Gazing at crystal balls: Electron backscatter diffraction pattern analysis and cross correlation on the sphere
Introduction
Electron backscatter diffraction (EBSD) is a popular microscopy technique used to reveal crystallographic information about materials. Automated, quantitative, robust, and precise interpretation of each electron backscatter pattern (EBSP) (aka Kikuchi pattern) has long been a major advantage of the technique [1], especially when compared to other methods, such as transmission electron microscopy (TEM) imaging and until recently [2] S(canning)TEM based mapping. To advance the EBSD technique further, it is advantageous to improve the quality of the information captured and simultaneously improve how we interpret each pattern. The latter motivates our present work, where we consider the patterns as images on the sphere and analyse them using the spherical Radon transform and spherical cross correlation. In particular, both algorithms may be utilized for automatic orientation determination and band shape analysis.
The majority of existing algorithms for the analysis of EBSP consider them in the gnomonic projection, i.e., as they are measured at a flat 2D capturing screen [3]. In this manuscript, we exploit the fact that the EBSP is generated from a point source and is therefore more properly rendered onto the sphere [4]. For band localisation this is advantageous as Kikuchi bands have parallel edges on the sphere, but hyperbolic edges when considered in the plane. For the cross correlation method, the advantage of the spherical representation originates from the fact that different orientations differ just by a rotation of the spherical Kikuchi pattern, while their correspondence at a flat detector is more involved.
The EBSP is generated as electrons enter the sample, scatter, and dynamically diffract. For an introduction to conventional EBSD analysis, see the review article by Wilkinson and Britton [1]. In practice, diffracting electrons are captured using a flat screen inserted within the electron microscope chamber. The result of this dynamical diffraction process is the generation of an EBSP that contains bands of raised intensity which are called the “Kikuchi bands”. The centre line of each band corresponds to a plane that contains the electron source point and is parallel to the diffracting crystal plane. The edges of the bands are two Kossel conic sections separated by 2θ. The dynamical diffraction process is explained in greater detail in the work of Winkelmann et al. [5]. The corresponding software provides us with high quality simulations that contain significant crystallographic information, such as the intensity profile near a zone axes. These simulated patterns more accurately reproduce the intensity distributions found within experimentally captured patterns, as compared to simple kinematic models. This development has spurred an interest in using these patterns for direct orientation determination by pattern matching techniques [6].
The Hough transform has been used to render the bands within the EBSP as points within a transformed space for easy localisation using a computer [7]. In these conventional algorithms, it is assumed that the bands within the EBSP are near parallel. This renders localisation of the bands into the computationally simpler challenge of finding peaks of high intensity within a sparsely populated space. Unfortunately, within the gnomonic projection the edges of the bands are not parallel. Additionally, the Hough transform of the bands produces butterfly artefacts which makes precise and robust localisation of the bands challenging. However, if the bands are presented as rings on the sphere [4] there is potential to integrate intensity profiles more precisely. This is advantageous for geometries where there may be divergence of the bands (e.g. low voltage or where the pattern centre is less central).
To advance our analysis further, peak localisation and indexing may not be needed if we can efficiently directly compare and match the intensity distributions found within a high quality simulation against our experimental pattern. This can be performed with cross correlation (i.e. finding a peak in the associated cross correlation function), which underpins template matching based EBSD analysis, including the “dictionary indexing” method [6] and template matching approaches [8], [9].
Existing cross correlation methods [6], [8], [9], [10] are performed within the gnomonic projection of the detector. Hereby, each measured Kikuchi pattern is compared with a reference pattern according to a test orientation O. The fit between both images is commonly measured by their correlationwhere the sum is over all pixels in the pattern.
For template matching, reference patterns are tested according to multiple orientations. Sampling of the orientations is performed with a desired angular resolution (sufficient to find a peak and related to the ultimate angular sensitivity). This is computationally very expensive as the above sum has to be computed for a sufficiently large number of reference patterns S(Om), to have a good estimate of the true orientation of the measured Kikuchi pattern P. Recently, Foden et al. [8] have presented an alternative approach where a FFT-based cross correlation is combined with a subsequent orientation refinement step to interpolate between library patterns to provide a more computationally efficient method of template matching. However, that method still involves an expensive gnomonic based library search.
In this work we address this efficiency problem and perform the matching directly on the sphere. Therefore, we require only one spherical master pattern. In this paradigm different orientations results in different rotations of the spherical master pattern. The central idea of this paper is to represent the correlation function between the experimental Kikuchi pattern and all rotated versions of the spherical master pattern as a spherical convolution which can be computed using fast Fourier techniques.
In the case of plane images P and S it is well known [11] that the correlation imagewith respect to all shifts k, ℓ can be computed simultaneously using the fast Fourier transform . More precisely, we havewhere ⊙ denotes the pointwise product. Such Fourier based algorithms have approximately square root the number of operations compared with direct algorithms.
The match between two spherical diffraction patterns can be measured through spherical cross correlation resulting in a function on orientation space. The position of the maximum peak of this function directly gives the desired misorientation of the experimental pattern with respect to the master pattern. In order to speed up the computation of the spherical cross correlation function we apply the same Fourier trick as explained above. In short, we compute spherical Fourier coefficients of the experimental and the master pattern, multiply them and obtain a series representation of the cross correlation function with respect to generalised spherical harmonics. Computation of the spherical Fourier coefficients and evaluation of the generalised spherical harmonics is done using the nonequispaced fast Fourier transform (NFFT) which is at the heart of the MTEX toolbox used for crystallographic texture analysis. The NFFT builds upon significant research generalising the FFT to non Euclidean domains, e.g. to the sphere, cf. [12], [13], or the orientation space cf. [14] and to apply them to problems in quantitative texture analysis, cf. [15], [16], [17], [18]. Although our algorithms are theoretically fast the running times of our implementations are behind those of well established methods. The main reason for this is that our implementations are not yet optimised to take advantage of crystal symmetries, computing on the graphics card or any other kind of parallelisation. On the other hand, this keeps our proof of concept code very simple and allows for easy customisation.
Section snippets
Spherical diffraction patterns
The advantages of considering Kikuchi patterns as spherical functions have been explained very nicely by Day [4]. As an illustrative example of a Kikuchi pattern we consider a high quality dynamical simulation of α-Iron (BCC) generated within DynamicS (Bruker Nano GmbH) and project it onto the sphere (Fig. 1a). The commercial program uses dynamical theory presented by Winkelmann et al. [5] to calculate the intensity of electrons in the resultant diffraction pattern.
In the case of experimental
Harmonic approximation on the sphere
Simulated as well as experimental Kikuchi patterns can be interpreted as diffraction intensities fj with respect to discrete diffraction directions ξj that can be computed from the position within the pattern by the inverse gnonomic projection. For our algorithms, we are interested in approximating these intensities fj by a smooth spherical function represented by a series expansion of the formsuch that f(ξj) ≈ fj. Hereby, denotes the spherical
Spherical Radon transform based band detection
In conventional orientation determination from EBSD data the Kikuchi pattern is represented in the flat, gnomonic frame and summed up along all straight lines resulting in the Radon (or Hough) transform. Since in the Radon transform diffraction bands appear as local extrema they can be found by a peak detection algorithm. A severe problem of this approach is that due to the gnomonic projection bands, in the Kikuchi pattern, do not appear as straight features but have hyperbolic shape. As a
Spherical cross correlation based orientation determination
We have established that experimental and master pattern can be well represented by their harmonic expansion on the sphere and that this representation is useful for band detection. Now we present the use of this representation when computing the cross correlation between an experimental pattern with all possible rotations of a master pattern.
Template matching of EBSD patterns usually employs the following steps:
- 1.
Simulate a dynamical master pattern of all orientation vectors.
- 2.
Select a dense set
Experimental demonstration
We test our two spherical algorithms using the demonstration α-Iron data set as used in Britton et al. [26] for conventional indexing using a planar Radon transform and the AstroEBSD indexing algorithm. This data can be found on Zenodo https://zenodo.org/record/1214829 and consists of a 9 130 point EBSD pattern map. The AstroEBSD background correction was used with operations: hot pixel correction; resize to 300 pixels wide; low frequency Gaussian background division (sigma = 4), performed
Discussion
In this manuscript, we demonstrated that considering EBSD patterns as spherical images allows for elegant algorithms for band detection, band analysis and cross correlation. The reason behind this elegance is the fact that in its spherical representation symmetry operations and misorientations are simple rotations of the Kikuchi pattern.
Beside being elegant these algorithms can be implemented using fast Fourier techniques which makes them at least theoretically fast. In practice, our algorithms
Conclusion
We have outlined and demonstrated methods to perform analysis of EBSD patterns in a spherical frame. We can summarise our conclusions as:
- •
Simulated as well as experimental Kikuchi patterns can be well approximated by spherical functions. These approximations can be computed efficiently using the fast spherical Fourier transform.
- •
The choice of a suitable harmonic cut-off frequency is crucial for the approximation process.
- •
The spherical Radon transform and spherical convolution are efficient methods
Data statement
The example iron data set can be found on Zenodo (https://doi.org/10.5281/zenodo.1214828). Upon article acceptance the full code for this manuscript will be released to Zenodo.
Acknowledgements
TBB acknowledges funding of his research fellowship from the Royal Academy of Engineering. We thank Alex Foden for useful discussions regarding pattern matching. We thank Jim Hickey for assisting with the example iron data set which was captured in the Harvey Flower EM suite at Imperial College on equipment supported by the Shell AIMS UTC. We thank Aimo Winkelmann for assistance with the spherical reprojection and dynamical pattern generation. Finally, we thank the anonymous reviewers for their
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