Correlation analysis of structure images

A typical high resolution structure image contains a large amount of intensity information which is masked by both statistical and amorphous noise. One useful method of quantifying such images is to employ correlation techniques. When one seeks to quantify the atom column positions, correlation techniques can be used to decompose the image into separate motifs (of specific peak amplitudes and positions - each motif corresponding to a single column of atoms), thereby reducing the data to a more manageable form.This problem can be considered as the least squares minimization of the function: where I(r) is the image, and m(r) is the motif, and the unknowns are the positions, rj, of the motifs and their peak heights, αj. The standard approach is to look for peaks in the cross-correlation (equation 2) between the motif and image, to determine rj and αj