Spherical Harmonics(rhotter.com) |
Spherical Harmonics(rhotter.com) |
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.
[1] https://en.wikipedia.org/wiki/Cosmic_microwave_background
(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?
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.
Really the harmonics are best understood as something like "wave height on the surface of a sphere". They tell you how the electrons (or whatever) are going to distribute themselves radially, not where they're going in 3D space.
Also FWIW: the much harder thing to grok here (at least it was for me), and that no one tries to tackle, is why the "l" number corresponds directly to angular momentum. In particular "l==0" doesn't look like there's any rotation going on at all.
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.
ALso, I noticed that there is a real/imaginary/complex menu. I was looking at the real part of the complex versions, but it's necesary to look at the complete complex version to understand them.
Note that the graphic in Wikipedia is showing the real versions. In the real versions you for l=1 the functions in directions x, y and z[m=0], and all of them look identical except for the direction. But in the complex versions you have (x+iy)/sqrt(2)[m=1], (x-iy)/sqrt(2)[m=-1] and z[m=0], and when you show only the real part of them the first two are smaller.
I have also seen “triangular harmonics”, “zonal harmonics”, etc. in use in other materials.
Also lots of other cool research around SH in rendering, e.g. the recent ZH3 paper.
My experience is from cosmology (CMB) where they are heavily used just like Fourier transforms, I think they are also used in meteorology.
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!
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...
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.
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.
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.
> So the spherical harmonic functions are also the shape of the various electron orbitals that you see in a chemistry textbook.
Minor nitpick: Chemistry textbooks usually use the “cubic harmonics” instead of the “spherical harmonics”. They are real-valued linear combinations of the standard spherical harmonics, with the additional benefit that the basis set respects the Cartesian symmetries.
For example, the “p_z orbital” is a l=1 spherical harmonic and a cubic harmonic. But the cubic harmonics then add p_x and p_y orbitals as basis functions, whereas the spherical harmonics choose the chiral "p_x ± ip_y orbitals” as its basis instead.
Now take this plate and turn it into a 3d sphere and that's spherical harmonica more or less. Electronsike to form them
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
A more general definition is that they are eigenfunctions of the Laplacian operator on a sphere, which arise in many contexts.
The spherical harmonics in general are typically derived as part of the solution to the Laplace equation in spherical coordinates. A bit of a semantic point (though perhaps the distinction is important) though, since the Laplace equation's angular dependence is identical to that of the Schrödinger equation for the hydrogen atom.
Even more generally, see the huge body of beautiful mathematics that has arisen from the study of elliptic differential operators in general (de Rham cohomology, Hodge theory, the Atiyah-Singer index theorem, …).
As for why l==0 has no rotation going on at all, one would say that this should be expected. Qualitatively, the symmetric sphere does not change with rotation, so how would we tell if it is rotating or not? And perhaps the next step is controversial, but if there is no way to tell, maybe there is no dependence? This is a similar argument to why the electric field of an infinite plane is constant with respect to distance from it.
https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membr...