nu }_{\text{2D}}$ increases. Furthermore, the predictions become more and more accurate in the same limit, with a fractional error falling roughly as $5{\nu }_{\text{2D}}^{-1/2}$.

Figure 4

Top: Random distribution of randomly oriented monodisperse cuboids in space. The equivalent major axes (in the sense defined in the text) are $(A_0,B_0,C_0)=(9,3,1)$. Bottom: The same distribution cut by a plane, showing (in black) the polygonal [4] cross sections produced.

Figure 5

Reconstructed major axis lengths for monodisperse cuboids with equivalent major axes $(A_0,B_0,C_0)=(9,3,1)$. The vertical bars in the left panel show the range (mean plus or minus one standard deviation) for ten repeats. Right-hand panel shows the fractional error in the reconstructed axis lengths, in the sense of the greatest distance of the standard deviation bar from the true value, divided by the true value. We see that the standard deviation of predictions falls as the number of ellipses ${\nu }_{\text{2D}}$ increases. Furthermore, the predictions become more and more accurate in the same limit, with a fractional error falling roughly as $7{\nu }_{\text{2D}}^{-1/2}$. We expect that for very large ${\nu }_{\text{2D}}$ the errors would eventually cease to fall further.

Figure 6

Image of basaltic rock sample, segmented by mineral type. We apply the analysis here separately to the plagioclase and clinopyroxene grains.