sphere-lab

Matlab code for spherical harmonic computations

Some routines for spherical harmonic transform related work

Spherical Harmonics

• Spherical harmonics calculations use the Schmidt semi-normalized Legendre functions, which means degrees of the order of l=2000 can be handled. Using the unnormalized (regular) associated Legendre functions in the obvious way the calculations fail around l=150.
• Evaluations are done on a theta-phi mesh. When the mesh separates in theta then there are huge reductions in complexity (increases in speed). The non-separable mesh needs to be done point-by-point
• The spectral representation is a vector indexed by n and not two indices l and m. Converting between the two respresentations is easy.
• The inverse spherical harmonic transform takes the spectral vector and creates the spatial function on a mesh, returning both the function and the mesh. The spatial function is a weighted combination of spherical harmonics. If the spectral representation is a delta (a single non-zero weight) then the spherical harmonix is returned.

Integration on the Sphere

• Inner products and the Spherical Harmonic Transform need a double integral over the sphere in the natural measure. Numerically this is done using a trapezoid rule and is achieved in two lines of code.
• The region is defined through a mesh. To do: add a mask to mesh to have a irregular region.

Indices

• l: l>=0 is the degree
  • when bandlimited the maximum non-zero index is L_max, and L_tot=L_max+1
• m: abs(m)<=l is the order
• n: n>=0 is the single countable index
  • n=l*(l+1)+m
  • l=floor(sqrt(r-1)); m=r-1-l*(l+1);
  • when bandlimited the maximum non-zero index is N_max=(L_max+1)^2-1, and N_tot=(L_max+1)^2

Spherical Harmonics on a Rectangular Grid

Here because the samples are separable with iso-latitude rings then the Associated Legendre functions are minimized and the Spherical Harmonic computation is fast. The Spherical Harmonics are

• function [Ylm,theta,phi]=sphHarmGrid(l,m,tv,pv)
  • computes Y_l^m on the sample grid [theta,phi]=ndgrid(tv,pv)
• function [YB,theta,phi]=sphHarmGridBank(l,tv,pv)
  • generates an cell array "Bank" Y_l^0,Y_l^1,...,Y_l^m
• function [Ylm,theta,phi]=sphHarmGridBankm(l,m,tv,pv)
  • same interface as sphHarmGrid but computed via sphHarmGridBank
  • essentially Ylm=YB{m+1} where YB comes from sphHarmGridBank

Spherical Harmonics on an Unstructured or Non-separable Grid

Inverse Spherical Harmonic Transform