Apply DFT matrix along each axis of 3D array in NumPy?

Question

I can first obtain the DFT matrix of a given size, say n by

import numpy as np
n = 64
D = np.fft.fft(np.eye(n))

The FFT is of course just a quick algorithm for applying D to a vector:

x = np.random.randn(n)
ft1 = np.dot(D,x)
print( np.abs(ft1 - fft.fft(x)).max() )
# prints near double precision roundoff

The 2D FFT can be obtained by applying D to both the rows and columns of a matrix:

x = np.random.randn(n,n)
ft2 = np.dot(x, D.T) # Apply D to rows.
ft2 = np.dot(D, ft2) # Apply D to cols.
print( np.abs(ft2 - fft.fft2(x)).max() )
# near machine round off again

How do I compute this analogously for the 3 dimensional Discrete Fourier Transform?

Ie,

x = np.random.randn(n,n,n)
ft3 = # dot operations using D and x
print( np.abs(ft3 - fft.fftn(x)).max() )
# prints near zero

Essentially, I think I need to apply D to each column vector in the volume, then each row vector in the volume, and finally each "depth vector". But I'm not sure how to do this using dot .

Answer 1

You can use the einsum expression to perform the transformation on each index:

x = np.random.randn(n, n, n)
ft3 = np.einsum('ijk,im->mjk', x, D)
ft3 = np.einsum('ijk,jm->imk', ft3, D)
ft3 = np.einsum('ijk,km->ijm', ft3, D)
print(np.abs(ft3 - np.fft.fftn(x)).max())
1.25571216554e-12

This can also be written as a single NumPy step:

ft3 = np.einsum('ijk,im,jn,kl->mnl', ft3, D, D, D, optimize=True)

Without the optimize argument (available in NumPy 1.12+) it will be very slow however. You can also do each of the steps using dot , but it requires a bit of reshaping and transposing. In NumPy 1.14+ the einsum function will automatically detect the BLAS operations and do this for you.