Discrete Fourier Transform: Inverse of a 2D periodic signal results in doubled frequency-CodePudding

When converting a periodic 2D signal from image space to Fourier space and back, the reconstructed signal has twice the frequency of the original signal (see picture below). I tried to use the Discrete Fourier Transform from NumPy and OpenCV, both with the same result. The problem does not occur when using a 1D DFT and IDFT.

Do you have any ideas what is the source of this problem could be and how to fix it?

Here is a sample code in Python demonstrating the issue:

import matplotlib.pyplot as plt
import numpy as np

# image width and height
w = 320
h = 320

# frequency of cosine wave w.r.t. the image width
frequency = 1

# function to create a horizontal 2D cosine wave
def create_cos_horizontal(frequency, w, h):
    img = np.zeros((h,w),np.float32)
    base_period = w/(2*np.pi)
    print(frequency)
    for x in range(0, w):
        img[0:, x] = np.cos(x*frequency/base_period)      
    return img

img = create_cos_horizontal(frequency, w, h)

# transform from image space to fourier space
dft = np.fft.fft2(img)
# transform back from fourier space to image space
im_back = np.fft.ifft2(dft)

# show original and reconstructed image side by side
ax1 = plt.subplot(1,2,1)
ax1.imshow(img, cmap='gray')
ax1.set_title("original signal")
ax2 = plt.subplot(1,2,2)
ax2.imshow(np.abs(im_back), cmap='gray')
ax2.set_title("signal after DFT and IDFT")

Thank you so much in advance.

CodePudding user response：

The call to abs() in the second plot is rectifying the cosine, inverting the negative part of the curve. If you replace it with real(im_back) the plots match as expected.