Maybe I can help you on the topic of spherical harmonics:
Actually, you only have to know "what are they" and "what can they do". For this you do NOT have to understand all the equations IMHO. You just have to sort them into a bigger context.
So first question: What are they?
Spherical harmonics are just another set of orthonormal functions. The Fourier functions (the functions exp(i*w_n*t)) are orthonormal functions, wavelets are orthonormal functions (*), it's all the same and it works all the same! If you want to develop a function f into a orthonormal set of functions u_n, it always works like
f = sum (f,u_n) u_n,
where (,) is a certain scalar product. (f,u_n) are the coefficients of f in the orthonormal basis u_n. It's always the same. Just the space you're living on and the scalar product changes. Fourier transforms and wavelets approximate "usual" functions defined on a rectangle for example. The spherical harmonics live on the (3D) unit sphere and the scalar product is IIRC the standard-integral-scalar product, so you can approximate e.g. skyboxes or whatever lives on the unit sphere. It's exactly the same as the Fourier transform and behaves exactly the same.
So "what can they do"?
Normally, one wants to approximate some function in a certain basis with as few coefficients as possible. So the question is: How many coefficients do I need and what happens if I don't take enough. Spherical harmonics are like a Fourier basis, but just on the unit sphere. So you can say two things:
o The more well-behaved your function is (very smooth, no local spikes), the less coefficients you need.
o If you don't take enough, you will get Gibbs phenomena (also called ringing artifacts) near sharp edges.
That's all you need to know.
(*) For the specialists: I'm not talking about biorthogonal wavelets.
[edited by - Lutz on June 7, 2004 11:06:29 AM]
Understanding every single math detail behind this
spherical harmonics have to good taste of being rotational invariant in their behavior in the sense that
let f be a function defined on the unit sphere
let R be a rotation of that function
let SHTran be the projection on an SH basis
there is r such that
SHTran (R(f)) = r(SHTran(f))
(Note, r is an orthonormal transformation)
the function that makes r And R match is a group morphism.
An important implication of this is that it is very easy to compute the transform of a rotated function. Another important thing is that there is no favored direction.
(Note the same properties hold for the Fourier transform and translations)
THe cool thing with SH is that there are so many remarkably simple properties that emanate from it.
let f be a function defined on the unit sphere
let R be a rotation of that function
let SHTran be the projection on an SH basis
there is r such that
SHTran (R(f)) = r(SHTran(f))
(Note, r is an orthonormal transformation)
the function that makes r And R match is a group morphism.
An important implication of this is that it is very easy to compute the transform of a rotated function. Another important thing is that there is no favored direction.
(Note the same properties hold for the Fourier transform and translations)
THe cool thing with SH is that there are so many remarkably simple properties that emanate from it.
Author
Thanks a lot for the explanations guys. It makes me feel better than what I''m studying is applicable in other areas as well. Makes all the time with it worth it 
This topic is closed to new replies.
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