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I'm trying to analyze the power spectrum of a 2D map, which is a map of the orientation of a field. Before going to discrete problem, I had difficulties finding out the Fourier transform of such kind of functions.

Briefly speaking, the problem arises because orientations, which is described by "angles", are "cyclic", namely $\theta=\theta+2\pi$. So whenever I have two points with angles like $0^\circ$ and $360^\circ$, they should be treated the same. Moreover, the difference between, e.g., $20^\circ$ and $50^\circ$ is also equivalent to that of $10^\circ$ and $340^\circ$.

Mathematically, I think Fourier transforming angle-valued functions would be effectively doing FT on a closed manifold like a circle, so the function is actually a map from angles to points on the circle.

But that could be very complicated and probably even harder for discrete case. So instead I tried to construct the desired FT from composite functions:

For simplicity, below is for 1D case only. By introducing $\tan^{-1}$, I can transform the angle-valued function into real-valued function. That is,

Let $f(x)=\tan^{-1}(g(x))$ be the "angle-valued" function, and $g(x)$ is the real-valued function, with $0^\circ \leqslant f \leqslant 360^\circ$.

Because $\int \frac{1}{1+x^2}=\tan^{-1}(x)$, according to integral property of FT, we have

$$\mathfrak{F}\{\tan^{-1}(x)\}=\frac{\mathfrak{F}\{1/1+x^2\}}{i\omega}=\sqrt{\frac{\pi}{\omega}}\frac{e^{-|\omega|}}{i\omega}$$

where $\mathfrak{F}$ denotes the Fourier transform. Then for the composite function $\tan^{-1}(g(x))$, we have (ref here)

$$\mathfrak{F}\{\tan^{-1}(g(x))\}(k)=\int\sqrt{\frac{\pi}{\omega}}\frac{e^{-|\omega|}}{i\omega}\int \text{exp}\{i2\pi(\omega g(x)-kx)\}dx d\omega$$

If I've done it correctly, how do I extend it to 2-dimensional $g(x,y)$? And how do I take into account the discreteness of my data? Or is there any simpler way (or existing algorithms) to obtain the desired result?

(Since my goal is to do it numerically, I'm not sure if I should post this here or elsewhere, given that even mathematically I don't know how to solve this)

Thanks !

EIDT After careful inspection, I found that the above method still does not and will not work since for $-\infty\leqslant g(x)\leqslant\infty$, where the upper limit and lower limit corresponds to $\tan^{-1}=-\pi/2$ and $\tan^{-1}=\pi/2$ respectively, $g(x)$ itself should be "cyclic" as well.

So my approach involves merely a "scaling" (from angle to the entire real line), and by no means it solves my original question.

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shaopao
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Depends on what you want, but one "way out" is that you transform your angle-valued function into a complex field with an additional constraint of unit length. That is, just do a fourier transform on a function

$$f(x,y)=e^{i\theta(x,y)}$$

The meaning of the spectrum you get out will of course be something different, but will make much more sense in terms of what you want to observe. Now you don't have any discontinuities and ambiguities and the real and imaginary parts encode the components of the "vector field".

Additionally, using this function can accomodate topological defects in the angular order (there may be places where magnitude drops to 0 and the angle turns around for nontrivial amount).

orion
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