I know Fourier Transformations only a little bit, I guess/hope, I understood the concepts, at least. One scenario is, e.g., describing a signal in the so-called time domain in the so-called frequency domain. This means, when observing a sound wave, you can decompose it into single frequencies it consists of.
Now, I'm trying to understand how it is possible to connect (local) slopes of a light wavefront to topology data. For a bit more understanding: Measure light that leaves an eye and the wavefront provides information about the cornea in terms of (gradients of) a topology.
This is actually done by a Fourier Transformation.
I don't see any time or frequency domain, here. Or are they? In case not, this leads me to a more general basic question: What exactly is Fourier Transformation doing? Not in technical matters but more like "What could be done with it?"
Best Answer
Turning my comment into an answer:
The Fourier transform can be defined in a few different ways. In digital signal processing, one often works with time and (angular) frequency. In this setting, the Fourier transform takes the form
$$ \mathcal{F}f(\omega) = \int_{\mathbb{R}} \frac{1}{\sqrt{2\pi}} e^{-i\omega t} f(t)\,dt $$
depending on your Fourier transform convention (i.e. where you want your $2\pi$ to be).
In physics, geophysics, and elsewhere, you're likely to run into a slightly different notion of the Fourier transform: one that connects position with wavenumber. Since you can have multiple position variables ($x$, $y$, $z$, etc.), you have that many wavenumber variables to match ($k_x$, $k_y$, $k_z$, etc.). The Fourier transform takes the form
$$ \mathcal{F} f(\vec{k}) = \int_{\mathbb{R}^n} \frac{1}{(2\pi)^{n/2}} e^{-i \vec{k}\cdot\vec{x}} f(\vec{x})\,d\vec{x}. $$
In your case, you have a 2D image which means two spatial variables, so you have two wavenumber variables. The heat map is then generated by taking the modulus (possibly squared) of the Fourier transform to get rid of the imaginary part and only look at the magnitude.
For computational purposes, the $n$-dimensional Fourier transform can be discretized just like the 1D Fourier transform to give rise to DFT and FFT analogues.
As an aside to prevent any confusion: you can get a heat map, often called a spectrogram in digital signal processing, from a time-varying signal by way of what is called a short-time Fourier transform (taking the Fourier transform of your function with a moving window). What you have is not that as far as I can tell. You have a naturally 2D image from the fact that you have two wavenumber variables to match your two spatial variables.
As for when/why to use a Fourier transform, there's no one-size-fits-all answer. Sometimes Fourier transforms simplify a (partial) differential equation (e.g. the heat equation). Sometimes you want to remove or alter high/low frequency/wavenumber information from the data because it's not representative, e.g. removing a vocal miscue in a recording for a song or clearing out some ringing. Sometimes you want to attenuate certain frequencies/wavenumbers. Sometimes you want to identify certain phenomena that you can't see in the time or spatial domains. Short-time Fourier transforms offer a glimpse into both worlds simultaneously to help you identify (roughly) specifically when a frequency/wavenumber phenomenon happens.