I am following Robert Fisher's page on the Fourier transform. I understand the application of the Fourier transform behind image processing, but, right now, I am curious about the mathematics behind it, and it is giving me a bit of a hard time. For example,
$$ F (k,l) = \sum_{i=0}^{N-1} \sum_{j=0}^{N-1} f (i,j) \exp \left( - \iota 2 \pi \left( \frac{k i}{N} + \frac{l j}{N} \right) \right) $$
In this formula, where do all these equations come from? Could somebody please elaborate the mathematics behind the scene in layman's term?
I have given a similar explanation of Fourier series here and here.
The Fourier series of a function $f : \mathbb R / \mathbb Z \to \mathbb C$ is a sum $\sum_{k = -\infty}^\infty \hat{f}(e^{2 \pi i k x}) e^{2 \pi i k x}$. Here, $e^{2 \pi i k x} = \chi_k (x)$ denotes the $k$-th character of the topological group $\mathbb R / \mathbb Z $ and $\hat{f} : \widehat{\mathbb R / \mathbb Z} \to \mathbb C$ denotes the Fourier transform of $f$.
The setting is: You have a topological group $G$, then you consider a space of functions $G \to \mathbb C$ on it, say for example $L^2(G)$. This is a Hilbert space and comes with an inner product $\langle \cdot , \cdot \rangle$ which lets you define orthogonality of functions in the space, and characters in particular. Those characters form an orthonormal basis for your functions which means that you can approximate every function in $L^2(G)$ as $\| f - \sum_{k = -N}^N \hat{f}(\chi_k) \chi_k (x) \|_2 \xrightarrow{N \to \infty} 0$.