Attempting to confirm the middle pool is equal to resetting the high-frequency Fourier coefficients using numpy

Question

I was told that applying the middle pool to the matrix M is equivalent to dumping the high-frequency components of the Fourier representation of M. With the average pool, I mean the average value of 2 by 2, as shown in this image:

I wanted to check this out and see how it works with numpy. Therefore, I wrote a naive implementation of the middle pool and copied a function to accurately display the matrices from here :

def prettyPrintMatrix(m):
    s = [['{:.3f}'.format(e) for e in row] for row in m]
    lens = [max(map(len, col)) for col in zip(*s)]
    fmt = '\t'.join('{{:{}}}'.format(x) for x in lens)
    table = [fmt.format(*row) for row in s]
    print '\n'.join(table)

def averagePool(im):
    imNew = np.empty((im.shape[0] /2, im.shape[1]/2))
    for i in range(imNew.shape[0]):
        for j in range(imNew.shape[1]):
            imNew[i,j] = np.average(im[(2*i):(2*i+2), (2*j):(2*j+2)])
    return imNew

Now, to check what happens to the Fourier coefficients, I executed the following code:

M = np.random.random((8,8))
Mpooled = averagePool(M)

# print the original M
print('original M:')
prettyPrintMatrix(M)

# print Fourier coefficients of regular matrix
print('Fourier of M')
prettyPrintMatrix(np.fft.fft2(M))

# print Fourier coefficients of pooled matrix
print('Fourier of the pooled M')
prettyPrintMatrix(np.fft.fft2(Mpooled))

Sample output for this:

original M:
0.849   0.454   0.231   0.605   0.375   0.842   0.533   0.954
0.489   0.097   0.990   0.199   0.572   0.262   0.299   0.634
0.477   0.052   0.429   0.670   0.323   0.458   0.459   0.954
0.984   0.884   0.620   0.657   0.352   0.765   0.897   0.642
0.179   0.894   0.835   0.710   0.916   0.544   0.968   0.557
0.253   0.197   0.813   0.450   0.936   0.165   0.169   0.712
0.677   0.544   0.507   0.107   0.733   0.334   0.056   0.171
0.356   0.639   0.580   0.517   0.763   0.401   0.771   0.219

Fourier of M
34.680+0.000j   -0.059-0.188j   0.076+1.227j    -1.356+1.515j   2.101+0.000j    -1.356-1.515j   0.076-1.227j    -0.059+0.188j
-1.968-1.684j   2.125-0.223j    2.277+1.442j    1.629-0.795j    -0.141+1.460j   0.694-2.363j    -0.627+0.971j   -0.847-2.094j
3.496+2.808j    -1.099+1.260j   0.921-0.814j    2.499+0.283j    -1.048-1.206j   -3.228+2.435j   -2.934+0.030j   0.386-0.015j 
0.451-0.301j    0.791-0.143j    -0.463-0.031j   1.841+0.032j    -1.979-1.066j   1.344-1.229j    3.487-1.297j    2.105-2.455j 
0.111+0.000j    0.166+1.317j    0.946-0.016j    0.587-0.443j    -2.710+0.000j   0.587+0.443j    0.946+0.016j    0.166-1.317j 
0.451+0.301j    2.105+2.455j    3.487+1.297j    1.344+1.229j    -1.979+1.066j   1.841-0.032j    -0.463+0.031j   0.791+0.143j 
3.496-2.808j    0.386+0.015j    -2.934-0.030j   -3.228-2.435j   -1.048+1.206j   2.499-0.283j    0.921+0.814j    -1.099-1.260j
-1.968+1.684j   -0.847+2.094j   -0.627-0.971j   0.694+2.363j    -0.141-1.460j   1.629+0.795j    2.277-1.442j    2.125+0.223j 

Fourier of the pooled M
8.670+0.000j    -0.180+0.019j   -0.288+0.000j   -0.180-0.019j
-0.228-0.562j   0.487+0.071j    0.156+0.638j    -0.049-0.328j
0.172+0.000j    -0.421-0.022j   -0.530+0.000j   -0.421+0.022j
-0.228+0.562j   -0.049+0.328j   0.156-0.638j    0.487-0.071j 

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Answer 1

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