**Will Lenthe, Principal Software Engineer, EDAX**

Dictionary indexing compares experimental electron backscatter diffraction (EBSD) patterns against a dictionary of simulated patterns for each orientation on a uniform grid in orientation space [1,2]. Synthetic patterns are generated by rotating the Kikuchi sphere by the crystal orientation and projecting onto a plane using the experimental geometry. Comparison against a physics-based forward model gives excellent precision and noise tolerance at the cost of significant computational overhead. Spherical harmonic-based indexing uses the same Kikuchi sphere or ‘master pattern,’ but back projects experimental patterns onto the sphere instead. The orientation is indexed using the maximum spherical cross-correlation between the back-projected pattern and the Kikuchi sphere [3,4]. Mathematically, dictionary and spherical indexing are extremely similar, but the spherical approach is more numerically efficient since it can leverage fast Fourier transforms for the computations. In practice, spherical indexing provides similar precision [5] and noise tolerance to dictionary indexing but at much faster speeds.

A GPU implementation of spherical harmonic-based EBSD indexing implemented in OIM Analysis™ as part of the OIM Matrix module provides excellent indexing quality at hundreds or thousands of patterns per second. Here, we applied it to a range of scans to demonstrate the indexing quality and user parameters.

Spherical harmonic indexing has two parameters: bandwidth and grid size. Bandwidth is how far in frequency space to compute harmonics (analogous to a low pass filter on the EBSD pattern). Grid size is the correlation resolution with an Euler angle cube of (grid size)^{3} used for correlation (i.e., 0 – 360 for phi1, Phi, and phi2). In general, computation time scales with the number of Euler angle grid points, and a reasonable bandwidth is one less than half the grid size. For example, the following are some reasonable pairs of values:

Bandwidth | Grid Size |

63 | 128 |

95 | 192 |

127 | 256 |

Once the best Euler grid point (maximum cross-correlation) is selected, subpixel resolution can be achieved through a refinement step.

## Ni Sequence

This dataset is a scan of the same region at different camera gains to intentionally produce corresponding sets of low and high-quality patterns.

Figure 1 shows a) the result of indexing high-quality patterns, b) spherical harmonic indexing at a bandwidth of 63 and Euler grid of 128^{3} without refinement, and c) at a bandwidth of 63 with refinement. Note that since grid point spacing is ~2.8° (360° / 128), the unrefined result has a stepped appearance due to the discrete orientation possibilities. After refinement, any orientation is possible, providing smooth results.

In Figure 2, KAM maps are shown for the same region at a) 0°, b) 1°, and c) 2°. Notice that without refinement, there is no misorientation within a patch and a sharp spike between them. Even though both the Hough and refined spherical appear smooth, the slight orientation noise in the Hough indexing is visible using KAM.

With low-quality patterns, Hough indexing a) starts to fail, but b) spherical indexing still provides robust solutions and c) accurately captures continuous orientation gradients after refinement (Figure 3).

For very low-quality patterns, higher bandwidths may be required for better indexing results. In Figure 4, a) bandwidths of 63, b) 95, and c) 127 are compared before (a – c) and after (d – f) refinement. Note that the discrete steps in orientations before refinement become smaller with increased Euler angle grid resolution, but they refine to similar orientations. For all three bandwidths, the grid size is 2 * (bandwidth + 1).

With spherical indexing integrated into OIM Analysis, existing image processing algorithms can be used for especially difficult patterns. At extremely high noise levels, Hough indexing cannot index any points, and the spherical indexing begins to fail for some points. NPAR trades spatial resolution for pattern quality by averaging each pattern with its neighbors. The improved patterns can be indexed reliably by both methods but Hough indexing struggles with the resulting overlap patterns near grain boundaries (Figure 5).

## Hot Rolled Mg

Hough indexing struggles to index when pattern quality is reduced by a) high deformation, but b) spherical indexing is robust against significantly degraded pattern quality. Note that the d) spherical indexing confidence index strongly correlates with c) image quality but is high even in some regions with extremely low IQ (Figure 6).

## Rutile

Spherical indexing can use a unique pattern center for each point at no extra cost for large fields of view. Excellent results are possible even with a single pattern center used for the entire dataset, as shown in Figure 7.

## Deformed Duplex Steel

Spherical indexing can be applied to multiple phases in the same way as any other indexing technique. Phase discrimination depends on the similarity of the phases with a two-phase steel shown in Figure 8. In addition to the quality in orientation results with d – f) spherical indexing vs. a – c) Hough indexing, b – c & e – f) phase discrimination is improved with spherical BCC and FCC iron well separable. Real space refinement may be required for particularly difficult cases in addition to the spherical harmonic refinement shown.

Again, spherical indexing’s confidence index correlates well with pattern quality. In Figure 9, a) spherical CI + IPF shows similar trends as b) Hough IQ + IPF.

## References

- Callahan, P. G., & De Graef, M. (2013). Dynamical electron backscatter diffraction patterns. Part I: Pattern simulations.
*Microscopy and Microanalysis*,**19**(5), 1255-1265. - Callahan, P. G., & De Graef, M. (2013). Dynamical electron backscatter diffraction patterns. Part I: Pattern simulations.
*Microscopy and Microanalysis*,**19**(5), 1255-1265. - Lenthe, W. C., Singh, S., & De Graef, M. (2019). A spherical harmonic transform approach to the indexing of electron backscattered diffraction patterns.
*Ultramicroscopy*,**207**, 112841. - Hielscher, R., Bartel, F., & Britton, T. B. (2019). Gazing at crystal balls: Electron backscatter diffraction pattern analysis and cross-correlation on the sphere.
*Ultramicroscopy*,**207**, 112836. - Sparks, G., Shade, P. A., Uchic, M. D., Niezgoda, S. R., Mills, M. J., & Obstalecki, M. (2021). High-precision orientation mapping from spherical harmonic transform indexing of electron backscatter diffraction patterns.
*Ultramicroscopy*,**222**, 113187.