CITE THIS NOTEBOOK: Amplitude and phase of a Fourier-transformed image by Markus van Almsick. Wolfram Community DEC 28 2022.
A centered spike as a test image (single channel):
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spike=
;
Fourier amplitude and phase for an image with the origin at the top left corner:
fourier=Fourier[ImageData[spike]];
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Image[Abs@fourier]//ImageAdjust
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Image[Arg@fourier]/(2π)+1/2
You can move the origin to the image center by multiplying the Fourier transformed image with the corresponding phase factors:
fourierCentered=Exp[-πOuter[Plus,N[(#2-1)/#2]Range[0,#2-1],N[(#1-1)/#1]Range[0,#1-1]]&@@ImageDimensions[spike]]*Fourier[ImageData[spike]];
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Image[Abs@fourierCentered]//ImageAdjust
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Image[Arg@fourierCentered]/(2π)+1/2
Finally, to display the 0-frequency at the center of the spectrum, we move all the Fourier coefficients with RotateLeft:
CenteredfourierCentered=RotateLeft[Exp[-πOuter[Plus,N[(#2-1)/#2]Range[0,#2-1],N[(#1-1)/#1]Range[0,#1-1]]&@@ImageDimensions[spike]]*Fourier[ImageData[spike]],Reverse@Ceiling[ImageDimensions[spike]/2]];
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Image[Abs@CenteredfourierCentered]//ImageAdjust
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Image[Arg@CenteredfourierCentered]/(2π)+1/2
Test with a spike translated by 2 pixels to the right:
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spike2=
;
CenteredfourierCentered=RotateLeft[Exp[-πOuter[Plus,N[(#2-1)/#2]Range[0,#2-1],N[(#1-1)/#1]Range[0,#1-1]]&@@ImageDimensions[spike2]]*Fourier[ImageData[spike2]],Reverse@Ceiling[ImageDimensions[spike2]/2]];
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Image[Abs@CenteredfourierCentered]//ImageAdjust
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Image[Arg@CenteredfourierCentered]/(2π)+1/2
