pix2alm

Find spherical harmonic decomposition of function on sphere

Contents

Syntax

alm = pix2alm(v)
alm = pix2alm(v, lmax)

Input Arguments

v       array of pixel values
lMax    (optional) max order of harmonic to calculate
nPix = numel(v), with nPix = 12*nSide^2 for nSide a power of 2.
lMax defaults to 2*floor(nSide/3)

Return Arguments

alm     coefficients of spherical harmonic expansion

Description

Let $x_k$ denote the location of pixel $k$ and $v_k$ the function value at $x_k$. Then

$$alm(j) = \frac{4\pi}{N}\sum_{k=0}^{N-1} Y_{LM}^{\dagger}(x_k) v_k$$

where $j=(L+1)^2+M-L$ and $N$ is the number of pixels (12*nSide^2)

Example

estimate alm of dummy data

ns = 2^4;
np = 12*ns^2;
v = ylm(ns,1,1) + ylm(ns,2,-2) + ylm(ns,3,0) + rand(1,np)/10;
lMax = 4;
alm = pix2alm(v,lMax);
for L = 0:3
  fprintf('L = %d: ',L);
  fprintf('%7.3f ',abs(alm((L+1)^2+(-L:L)-L)));
  fprintf('\n');
end
L = 0:   0.176 
L = 1:   0.001   0.002   1.000 
L = 2:   1.001   0.002   0.001   0.002   0.003 
L = 3:   0.003   0.003   0.002   0.998   0.002   0.003   0.003 

See also

alm2pix

Requires

ylm

Copyright 2010-2011 Lee Samuel Finn. Terms of Use.