What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?
I searched on Google but couldn't find a satisfiable answer.
Thanks in advance :)
2 Answers
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The usual (discrete) Fourier transform expands a function in the basis set $e^{i n\phi}$, $\phi\in(0,2\pi)$, $n\in\mathbb{Z}$. The $\text{SO}(3)$ Fourier transform uses as basis the Wigner D-functions $D_{\ell}^{m,n}$, which are an orthogonal basis for the rotation group, $$f=\sum_{\ell=1}^L\sum_{m,n=-\ell}^\ell f_{\ell,m,n}D_\ell^{m,n}.$$ The integer $L$ is the degree of the transform. It is possible to rewrite this as a usual Fourier transform by expanding the Wigner-D function into a Fourier sum and then one can apply the Fast Fourier Transform algorithm, see A Fast Fourier Algorithm on the Rotation Group. The complexity of this operation scales as $L^3\log L$.
This is an example of noncommutative harmonic analysis.
answered Jun 4, 2022 at 6:53
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2$\begingroup$ Why the scaling is not $\mathcal{O}(\log(L)L^3)$? $\endgroup$– yarchikCommented Jun 5, 2022 at 9:34
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$\begingroup$ indeed it does, corrected --- thanks. $\endgroup$ Commented Jun 5, 2022 at 11:27
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In addition to what Carlo Beenakker stated, you can find a detailed introduction to Fourier Transforms, Spherical Harmonics and Wigner-D matrices in Chapter 3 of a 2017 PhD thesis at Bielefeld University linked here.
It focuses on applications for visual robot navigation (and therefore especially on real-valued implementations), but a brief summary of the general idea is given. For a purely math based focus, you can check the references.
