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.
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).
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?
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.
For those of you curious about WHY these shapes look like the do (e.g. "why does l=0, m=0 have a donut in the middle of two lobes?"), this video from Münster University finally gave me an intuitive understanding of how these shapes arise.
Very very high level explanation that I'm trying to paraphrase from memory, so there's a good chance parts of it are wrong or use the wrong terminology:
A polynomial is a function like `f(x) = Ax^3 + Bx^2 + Cx^1 + Dx^0`
You can approximate most(all?) continuous functions using polynomials. The more "parts" (e.g. Bx^2 is one part) of the polynomial you have, the better you can represent a given function.
The previous example was a 1d function, but polynomials can be over any number of dimensions. E.g. a 3d polynomial `f(x, y, z)`.
Spherical harmonics are just a form of 3d polynomials, but with some special "parts" (called a basis function), with the amount of parts you have called a "band".
As for what they're good for, they're a fairly compact way of representing and filtering 3d signals.
In 3d rendering, they're really good at storing light hitting a point from different directions. You have an incoming ray of light on a unit sphere/hemisphere with origin x, y, z going towards the given point. You can then take your list of light rays with various (x,y,z) origins and (r,g,b) intensities, and then form a spherical harmonics approximation over it (basically a fitted 3d polynomial), which can be stored as just a few coefficients (A, B, C, D, etc...) using 1 or 2 bands, and cheaply computed (querying the light value r,g,b for a given ray) by plugging in the ray's x, y, z into the formula.
Besides being cheap to store and query, because you're only using 1-2 bands, you only capture the "low frequency" of the lighting signal, e.g. large changes in the light value get dropped, since it's just an approximation of the original signal. While this is normally bad, for 3d rendering, it's free denoising, giving you a smoother output image!
Coming at them from practical applications is one approach.
I've used spherical harmonics to model earth centric "surfaces" and fields - magnetics and gravity, etc.
You might think of them as a stacked sine and cosine waves (like a fourier transform breaking a continuous function into sin and cosine components) on a directional vector radiating outwards from the centre point of sphere.
I think of it as a good basis for functions on a perfectly spherical surface. Going down in levels of "l", you describe more and more details in terms of angular scale.
Thus, it's widely used in earth science and astrophysics, and anything that involves spherical symmetry (like a Hydrogen atom) -- in reality, nothing is a perfect sphere, but that's a very good approximation.
It's the list of "waves" that can propagate around the surface of a sphere without interfering with each other. They are self-reinforcing modes. So you can represent any function of values on the surface as a combination of these, the same way you do with e.g. FFT coefficients in a JPEG file.
And it turns out that this "self-reinforcing" property is critically important for quantum mechanics, as each of the functions defines a different "state" from the perspective of a "particle". So we can do a lot of good physics work[1] by pretending[2] that all electrons exist in one of these states.
[1] Like, y'know, explaining chemistry.
[2] The details are always harder, because the electrons interact with other ways than just flying around the sphere, so the math isn't tractable in an absolute sense. But as long as you pretend that this is mostly right you can treat the remainder as just "fixups" in a giant framework called perturbation theory.
Or more generally a sum over all real values, f(x) = ∫ a(k) cos(kx) + b(k) sin(kx) dk, since signals can have fractional frequencies.
And in many cases the two sides are the same. Many operations in math and in ph operate on functions in such a way that they can distribute over their behavior on different frequencies, which is why Fourier transforms are really useful. For instance we hear different frequencies in different ways so if you Fourier-transform an audio waveform you can turn some frequencies up and others down (or drop them entirely, which is more-or-less what mp3 compression is).
This also works in 2d or 3d, where now you have frequencies in all three directions:
f(x,y,z) = f(0) + a_(100) cos(x) + a_(110) cos(x) cos(y) + a_(111) cos(x) cos(y) cos(z) + (all the sine terms and negative frequencies and everything else)
and this is useful in all kinds of ways also, e.g. taking Fourier transforms of a 2d image and then dropping the high frequency components that are hard to see is basically what JPEG compression is. As before it helps a lot that you can interact with the different frequency components separately.
But. Sometimes what you want is not the frequency in linear space (e.g. the frequency in x or y) but the frequency in angular coordinates: to ask "how many times does this variable change as you go around a circle in the (xy) plane?" Which is to say, you want to know the Fourier component in term like cos(ϕ), cos(2ϕ), sin(ϕ), etc. That looks like
Which is what we would call a "circular harmonic" (with ϕ=ϕ(x,y)=arctan(y/x)), each coefficient a_i is a function of (r). Unlike the linear Fourier transform, there can be no "fractional" frequencies---since ϕ=0 and ϕ=2pi are the same point, all the circular frequencies have to be integers.
When you try to do this in 3d using two angular coordinates ϕ and θ, it gets a lot funkier. Now you can't really write it as a simple series of the two frequencies separately; they kinda "step on each other", because a rotation in (xy) can be written as a sum of rotations in (yz) and (zx).
But you can do it in terms of a different, slightly stranger series. One variable L=0,1,2,3... will describe the "total" frequency on any plane, and another variable m=-L,-L+1,...0,...L-1,L describes how many rotations happen in your favorite choice of (xy) plane. m is allowed to range from -L to L because we have already said that L is the total frequency on any axis, so we just have to say whether they're happening in our chosen axis or not. So the series becomes
The functions Y_(L,m) (θ,ϕ) are the "spherical harmonics". They serve the role of sin(ωx) and cos(ωx) when you Fourier-expand a function in terms of spherical coordinates ϕ and θ.
There are lots of reasons that that's useful, but the case that is most well-known is that the state of an electron wave function in an atom can be indexed in terms of which spherical harmonic it's in, and only two electrons are allowed to be in each one (one spin up and one spin down, for much-more-bizarre reasons). So the spherical harmonic functions are also the shape of the various electron orbitals that you see in a chemistry textbook.
You know if you take a metal plate bolted down at it's center, throw some salt on it and bow the edge with a violin bow. It makes stable patterns depending on the frequency of the tone playing
Now take this plate and turn it into a 3d sphere and that's spherical harmonica more or less. Electronsike to form them
An one dimensional wave can be decomposed as a sum of sines/cosines. So sines form a vector basis in the space of periodic functions
A 3D wave coming from a single point can be decomposed as a sum of spherical harmonics. It's a basis for waves, but ones expanding radially in 3D space
Spherical harmonics are essentially orthogonal basis functions on the sphere. They are popular in 3D graphics because you can very quickly compute the sphere integral over the product of two functions, as the dot product of their basis function coefficients(!). This can be used for real-time complex lighting on relatively modest hardware.
See also the “cubic harmonics”, which is an equivalent basis to spherical harmonics but they are real instead of complex, and also more natural to use in cubic crystals due to their symmetries.
I have also seen “triangular harmonics”, “zonal harmonics”, etc. in use in other materials.
Scene_Cast2|1 year ago
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.
ziotom78|1 year ago
[1] https://en.wikipedia.org/wiki/Cosmic_microwave_background
RossBencina|1 year ago
xeonmc|1 year ago
(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?
lloda|1 year ago
dustingetz|1 year ago
dwallin|1 year ago
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.
rzzzt|1 year ago
unknown|1 year ago
[deleted]
JeremyHerrman|1 year ago
https://youtu.be/Opufc3onVow
JeremyHerrman|1 year ago
vecter|1 year ago
aeve890|1 year ago
momoschili|1 year ago
pletnes|1 year ago
https://github.com/SHTOOLS/SHTOOLS
ghostpepper|1 year ago
jms55|1 year ago
A polynomial is a function like `f(x) = Ax^3 + Bx^2 + Cx^1 + Dx^0`
You can approximate most(all?) continuous functions using polynomials. The more "parts" (e.g. Bx^2 is one part) of the polynomial you have, the better you can represent a given function.
The previous example was a 1d function, but polynomials can be over any number of dimensions. E.g. a 3d polynomial `f(x, y, z)`.
Spherical harmonics are just a form of 3d polynomials, but with some special "parts" (called a basis function), with the amount of parts you have called a "band".
As for what they're good for, they're a fairly compact way of representing and filtering 3d signals.
In 3d rendering, they're really good at storing light hitting a point from different directions. You have an incoming ray of light on a unit sphere/hemisphere with origin x, y, z going towards the given point. You can then take your list of light rays with various (x,y,z) origins and (r,g,b) intensities, and then form a spherical harmonics approximation over it (basically a fitted 3d polynomial), which can be stored as just a few coefficients (A, B, C, D, etc...) using 1 or 2 bands, and cheaply computed (querying the light value r,g,b for a given ray) by plugging in the ray's x, y, z into the formula.
Besides being cheap to store and query, because you're only using 1-2 bands, you only capture the "low frequency" of the lighting signal, e.g. large changes in the light value get dropped, since it's just an approximation of the original signal. While this is normally bad, for 3d rendering, it's free denoising, giving you a smoother output image!
defrost|1 year ago
I've used spherical harmonics to model earth centric "surfaces" and fields - magnetics and gravity, etc.
You might think of them as a stacked sine and cosine waves (like a fourier transform breaking a continuous function into sin and cosine components) on a directional vector radiating outwards from the centre point of sphere.
https://geomag.bgs.ac.uk/research/modelling/IGRF.html
https://en.wikipedia.org/wiki/International_Geomagnetic_Refe...
https://en.wikipedia.org/wiki/World_Magnetic_Model
lizmutton|1 year ago
Thus, it's widely used in earth science and astrophysics, and anything that involves spherical symmetry (like a Hydrogen atom) -- in reality, nothing is a perfect sphere, but that's a very good approximation.
ajross|1 year ago
And it turns out that this "self-reinforcing" property is critically important for quantum mechanics, as each of the functions defines a different "state" from the perspective of a "particle". So we can do a lot of good physics work[1] by pretending[2] that all electrons exist in one of these states.
[1] Like, y'know, explaining chemistry.
[2] The details are always harder, because the electrons interact with other ways than just flying around the sphere, so the math isn't tractable in an absolute sense. But as long as you pretend that this is mostly right you can treat the remainder as just "fixups" in a giant framework called perturbation theory.
xeonmc|1 year ago
ajkjk|1 year ago
They are relatively easy to understand if you already understand Fourier transforms.
In a Fourier transform you can write some (suitably well-behaved) function f(x) as a sum of a bunch of sinusoids of different frequencies:
f(x) = a_0 + a_1 cos(x) + a_2 cos(2x) + ... + b_1 sin(x) + b_2 sin(2x) + ...
Or more generally a sum over all real values, f(x) = ∫ a(k) cos(kx) + b(k) sin(kx) dk, since signals can have fractional frequencies.
And in many cases the two sides are the same. Many operations in math and in ph operate on functions in such a way that they can distribute over their behavior on different frequencies, which is why Fourier transforms are really useful. For instance we hear different frequencies in different ways so if you Fourier-transform an audio waveform you can turn some frequencies up and others down (or drop them entirely, which is more-or-less what mp3 compression is).
This also works in 2d or 3d, where now you have frequencies in all three directions:
f(x,y,z) = f(0) + a_(100) cos(x) + a_(110) cos(x) cos(y) + a_(111) cos(x) cos(y) cos(z) + (all the sine terms and negative frequencies and everything else)
and this is useful in all kinds of ways also, e.g. taking Fourier transforms of a 2d image and then dropping the high frequency components that are hard to see is basically what JPEG compression is. As before it helps a lot that you can interact with the different frequency components separately.
But. Sometimes what you want is not the frequency in linear space (e.g. the frequency in x or y) but the frequency in angular coordinates: to ask "how many times does this variable change as you go around a circle in the (xy) plane?" Which is to say, you want to know the Fourier component in term like cos(ϕ), cos(2ϕ), sin(ϕ), etc. That looks like
f(x,y) = a_1 cos(ϕ) + a_2 cos(2ϕ) + b_1 sin(ϕ) + ...
Which is what we would call a "circular harmonic" (with ϕ=ϕ(x,y)=arctan(y/x)), each coefficient a_i is a function of (r). Unlike the linear Fourier transform, there can be no "fractional" frequencies---since ϕ=0 and ϕ=2pi are the same point, all the circular frequencies have to be integers.
When you try to do this in 3d using two angular coordinates ϕ and θ, it gets a lot funkier. Now you can't really write it as a simple series of the two frequencies separately; they kinda "step on each other", because a rotation in (xy) can be written as a sum of rotations in (yz) and (zx).
But you can do it in terms of a different, slightly stranger series. One variable L=0,1,2,3... will describe the "total" frequency on any plane, and another variable m=-L,-L+1,...0,...L-1,L describes how many rotations happen in your favorite choice of (xy) plane. m is allowed to range from -L to L because we have already said that L is the total frequency on any axis, so we just have to say whether they're happening in our chosen axis or not. So the series becomes
f(x,y,z) = a_(0, 0) + a_(1,-1) Y_(1,-1) + a_(1,0) Y_(1,0) + a_(1,1) Y_(1,1) + a_(2,-2) Y_(2,-2) + ... = ∑ a_(L,m) Y_(L,m) (θ,ϕ)
The functions Y_(L,m) (θ,ϕ) are the "spherical harmonics". They serve the role of sin(ωx) and cos(ωx) when you Fourier-expand a function in terms of spherical coordinates ϕ and θ.
There are lots of reasons that that's useful, but the case that is most well-known is that the state of an electron wave function in an atom can be indexed in terms of which spherical harmonic it's in, and only two electrons are allowed to be in each one (one spin up and one spin down, for much-more-bizarre reasons). So the spherical harmonic functions are also the shape of the various electron orbitals that you see in a chemistry textbook.
xkcd-sucks|1 year ago
Now take this plate and turn it into a 3d sphere and that's spherical harmonica more or less. Electronsike to form them
nextaccountic|1 year ago
A 3D wave coming from a single point can be decomposed as a sum of spherical harmonics. It's a basis for waves, but ones expanding radially in 3D space
raffihotter|1 year ago
bjornsing|1 year ago
quantum_state|1 year ago
NotYourLawyer|1 year ago
unknown|1 year ago
[deleted]
openrisk|1 year ago
https://en.m.wikipedia.org/wiki/Spinor_spherical_harmonics
tomxor|1 year ago
djmips|1 year ago
gus_massa|1 year ago
esperent|1 year ago
Compare them to the image on this page:
https://en.m.wikipedia.org/wiki/Spherical_harmonics
Suggestion to OP: this would be much more useful if you add a button to stop the rotation.
setopt|1 year ago
I have also seen “triangular harmonics”, “zonal harmonics”, etc. in use in other materials.
jms55|1 year ago
Also lots of other cool research around SH in rendering, e.g. the recent ZH3 paper.
raffihotter|1 year ago
ykonstant|1 year ago
lizmutton|1 year ago
liontwist|1 year ago
raffihotter|1 year ago
raffihotter|1 year ago
dagss|1 year ago
My experience is from cosmology (CMB) where they are heavily used just like Fourier transforms, I think they are also used in meteorology.