EBSD.com is an educational site by Oxford Instruments Once an EBSD system has been calibrated, it is possible to automatically index the diffraction patterns and calculate the crystal orientation. This is typically accomplished using the following steps:

1. The diffraction pattern is transferred from the detector to the EBSD software.
2. A Hough transform is used to identify the positions of the Kikuchi bands. The bands are seen as peaks in Hough space.
3. Having identified the Kikuchi band positions and from knowing the calibrated geometry, it is possible to calculate the angles between the detected bands.
4. The calculated angles are compared with a list of interplanar angles for the analysed structure(s) based on a selected number of reflecting planes (reflectors) in the reference structure.
5. The possible solutions are sorted to find the best fit and the orientation matrix is calculated.

This whole process is automatic and takes less than a few milliseconds on modern computers.

### The Hough transform

The Hough transform, which identifies the positions of the Kikuchi bands, converts the image from the EBSD camera into a representation in Hough space, by using the following relation between the points (x, y) in the diffraction pattern and the coordinates (ρ, θ) of the Hough space: ρ = x cosθ + y sinθ. A straight line in the image space (x, y) can be characterised by ρ, the perpendicular distance from the line to origin and θ, the angle made with the x-axis, and can be presented by a single point (ρ, θ) in Hough space (Figure 1). The Kikuchi bands appear as bright regions or peaks in Hough space, which are easily detected and used to calculate the original band positions (Figure 2).

Figure 1 The Hough transform converts lines into points in Hough space.

Figure 2 Finding the position of the Kikuchi bands in the diffraction pattern using the Hough transform. (a) Diffraction pattern collected from silicon at 20kV accelerating voltage; (b) The peaks in the Hough transform identified and coloured; (c) The bands in the original diffraction pattern corresponding to the peaks found in the Hough transform and coloured similarly; (d) The indexed diffraction pattern with the blue cross indicating the position of the pattern centre.

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