Why real part of FFT converts image into rotation + original?

Question

I have read this image:

taken its FFT (2D) and then Inverse FFT to get exactly the image back. Code is provided for reference:

imfft = fft2(photographer);
im = uint8(ifft2(imfft));

imshow(im); %Output is same image

But when I change the fourier and take only the real part,

imfft = real(fft2(photographer));
im = uint8(ifft2(imfft));
imshow(im);

I get an image like this (note that size change is irrelevant and only due to saving it from Matlab figure handler):

Can someone explain me the theory (maths) behind it? Thanks

Answer 1

Let's say your image is given by $I(x,y)$. Then its Fourier transform is given by $$ I^f(\omega_x,\omega_y) = \int_x\int_yI(x,y)e^{j\omega_xx}e^{j\omega_yy}dxdy $$

Now you take the real part and perform the inverse:

\begin{align} I_m(\alpha,\beta) &= \int_{\omega_x}\int_{\omega_y}\Re\left\{I^f(\omega_x,\omega_y)\right\} e^{j\omega_x\alpha}e^{j\omega_y\beta}d\omega_xd\omega_y \\ &= \int_{\omega_x}\int_{\omega_y}\Re\left\{\int_x\int_yI(x,y)e^{j\omega_xx}e^{j\omega_yy}dxdy\right\} e^{j\omega_x\alpha}e^{j\omega_y\beta}d\omega_xd\omega_y\\ &= \int_x\int_yI(x,y)\int_{\omega_x}\int_{\omega_y}\Re\left\{e^{j\omega_xx}e^{j\omega_yy}\right\} e^{j\omega_x\alpha}e^{j\omega_y\beta}d\omega_xd\omega_ydxdy \end{align}

You can clearly see that the inner integral is the 2D Fourier transform of $$ \cos(\omega_xx)\cos(\omega_yy) + \sin(\omega_xx)\sin(\omega_yy) $$ which is $$ \frac{1}{2}\left[\delta(x-\alpha)\delta(y-\beta) + \delta(x+\alpha)\delta(y+\beta)\right] $$

Substituting the result to $I_m$ yields $$ I_m(x,y) = \frac{1}{2}\left[I(x,y)+I(-x,-y)\right] $$

Of course in your case $x,y>0$, however the discrete Fourier transform assumes your signal is $N$-periodic and you get $$ I_m(x,y) = \frac{1}{2}\left[I(x,y)+I(N-x,M-y)\right] $$ where $N,M$ are the dimensions of your image. I think you can see now why you got that result.

Answer 2

The result ThP provided can also be stated in very simple terms: If you have a data set which is purely real, its (inverse) Fourier transform will have Hermitian symmetry: If you find the value $z$ at position $(x,y)$, then you will find the complex conjugate value $z^*$ at the point-reflected position $(-x,-y)$ about the origin. Note that the origin here would be the center of Fourier-space. This can be reformulated, of course, if the DC component is not in the center of your FFT implementation. And this is what you see in your image: A point-reflected version is overlaying the true image - because you forced one space to be real valued.

This property is actually being used for accelerating magnetic resonance imaging (MRI) in some cases: MRI acquires the data directly in Fourier-space. Since an ideal MR image can be described by real values only (all excited magnetization vectors have phase 0), you only have to acquire half of the data space, which saves you half of the imaging time. Of course, MR images are not completely real valued due to the limiations of reality... but with a few tricks you can still use this technique advantageously.