The following code shows some of these routines in action when applied to a checkerboard:
import numpy
from scipy.fftpack import fft,fft2, fftshift
import matplotlib.pyplot as plt
B=numpy.ones((4,4)); W=numpy.zeros((4,4))
signal = numpy.bmat("B,W;W,B")
onedimfft = fft(signal,n=16)
twodimfft = fft2(signal,shape=(16,16))
plt.figure()
plt.gray()
plt.subplot(121,aspect='equal')
plt.pcolormesh(onedimfft.real)
plt.colorbar(orientation='horizontal')
plt.subplot(122,aspect='equal')
plt.pcolormesh(fftshift(twodimfft.real))
plt.colorbar(orientation='horizontal')
plt.show()
Note how the first four rows of the one-dimensional transform are equal (and so are the last four), while the two-dimensional transform (once shifted) presents a peak at the origin and nice symmetries in the frequency domain.
In the following screenshot, which has been obtained from the previous code, the image on the left is the fft and the one on the right is the fft2 of a 2 x 2 checkerboard signal:
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