rotation in frequency domain - question about question 64441200 - rotation

Question

Following : How to rotate a non-squared image in frequency domain
I just ran this exact code (copy pasted from the author's code with just an additional normalization of the resulting image between 0 an 255 as shown)
... but I get horrible "aliasing" artifacts ... How is this possible ? I see that the OP shows nice unartifacted images from the rotation in frequency space... I would be very curious to know how to obtain that, surely you did not show all your code?
import numpy as np
import cv2
from numpy.fft import fftshift as fftshift
from numpy.fft import ifftshift as ifftshift
angle = 30
M = cv2.imread("phantom.png")
M = cv2.cvtColor(M, cv2.COLOR_BGR2GRAY)
M=np.float32(M)
hanning=cv2.createHanningWindow((M.shape[1],M.shape[0]),cv2.CV_32F)
M=hanning*M
sM = fftshift(M)
rotation_center=(M.shape[1]/2,M.shape[0]/2)
rot_matrix=cv2.getRotationMatrix2D(rotation_center,angle,1.0)
FsM = fftshift(cv2.dft(sM,flags = cv2.DFT_COMPLEX_OUTPUT))
rFsM=cv2.warpAffine(FsM,rot_matrix,(FsM.shape[1],FsM.shape[0]),flags=cv2.INTER_LINEAR, borderMode=cv2.BORDER_CONSTANT)
IrFsM = ifftshift(cv2.idft(ifftshift(rFsM),flags=cv2.DFT_REAL_OUTPUT))
x = IrFsM
x = ((x-np.min(x[:]))/(np.max(x[:])-np.min(x[:])))*255.0
cv2.imwrite('rotated_phantom.png',x)
the output image is:
Also, i ve always been told that it was impossible to correctly rotate (in discrete, not continuous) Fourier space because of interpolation, so how can you explain that?

Answer 1

The DFT imposes a periodicity to the image in both the frequency and the spatial domain (some people disagree with this, but still agree that this view is a good way to explain just about everything that happens in the DFT...). So imagine that your input is not the Shepp-Logan phantom, but an infinite repetition of it. When manipulating the data in the frequency domain, you affect not just the one copy of the image you see, but all of them, and not always in intuitive ways.
One of the consequences is that neighboring copies of your image in the spatial domain rotate, but also expand and come into you image.
The simplest way to avoid this is to pad the image with zeros to double its size.
import numpy as np
import cv2
from numpy.fft import fftshift as fftshift
from numpy.fft import ifftshift as ifftshift
angle = 30
M = cv2.imread("shepp-logan-small.tif")
M = cv2.cvtColor(M, cv2.COLOR_BGR2GRAY)
M = np.float32(M)
# Pad
v, h = M.shape
v //= 2
h //= 2
M = cv2.copyMakeBorder(M, v, v, h, h, cv2.BORDER_CONSTANT)
Now you can apply the rotation in the frequency domain like before. But note that the center of rotation should the pixel at shape//2, do not use a true division! Also note that we no longer need to apply a window function.
sM = fftshift(M)
rotation_center = (M.shape[1]//2,M.shape[0]//2)
rot_matrix = cv2.getRotationMatrix2D(rotation_center,angle,1.0)
FsM = fftshift(cv2.dft(sM,flags = cv2.DFT_COMPLEX_OUTPUT))
rFsM = cv2.warpAffine(FsM,rot_matrix,(FsM.shape[1],FsM.shape[0]),flags=cv2.INTER_LINEAR, borderMode=cv2.BORDER_CONSTANT)
IrFsM = ifftshift(cv2.idft(ifftshift(rFsM),flags=cv2.DFT_REAL_OUTPUT))
Finally, crop the result back to its original size.
# Crop
IrFsM = IrFsM[v:-v, h:-h]
Do note that the result is not pretty. It is much better to rotate in the spatial domain because, as you said, interpolation in the frequency domain is not really sensical.