Cross-correlation
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In signal processing, cross-correlation or sometimes simply correlation is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time between the signals, is sometimes called the sliding dot product, and has applications in pattern recognition and cryptanalysis.
For discrete functions f i and g i the cross correlation is defined as
- <math>(f\star g)_i \equiv \sum_j f^*_j\,g_{i+j}<math>
where the sum is over the appropriate values of the integer j and an asterisk indicates the complex conjugate. For continuous functions f (x) and g i the cross correlation is defined as
- <math>(f\star g)(x) \equiv \int f^*(t) g(x+t)\,dt<math>
where the integral is over the appropriate values of t.
The cross-correlation is similar in nature to the convolution of two functions.
Properties
The cross-correlation is related to the convolution by:
- <math>f(t)\star g(t) = f^*(-t)*g(t)<math>
so that
- <math>(f\star g) = f*g<math>
if either f or g is an even function. Also:
- <math>(f\star g)\star(f\star g)=(f\star f)\star (g\star g)<math>
See also
External links
- Cross Correlation from Mathworld (http://mathworld.wolfram.com/Cross-Correlation.html)
- http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:physics/0405041
- http://www.idiom.com/~zilla/Work/nvisionInterface/nip.html
- http://www.phys.ufl.edu/LIGO/stochastic/sign05.pdf
- http://archive.nlm.nih.gov/pubs/hauser/Tompaper/tompaper.php
- http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf
- http://www.is.ac.cn/China-Bejing2.pdf
- Cross correlation examples including 2D pattern identification (http://astronomy.swin.edu.au/~pbourke/analysis/correlate/)