IFFT of FFT-image results in weird upside-down overlap of original image

Question

In the following code I have a function which returns an image cropped to a centered circle of a certain radius. Then I perform fourier-tranformation of an image, and reverse fourier-transformation again, to restore the image, which works fine.

Then I have calculated, that a centered circle with radius of 43 of the energy spectrum (excluded here) will yield 99% of the energy of the original image. But when I try to apply my function "imgRadius" to the amplitude-spectrum (the fourier-transformed image), and then perform inverse-fourier-tranformation on that image, I get this weird upside-down overlap of the original image.

def imgRadius(img, radius):
    result = np.zeros(img.shape,np.float64)
    centerX = (img.shape[0])/2
    centerY = (img.shape[1])/2
    for m in range(img.shape[0]):
        for n in range(img.shape[1]):
            if math.sqrt((m-centerX)**2+(n-centerY)**2) < radius:
                result[m,n] = img[m,n]
    return result

imgpath = directory+"NorbertWiener.JPG"
img = cv.imread(imgpath, cv.IMREAD_GRAYSCALE)
norm_image = cv.normalize(img, None, alpha=0, beta=1, norm_type=cv.NORM_MINMAX, dtype=cv.CV_32F)
amp = (np.fft.fftshift(np.fft.fft2(norm_image)))
amp_log = np.log(np.abs(amp))
norm_amp = cv.normalize(amp_log, None, alpha=0, beta=1, norm_type=cv.NORM_MINMAX, dtype=cv.CV_32F)
restoredAMP = np.abs(np.fft.ifft2(np.fft.ifftshift(amp)))
radamp = imgRadius(norm_amp,43)
rest99 = np.abs(np.fft.ifft2((imgRadius(amp,43))))

plt.subplot(231),plt.imshow(norm_image, "gray", vmin=0, vmax=1),plt.title('Image')
plt.xticks([]), plt.yticks([])
plt.subplot(232),plt.imshow(norm_amp, "gray", vmin=0, vmax=1),plt.title('Amplitude')
plt.xticks([]), plt.yticks([])
plt.subplot(233),plt.imshow(restoredAMP, "gray", vmin=0, vmax=1),plt.title('Restored from amplitude')
plt.xticks([]), plt.yticks([])
plt.subplot(235),plt.imshow(rest99, "gray", vmin=0, vmax=1),plt.title('Restored from r=43')
plt.xticks([]), plt.yticks([])
plt.subplot(234),plt.imshow(radamp, "gray", vmin=0, vmax=1),plt.title('Amplitude r=43')
plt.xticks([]), plt.yticks([])
plt.show()

Does it somehow have anything to do with the fact that I use absolute values? I don't see how I should be able to plot the image without getting rid of the imaginary parts of the images, but I can see how maybe some information gets lost in the inverse fft.

I just don't get why I don't get problems when performing IFFT on the original amplitude spectrum.

Answer 1

The problem happens here:

def imgRadius(img, radius):
    result = np.zeros(img.shape,np.float64)

You are creating a real-valued array, and copying over the complex values. Likely either the real component or the magnitude is written to the array. In any case, the complex-valued frequency domain data becomes real-valued. The inverse transform is a symmetric matrix.

To solve the problem, initialize result as a complex-valued array.

After this, make sure to use the real component of the inverse transform, not the magnitude, as Gianluca already suggested in their answer.