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Spherical Harmonics and Transfer Function

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Hello!

 

I am going through this aritcle by Ramamoorthi and Hanrahan on "Irradiance Environment Maps"

https://cseweb.ucsd.edu/~ravir/papers/envmap/envmap.pdf.

 

I am having difficulties trying to understand how they approximate transfer function using spherical harmonics.

?

The transfer function is a clamped cosine function A(surface normal, incoming light direction) = dot(surface normal, incoming light direction).

It could also be rewritten as A(theta) = max(cos(theta), 0), where theta is an angle between surface normal and incoming light direction.

 

Let c(l, m) be spherical harmonics coefficients of A(theta) projected into spherical harmonic function Y(l, m, theta)

They claim in the article: "Since A has no azimuthal dependence, then m = 0. A(theta) = sum of all c(l, 0) * Y(l, 0, theta)".

 

I do not understand what happens to Y(l, m, theta) when m is not equal to 0. They seem to skip them.

Could you please elaborate on this please?

 

Many thanks indeed!

Edited by _void_

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OK, I have calculated manually integral to find coefficient for the case l = 1, m = -1. It is equal to 0 ohmy.png

I guess, for the rest of the cases, when m is not equal to 0,  the integral will also be 0.

This explains everything rolleyes.gif.

?

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I didn't bother reading all the details and the naming conventions used in the paper, but it's probably the case that some symmetry of A(theta) allows you to prove that all those integrals are 0. Like if you try to find the Fourier coefficients of an even function (i.e., F(x) = F(-x)), you'll find that all the integrals of the form Integral[sin(k*x)*F(x), {x, -Pi,+Pi}] are 0. It must be an analogous phenomenon.

Edited by Álvaro

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