Editing Cross-correlation (section)

==Time delay analysis==
Cross-correlations are useful for determining the time delay between two signals, e.g., for determining time delays for the propagation of acoustic signals across a microphone array.<ref>{{cite conference|last=Rhudy|first=Matthew|author2=Brian Bucci |author3=Jeffrey Vipperman |author4=Jeffrey Allanach |author5=Bruce Abraham |title=Microphone Array Analysis Methods Using Cross-Correlations|conference=Proceedings of 2009 ASME International Mechanical Engineering Congress, Lake Buena Vista, FL|pages=281–288|date=November 2009 |doi=10.1115/IMECE2009-10798|isbn=978-0-7918-4388-8}}</ref><ref>{{cite thesis |last=Rhudy|first=Matthew|title=Real Time Implementation of a Military Impulse Classifier |type=MS thesis |publisher=University of Pittsburgh |date=November 2009|url=http://d-scholarship.pitt.edu/9773/}}</ref>{{clarify|reason=I doubt that this definition is used for microphone arrays, since it involves an integral over all time. Perhaps integration over a time-window?|date=May 2015}}  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 in<math display="block">\tau_\mathrm{delay}=\underset{t \in \mathbb{R}}{\operatorname{arg\,max}}((f \star g)(t))</math>Terminology in image processing
===Zero-normalized cross-correlation (ZNCC)===

For [[Digital image processing|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  <math>t(x,y)</math> with a subimage <math>f(x,y)</math> is

<math display="block">\frac{1}{n\sigma_f \sigma_t} \sum_{x,y}\left(f(x,y) - \mu_f \right)\left(t(x,y) - \mu_t \right)</math>

where <math>n</math> is the number of pixels in <math>t(x,y)</math> and <math>f(x,y)</math>,
<math>\mu_f</math> is the average of <math>f</math> and <math>\sigma_f</math> is [[standard deviation]] of <math>f</math>.

In [[functional analysis]] terms, this can be thought of as the dot product of two [[Unit vector|normalized vectors]]. That is, if<math display="block">F(x,y) = f(x,y) - \mu_f</math>and<math display="block">T(x,y) = t(x,y) - \mu_t</math>then the above sum is equal to<math display="block">\left\langle\frac{F}{\|F\|},\frac{T}{\|T\|}\right\rangle</math>where <math>\langle\cdot,\cdot\rangle</math> is the [[inner product]] and <math>\|\cdot\|</math> is the [[Lp space|''L''² norm]]. [[Cauchy–Schwarz]] then implies that ZNCC has a range of <math>[-1, 1]</math>.

Thus, if <math>f</math> and <math>t</math> are real matrices, their normalized cross-correlation equals the cosine of the angle between the unit vectors <math>F</math> and <math>T</math>, being thus <math>1</math> if and only if <math>F</math> equals <math>T</math> 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:<math display="block">\frac{1}{n\sigma_f \sigma_t} \sum_{x,y}f(x,y) t(x,y)</math>