# Autocorrelation Coefficients

The algorithm calculates autocorrelation coefficients for color and black and white, 3D and 4D images.

## Background

Visualization of ACF

In any time series containing non-random patterns of behavior, it is likely that any particular item in the series is related in some way to other items in the same series. This can be described by the autocorrelation function and/or autocorrelation coefficient.

The correlation of two continuous functions f(x) and g(x) can be defined as

$f(x)\bullet g(x)= \int (f(a)*g(x+a))dx$

where, $\bullet$ represents the complex conjugate. If f(x) and g(x) are the same functions, $f(x)\bullet g(x)$ is called autocorrelation function.

For a 2D image, its autocorrelation function (ACF) can be calculated as

$f(x,y)\bullet g(x,y)= \int (f(a,b)*g(x+a, y+b))dx$

where f(x,y) is the two-dimensional brightness function that defines the image, and a and b are the variables of integration. Like the original image, the auto-correlation function (ACF) is a 2D function. Although the dimensions of the ACF and the original image are exactly the same, they have different meaning. In the original image, a given coordinate point (x,y) denotes a pixel position, while in the ACF, a given coordinate point (a,b) denotes the endpoint of a neighborhood vector.

The ACF describes how well an image correlates with itself under conditions where the image is displaced with respect to itself in all possible directions

For a given series of points {xi}, the autocorrelation coefficient can be described as follows:

<a measure of covariance among the {xi}>/<signal variance>

Or, for a given autocorrelation function f(x, y) autocorrelation coefficient A can be described as:

$\frac {A(dx,dy)}{A(0,0)}=\frac {\sum _{dx=0} ^{dx=xDim-1-dx} \sum_{dy=0} ^{dy=yDim-1-dy} *f(x+dx, y+dy)}{(xDim-dx)*(yDim-dy)}$

## Image types

Color and black and white 3D and 4D images

## Applying Autocorrelation Coefficients

Calculating autocorrelation for 2D grayscale image

### Autocorrelation Coefficients dialog box

OK - Applies the algorithm according to the specifications in this dialog box.

Cancel - Disregards any changes that you made in the dialog box and closes this dialog box.

### To use this algorithm, do the following:

• Open an image of interest.
• Select Algorithms > AutoCorrelation Coefficients in the MIPAV window. The AutoCorrelation Coefficients dialog box appears.
• Click OK. The algorithm begins to run, and a status bar appears with the status. When the algorithm finishes running, the progress bar disappears, and the result image replaces the original one.

## References

Digital Image Processing, Second Edition by Rafael C. Gonzalez and Richard C. Woods, Prentice-Hall, Inc., 2002, pp. 205 - 208 and pp. 414-417.