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```# 2D Discrete Fourier Transform (DFT) and its inverse
# Warning: Computation is slow so only suitable for thumbnail size images!
# FB - 20150102
from PIL import Image
import cmath
pi2 = cmath.pi * 2.0

def DFT2D(image):
global M, N
(M, N) = image.size # (imgx, imgy)
dft2d_red = [[0.0 for k in range(M)] for l in range(N)]
dft2d_grn = [[0.0 for k in range(M)] for l in range(N)]
dft2d_blu = [[0.0 for k in range(M)] for l in range(N)]
for k in range(M):
for l in range(N):
sum_red = 0.0
sum_grn = 0.0
sum_blu = 0.0
for m in range(M):
for n in range(N):
(red, grn, blu, alpha) = pixels[m, n]
e = cmath.exp(- 1j * pi2 * (float(k * m) / M + float(l * n) / N))
sum_red += red * e
sum_grn += grn * e
sum_blu += blu * e
dft2d_red[l][k] = sum_red / M / N
dft2d_grn[l][k] = sum_grn / M / N
dft2d_blu[l][k] = sum_blu / M / N
return (dft2d_red, dft2d_grn, dft2d_blu)

def IDFT2D(dft2d):
(dft2d_red, dft2d_grn, dft2d_blu) = dft2d
global M, N
image = Image.new("RGB", (M, N))
for m in range(M):
for n in range(N):
sum_red = 0.0
sum_grn = 0.0
sum_blu = 0.0
for k in range(M):
for l in range(N):
e = cmath.exp(1j * pi2 * (float(k * m) / M + float(l * n) / N))
sum_red += dft2d_red[l][k] * e
sum_grn += dft2d_grn[l][k] * e
sum_blu += dft2d_blu[l][k] * e
red = int(sum_red.real + 0.5)
grn = int(sum_grn.real + 0.5)
blu = int(sum_blu.real + 0.5)
pixels[m, n] = (red, grn, blu)
return image

# TEST
# Recreate input image from 2D DFT results to compare to input image
image = IDFT2D(DFT2D(Image.open("input.png")))
image.save("output.png", "PNG")
```