Editing Cross-correlation (section)

===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]].