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Comment by Scene_Cast2 | Hacker News Reader

If anyone is curious about applications - these can be used to approximate low-frequency components of a point's surroundings. They were used in Halo 3 to do real-time HDRI lighting and shadowing (see "Lighting and Material of Halo 3" from Siggraph 2008).

After the success of this method, there was a fairly long stretch of researchers looking for a better orthonormal basis (such as 2D Haar wavelents, as spherical harmonics is basically a Fourier Transform on a spherical basis). I think the pinnacle of this direction was Anisotropic Spherical Gaussians from 2013.

These days though, you'd at least use a neural net to learn a basis (or use a neural net to learn something else entirely). And of course, Gaussian Splats are the technique du jour for realtime relighting.

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They are used also to characterize the statistical properties of fields over the sphere. A notable example is the pattern of hot/cold spots in the Cosmic Microwave Background Radiation (CMBR, [1]). They are distributed stochastically, and the best way to fit cosmological models against the measurements is to decompose the temperature/polarization fields into spherical harmonics and compute the power spectrum associated with each ℓ (which plays the role of a “spatial frequency” over the sky sphere).

[1] https://en.wikipedia.org/wiki/Cosmic_microwave_background

Another application is Ambisonic sound spatialisation formats, where a finite set of signals corresponding to spherical harmonics are used to encode spatial sound fields. https://en.wikipedia.org/wiki/Ambisonics

I wonder if Cartesian-basis multipole expansion could get the best of both worlds of GS and SH, as the former basis captures anisotropy but not detail while the latter captures detail but not anisotropy, whereas Cartesian multipole expansion naturally captures both right from the low orders, and is much easier to align to game worlds.

(to be precise, both can be captured by either if you include enough orders, what I mean is mainly how the information distribution scales with respect to each attribute)

Also, the age-old physics question: what is the minimum order of spherical harmonics required to approximate a cow?

The vector version of these is used in antenna theory to represent and transform the radiation of finite sources, like when measuring antenna patterns.

in QM the spherical harmonics are more of a basis space for electronic state and not the actual electronic state, right? So does that mean there are other ways to think about electron configurations that satisfy Shrodinger etc?

Yes - for example when modeling solid materials it’s common to use a plane wave basis set for the electronic wave function instead of atomic orbitals

Spherical harmonics are exact solutions to electronic states for hydrogen-«style» atoms.

... for the angular dependence; there is also radial dependence.

I do want to point out, gaussian splats don't really offer anything in particular for realtime relighting, if anything it adds additional challenges. Under the hood, most implementations leverage spherical harmonics for baked-in lighting.

Did you by chance mean anisotropic spherical gaussians? ASG is a new-ish technique often used to model specular lighting but is unrelated to gaussian splatting.

I have "Spherical Harmonic Lighting: The Gritty Details" in my PDF stash: https://www.semanticscholar.org/paper/Spherical-Harmonic-Lig...

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