2-D Fourier Transforms
Recently, I have encountered an issue with ArrayPlot after performing a Fourier transform of a table. Picture presented above is an ArrayPlot of a 2D table. Using Fourier on this 2D table I obtain the following ArrayPlot: After this procedure one ends up with the result. Fourier Transforms Properties - Here are the properties of Fourier Transform. SignalProcessingFFT: Similar to the SignalProcessingDFT command, SignalProcessingFFT computes the discrete Fourier transform of an Array of signal data points. The difference between the two commands is that the SignalProcessingFFT command uses the fast Fourier transform algorithm. Note: SignalProcessingFFT requires that the size of the Array must be a power of 2, greater than 2. 1D, 2D, and 3D Half and Full spectrum transforms. They all are open-source and use the FFTw software for the transforms. The individual transforms should be applied only using the Ft Fourier Transform Suite script rather than from the Filter menu as a standard RGB file cannot hold the amount of data generated. Windows and Mac with Photoshop CS6. Students can load scanlines from common image patterns and see that scanline's Fourier Transform in real-time. You may want to check out more software, such as Fourier Painter, 1D-Nest or 1D Cutting Optimizer, which might be related to 1D Fast Fourier Transform.
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).').'
.![2d fourier transform matlab 2d fourier transform matlab](/uploads/1/2/6/0/126008137/412857284.png)
2-D Diffraction Pattern
2d Discrete Fourier Transform
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.2d Fourier Transform
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 Also
fft
| fft2
| fftn
| fftshift
| ifft2