Cross-correlation in the context of Electron tomography

Phase correlation is an approach to estimate the relative translative offset between two similar images (digital image correlation) or other data sets. It is commonly used in image registration and relies on a frequency-domain representation of the data, usually calculated by fast Fourier transforms. The term is applied particularly to a subset of cross-correlation techniques that isolate the phase information from the Fourier-space representation of the cross-correlogram.

Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at different points in time. The analysis of autocorrelation is a mathematical tool for identifying repeating patterns or hidden periodicities within a signal obscured by noise. Autocorrelation is widely used in signal processing, time domain and time series analysis to understand the behavior of data over time.

Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance.

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ${\displaystyle f}$ and ${\displaystyle g}$ that produces a third function ${\displaystyle f*g}$, as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other.

Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution ${\displaystyle f*g}$ differs from cross-correlation ${\displaystyle f\star g}$ only in that either ${\displaystyle f(x)}$ or ${\displaystyle g(x)}$ is reflected about the y-axis in convolution; thus it is a cross-correlation of ${\displaystyle g(-x)}$ and ${\displaystyle f(x)}$, or ${\displaystyle f(-x)}$ and ${\displaystyle g(x)}$. For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator.

In the analysis of data, a correlogram is a chart of correlation statistics. For example, in time series analysis, a plot of the sample autocorrelations ${\displaystyle r_{h}\,}$ versus ${\displaystyle h\,}$ (the time lags) is an autocorrelogram. If cross-correlation is plotted, the result is called a cross-correlogram.

The correlogram is a commonly used tool for checking randomness in a data set. If random, autocorrelations should be near zero for any and all time-lag separations. If non-random, then one or more of the autocorrelations will be significantly non-zero.