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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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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