Correlation functions revisited

Abstract Correlation functions (auto-correlations, cross-correlations, rotational correlations) are used extensively in general signal processing, in Fourier optics, and particularly in electron microscopical image processing. In spite of their widespread and successful use in a broad spectrum of pattern-matching procedures, we believe there is reason to reconsider their general use. Correlation functions have the intrinsic limitation that they are calculated by squaring or multiplication operations in Fourier space. This fact causes the strong frequency components in the data (typically the low-frequency ones) to become overwhelmingly stronger relative to the weaker components, typically the high-frequency components associated with the fine details in the data in which we are especially interested. By dividing Fourier components by the square roots of their amplitudes, these unfavorable effects can be avoided resulting in substantially better correlation functions. Our new mutual-correlation function (MCF) and self-correlation function (SCF), which relate to each other the way the conventional cross-correlation function (CCF) relates to the auto-correlation function (ACF), can supersede conventional correlation functions for most purposes. We have subjected the novel correlation functions to practical tests in routine image-alignment procedures and found the results satisfactory. The problems we experienced with conventional squared correlation functions are likely to hinder triple-correlation function applications to an even greater extent.