retrieving original data from phase and magnitude of Fourier transform

Question

I use this snippet of python code to transform data to Fourier phase and magnitude and then retrieving original data.

from random import randint as RI
import numpy.fft as FFT
import math

w = 4
h = 4
random_range = 255

vals = [[] for i in range(h)]

for i in range(h):
    for j in range(w):
        vals[i].append(RI(0,random_range))

fftc = FFT.fft2(vals) 

magnitudes = [[] for i in range(h)]
phases = [[] for i in range(h)]

for i in range(h):
    for j in range(w):
        magnitudes[i].append(math.sqrt(fftc[i][j].real**2+fftc[i][j].imag**2))
        phases[i].append(math.atan(fftc[i][j].imag/fftc[i][j].real))

for_ifft = [[] for i in range(h)]

for i in range(h):
    for j in range(w):
        rl = magnitudes[i][j]*math.cos(phases[i][j])
        im = magnitudes[i][j]*math.sin(phases[i][j])
        for_ifft[i].append(complex(rl,im))


ifftc = FFT.ifft2(for_ifft)


print vals
print ifftc

When i compare outputs, results are different. for example two typical outputs are:

//original data
[[115, 4, 33, 91], [228, 123, 46, 111], [83, 227, 81, 55], [229, 188, 138, 41]]

//retrieved data from phase and magnitude informations
[[ 309,182.25,105,57.25],[ 33,118.75,120.50,116.25],[ 148,128.75,79,94.75],[ 165,112.25,-6.50,29.75]]

I know the theory behind it. for example for calculating phase and magnitude we do this :

Magnitude = Square_Root(real*real+imaginary*imaginary)
Phase = Inverse_Tangent(imaginary/real)

and for calculating complex number from magnitude and phase we do this :

Real = Magnitude * cos(Phase)
Imaginary = Magnitude * sin(Phase)

So, what is the cause of problem. Is it a data scaling issue.

Answer 1

You need to change how you're calculating the phase from "atan" to "atan2". The reason is that atan doesn't work in all quadrants. More on atan2 here.