2D fourier transform

Question

I an new to image processing and I have some questions I want to understand:

What is the meaning of the imaginary and real part of the 2d fourier transform? What can it tell me about the image it transformed from? For example, what can I tell about an image that its fourier transform omaginary part is always 0? and its real part always 0? What can I tell about an image that it fourier transforn in (0,0) equals 0?

Answer 1

It has same meaning as the 1d Fourier transform: the magnitude and phase of waves that, when superimposed, reconstruct your original signal. In one dimension,

1. the signal is (usually) a function of time, $f(t)$,
2. the Fourier transform takes you into the frequency domain, a kind of "inverse time", and
3. each function value, $F(\omega)$, tells you how much of the wave with frequency $\omega$ you have. Additionally, since we admit positive and negative frequencies, we can think of those waves as propagating "forward" or "backwards" along the time axis depending on the sign.

Now, in two (or more) dimensions, all of that still holds:

1. The signal is a function of space, $f(\mathbf{r})$ (where $\mathbf{r}$ is vector-valued),
2. The Fourier transform takes you into a spatial frequency domain, a kind of "inverse space" (k-space),
3. Each function value, $F(\mathbf{k})$, tells you how much of the wave with that (spatial) frequency and propagating in that direction (along $\mathbf{k}$) you have.

What can I tell about an image that it fourier transforn in (0,0) equals 0?

Following from the above, if $\mathbf{k} = \mathbf{0}$ and $F(\mathbf{k})$ = 0, then

• you have no stationary signal component.
• The "amount of unpropagating wave" in your original signal is zero.
• The average value of your signal is zero.

All of these are equivalent statements.