Computing the magnitude and phase of a DFT
We'll use the house.png image as input and thus fft2() to get the real and imaginary components of the Fourier coefficients; after that, we'll compute the magnitude/spectrum and the phase and, finally, use ifft2() to reconstruct the image:
import numpy.fft as fp
im1 = rgb2gray(imread('../images/house.png'))
pylab.figure(figsize=(12,10))
freq1 = fp.fft2(im1)
im1_ = fp.ifft2(freq1).real
pylab.subplot(2,2,1), pylab.imshow(im1, cmap='gray'), pylab.title('Original Image', size=20)
pylab.subplot(2,2,2), pylab.imshow(20*np.log10( 0.01 + np.abs(fp.fftshift(freq1))), cmap='gray')
pylab.title('FFT Spectrum Magnitude', size=20)
pylab.subplot(2,2,3), pylab.imshow(np.angle(fp.fftshift(freq1)), cmap='gray')
pylab.title('FFT Phase', size=20)
pylab.subplot(2,2,4), pylab.imshow(np.clip(im1_,0,255), cmap='gray')
pylab.title('Reconstructed Image', size=20)
pylab.show()
You'll get this output:
As can be seen, the magnitude 
generally decreases with higher spatial frequencies and the FFT phase appears to be less informative.
Let's now compute the spectrum/magnitude, phase, and reconstructed image with another input image, house2.png:
im2 = rgb2gray(imread('../images/house2.png'))
pylab.figure(figsize=(12,10))
freq2 = fp.fft2(im2)
im2_ = fp.ifft2(freq2).real
pylab.subplot(2,2,1), pylab.imshow(im2, cmap='gray'), pylab.title('Original Image', size=20)
pylab.subplot(2,2,2), pylab.imshow(20*np.log10( 0.01 + np.abs(fp.fftshift(freq2))), cmap='gray')
pylab.title('FFT Spectrum Magnitude', size=20)
pylab.subplot(2,2,3), pylab.imshow(np.angle(fp.fftshift(freq2)), cmap='gray')
pylab.title('FFT Phase', size=20)
pylab.subplot(2,2,4), pylab.imshow(np.clip(im2_,0,255), cmap='gray')
pylab.title('Reconstructed Image', size=20)
pylab.show()
The output that you should get is as follows:
Although it's not as informative as the magnitude, the DFT phase is also important information and an image can't be reconstructed properly if the phase is not available or if we use a different phase array.
To witness this, let's see how a reconstructed output image gets distorted if we use the real components of the frequency spectrum from one image and imaginary components from another one:
pylab.figure(figsize=(20,15))
im1_ = fp.ifft2(np.vectorize(complex)(freq1.real, freq2.imag)).real
im2_ = fp.ifft2(np.vectorize(complex)(freq2.real, freq1.imag)).real
pylab.subplot(211), pylab.imshow(np.clip(im1_,0,255), cmap='gray')
pylab.title('Reconstructed Image (Re(F1) + Im(F2))', size=20)
pylab.subplot(212), pylab.imshow(np.clip(im2_,0,255), cmap='gray')
pylab.title('Reconstructed Image (Re(F2) + Im(F1))', size=20)
pylab.show()
Here are the reconstructed images with real and imaginary frequency components mixed with each other:
Frame two is as follows:
