across a microphone array. After calculating the cross-correlation between the two signals, the maximum (or minimum if the signals are negatively correlated) of the cross-correlation function indicates the point in time where the signals are best aligned; i.e., the time delay between the two signals is determined by the argument of the maximum, or arg max of the cross-correlation, as inTerminology in image processing Zero-normalized cross-correlation (ZNCC) For image-processing applications in which the brightness of the image and template can vary due to lighting and exposure conditions, the images can be first normalized. This is typically done at every step by subtracting the mean and dividing by the standard deviation. That is, the cross-correlation of a template with a subimage iswhere is the number of pixels in and , is the average of and is standard deviation of . In functional analysis terms, this can be thought of as the dot product of two normalized vectors. That is, ifandthen the above sum is equal towhere is the inner product and is the L² norm. Cauchy–Schwarz then implies that ZNCC has a range of . Thus, if and are real matrices, their normalized cross-correlation equals the cosine of the angle between the unit vectors and , being thus if and only if equals multiplied by a positive scalar. Normalized correlation is one of the methods used for template matching, a process used for finding instances of a pattern or object within an image. It is also the 2-dimensional version of Pearson product-moment correlation coefficient. Normalized cross-correlation (NCC) NCC is similar to ZNCC with the only difference of not subtracting the local mean value of intensities: Nonlinear systems Caution must be applied when using cross correlation for nonlinear systems. In certain circumstances, which depend on the properties of the input, cross correlation between the input and output of a system with nonlinear dynamics can be completely blind to certain nonlinear effects. This problem arises because some quadratic moments can equal zero and this can incorrectly suggest that there is little "correlation" (in the sense of statistical dependence) between two signals, when in fact the two signals are strongly related by nonlinear dynamics. See also Autocorrelation Autocovariance Coherence Convolution Correlation Correlation function Cross-correlation matrix Cross-covariance Cross-spectrum Digital image correlation Phase correlation Scaled correlation Spectral density Wiener–Khinchin theorem References Further reading External links Cross Correlation from Mathworld http://scribblethink.org/Work/nvisionInterface/nip.html http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf Bilinear maps Covariance and correlation