# Spherical Indexing

### Introduction

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 (e.g., 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, you can achieve subpixel resolution through a refinement step.

### Examples

##### Ni sequence

Spherical indexing can be used on EBSD patterns with a wide range of patten quality. In this example, a set of scans of the same region on a well-polished nickel alloy were collected at different camera gains to intentionally produce corresponding sets of low and high-quality patterns with varying pattern signal-to-noise ratios (SNR). Spherical indexing was then performed and compared with standard Hough-based indexing to illustrate how spherical indexing can better index patters with lower SNR values and obtain higher quality data compared to conventional indexing.

**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.**

Figure 1 shows a) the result of indexing high SNR 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.

**Figure 2. KAM maps are shown for the same region at a) 0°, b) 1°, and c) 2°.**

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.

**Figure 3. 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.**

With low SNR 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).

**Figure 4. a) bandwidths of 63, b) 95, and c) 127 are compared before (a – c) and after (d – f) refinement.**

For very low SNR 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).

**Figure 5. a) Raw pattern and b) NPAR pattern using Hough indexing and c) raw pattern and d) NPAR pattern using spherical indexing with a bandwidth of 127.**

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

**Figure 6. 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.**

Magnesium can be a difficult material for EBSD analysis, as it is a light element with a lower backscatter signal resulting in lower SNR values for the EBSD patterns. This challenge is compounded when the magnesium sample is deformed, as the resultant lattice distortion will degrade the overall pattern quality. Hough indexing struggles to index these types of patterns (a) but spherical indexing is robust against significantly degraded pattern quality (b), allowing better indexing as deformation levels increase. 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).

##### Deformed duplex steel

**Figure 7. Phase discrimination depends on the similarity of the phases with a two-phase steel. 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.**

The previous examples have shown how spherical indexing can improve EBSD pattern indexing when the EBSD pattern quality decreases due to lower SNR values or increased sample deformation. In this example, spherical indexing is used of a deformed multiphase sample, showing how the indexing improvements can also be achieved even when considering multiple phases. 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 7. 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.

**Figure 8. a) spherical CI + IPF shows similar trends as b) Hough IQ + IPF.**

Again, spherical indexing’s confidence index correlates well with pattern quality. In Figure 8, a) spherical CI + IPF shows similar trends as b) Hough IQ + IPF.

### Conclusion

Forward model indexing allows users to maximize the information extracted from their data but historically has a steep learning curve. Hough indexing is well established but susceptible to poor pattern quality. Spherical harmonic-based indexing combines the robustness of forward model indexing with the speed and ease of use of Hough indexing.

### 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.