12/25/2020 2d Fourier Transform Software For Mac
Next: Two-dimensional Fourier Filtering Up: ImageProcessing Previous: Fast Fourier Transform Two-Dimensional Fourier Transform. Fourier transform can be generalized to higher dimensions. For example, many signals are functions of 2D space defined over an x-y plane. Two-dimensional Fourier transform also has four different forms depending on.
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2-D Fourier Transforms
The
fft2 function transforms2-D data into frequency space. For example, you can transform a 2-Doptical mask to reveal its diffraction pattern.
Two-Dimensional Fourier Transform
The following formula defines the discrete Fourier transform Y ofan m-by-n matrix X.
ωm and ωn arecomplex roots of unity defined by the following equations.
i is the imaginary unit, p and j areindices that run from 0 to m–1, and q and k areindices that run from 0 to n–1. The indicesfor X and Y are shifted by 1in this formula to reflect matrix indices in MATLAB®.
Computing the 2-D Fourier transform of X isequivalent to first computing the 1-D transform of each column of X,and then taking the 1-D transform of each row of the result. In otherwords, the command
fft2(X) is equivalent to Y= fft(fft(X).').' .
2-D Diffraction Pattern![]()
In optics, the Fourier transform can be used to describe the diffraction pattern produced by a plane wave incident on an optical mask with a small aperture [1]. This example uses the
fft2 function on an optical mask to compute its diffraction pattern.
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Create a logical array that defines an optical mask with a small, circular aperture.
Use
fft2 to compute the 2-D Fourier transform of the mask, and use the fftshift function to rearrange the output so that the zero-frequency component is at the center. Plot the resulting diffraction pattern frequencies. Blue indicates small amplitudes and yellow indicates large amplitudes.
To enhance the details of regions with small amplitudes, plot the 2-D logarithm of the diffraction pattern. Very small amplitudes are affected by numerical round-off error, and the rectangular grid causes radial asymmetry.
References
[1] Fowles, G. R. Introductionto Modern Optics. New York: Dover, 1989.
See Alsofft | fft2 | fftn | fftshift | ifft2
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