Approximating radii of a sphere using spherical harmonics with an uneven distribution

I am using spherical harmonics to approximate the radii of NN points on a sphere:
Lmax∑l=0l∑m=−lˆamlYml(θi,ϕi)≈r(θi,ϕi)where i=(1,…,N)\sum_{l=0}^{L_{max}}\sum_{m=-l}^{l}\hat{a}_l^mY_l^m(\theta_i,\phi_i) \approx r(\theta_i,\phi_i)\quad \text{where } i =(1,\dots,N)

However the points are unevenly distributed (randomly, to be specific). Does this influence the quality of my solution?

I already thought about approximating r(θi,ϕi)⋅sir(\theta_i, \phi_i)\cdot s_i instead, where sis_i is an estimate of the surface area of the points. But I don’t know if this is right and sufficient, or if there is some better way.

Edit: Sorry, I probably should have provided more context to make clear why I’m concerned about spherical harmonics in particular:

I’m approximating spheres with slightly noisy points (thus slightly different radii). Then I calculate the amplitudes by taking the L2-Norm ||ˆal||||\hat{a}_l|| with respect to mm. By looking at the amplitudes I want to be able to tell a very smooth surface from a very noisy one.

But right now, the amplitude curves in a plot (over l=0…Lmax)l=0\dots L_{max}) look very similar for different surfaces. That’s why I wondered how the distribution of the points affects my result. A large distortion in a very dense surrounding should have a stronger effect on the amplitude than one in a very sparsely sampled area. The other reason, I can think of why it doesn’t quite work is that my amplitude calculation is possibly problematic.



1 Answer


Independent of the fact that they’re spherical harmonics: you have a system of NN equations in a set of unknowns ˆaml\hat{a}^m_l. There are (Lmax+1)2(L_\max + 1)^2 such unknowns, so unless N≤(Lmax+1)2N \leq (L_\max+1)^2, your system will likely be underdetermined (multiple solutions will exist); and if N>(Lmax+1)2N > (L_\max+1)^2, you run the risk of having your system be overdetermined (no solutions will exist.)

Viewed this way, using randomly selected points on the sphere is probably a better choice, since you’ll be less likely to run into “accidental” degeneracies in your system of equations. As (somewhat trivial) example of this: suppose that instead of picking your point all over the surface of the sphere, you happened to only pick points along the equator. Then none of the coefficients ˆa0l\hat{a}_l^0 (where ll is odd) would appear in the system of equations, since since Y0l(π/2,ϕ)=0Y_l^0(\pi/2, \phi) = 0 for all odd values of ll. Thus, these coefficients would be completely undetermined, no matter how many points you had on (the equator of) the sphere.