XCF: The Cross Correlation Function
At the core of the HR EBSD method is cross correlation. The Cross-Correlation function (or XCF) is a well-established mathematical technique used in many fields from signal processing to astronomy. As used in HREBSD it provides a comparative measure of the distortion between a test pattern and a reference pattern. Cross correlation is performed over numerous regions of interest (ROI) common to both patterns and the average shift within each of the selected ROIs between test and reference patterns is measured. These are used to calculate a relative distortion matrix which in turn is used to elucidate the stress/strain state of the material under study.
Typically an ROI is square and covers around 1/16th of the total area of the pattern. They are distributed so as to sample predominately the areas towards the outer edges of the pattern which because of the gnomonic projection form of the pattern are the most sensitive to strain. The XCF is based upon Fourier Transforms and these are best calculated using the Fast Fourier transform or FFT – this constrains the size (width and height) of the ROI to be a number that is a power of 2 (e.g. 128, 256, 512 etc.)
The result of the XCF of an ROI from 2 different patterns of similar orientation is a square array of real numbers. The value of the number at each point in the XCF array represents the degree of correlation between the 2 patterns if the test pattern had been offset so that it is centred at that point. These numbers are largest at the point where the 2 patterns best overlap (i.e. correlate). For example, if the contents of the ROI are identical apart from the test pattern being shifted one pixel to the right compared to the reference pattern, the XCF will have its brightest pixel at the position 1,0.
By using a peak fitting routine to interpolate the position of the XCF peak, it is possible to measure sub-pixel shifts. In astronomy, for example, they can track the movement of a star to at least 1/100 of a pixel sensitivity. In HREBSD the presence of noise and a background signal limit the sensitivity to 1/10 to 1/20 of a pixel.