autocorrelation function

where f(x,y) is the two-dimensional brightness function that defines the image, and x' and y' are the dummy variables of integration.

Like the original image, the ACF is a two-dimensional 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 position, whereas in the ACF, a given coordinate point (x',y') denotes the endpoint of a neighbourhood 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.

AUTOCORRELATION FUNCTION (ACF)

autocorrelation function (ACF)

image

ACF

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autocorrelation function (ACF)

Visualization of ACF

image

ACF

autocorrelation function (ACF)

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Quantification of the ACF

Image of circle

ACF of circle

Thresholded ACF of circle ( level = 50% )

39%

autocorrelation function (ACF)

Quantification of the ACF

Image of white ellipse and corresponding ACF (halftone and thresholded)

autocorrelation function (ACF)

CALCULATION OF THE ACF

Since a correlation (convolution) in object space "reduces" to a multiplication in frequency space, Fourier transforms may be used: the product of the Fourier transform of an image times its complex conjugate is the Fourier transform of the ACF. The most efficient way of calculating Fourier transforms is to use FFT methods.

image (object space) FFT (frequency space) ACF (object space)

autocorrelation function (ACF)

ACF of synthetic fabrics

autocorrelation function (ACF)

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ACF TESSELATIONS

It is possible to calculate one ACF per image:

>>> “bulk” ACF: descriptor of bulk size and shape

or to calculate a number of (smaller) ACFs for subregions of the image:

>>> "local" ACFs: descriptors of local size and shape, in other words, descriptors of size and shape variations

The images of the "local ACFs" can be summed and averaged to yield the “bulk” ACF of the entire image.

A set of macros, "Lazy ACF tiles", written for NIH Image, is designed

• to perform ACF tesselations of large images,• to calculate the bulk ACF for non-square images, and • to analyze size, shape and anisotropy at the local and the global scale.

autocorrelation function (ACF)

“LOCAL” AND “BULK” ACF

ACF tesselations are particularly useful if the image is not square or if variations of local shape and (grain) size are to be monitored

384*256 image

one 256*256 ACF of central part of image

tesselation of six 128*128 local ACFs

autocorrelation function (ACF)

application

EXAMPLE: experimentally deformed Black Hills Quartzite

application of the ACF

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EXAMPLE: experimentally deformed Black Hills Quartzite

ACF image = same size as image = half size (centre)

ACF peak sizedepends on size offeature or “grain size”

application of the ACF

image ACF centers thresholded ACFs

application of the ACF

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<-- high deformation ------- site in sample ------- low deformation -->

axia

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ong

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28· 20 ACFs of Black Hills Quartzite

application of the ACF

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<-- high deformation ------- site in sample ------- low deformation -->

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28· 20 ACFs of Black Hills Quartzite

application of the ACF

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