Recently I decided to finally get the hang of spherical harmonics (in the context of computer graphics, of course). I started with the standard reading on the subject, which is

Robin Green's paper. It's nice but I found an even better reading in

A Gentle Introduction to PRT (and this paper as a bit of basis for this note). Both papers nicely explain how to integrate numerically the lighting environment and project it into SH. What they don't talk about is how to project analytical, directional lights into SH basis. One article I found on this subject is chapter 2.15 in ShaderX3 but (to me) it's not thorough enough, or at least it doesn't discuss some details I was interested in. A lot of light was shed on me

here and

here.

##
SH Formulas

Here come the first three bands of SH functions (both in spherical and cartesian coords, with Condon-Shortley phase). I've seen them in various places but often with typos. The formulas here are taken from chapter 3.2 from GPU Pro 2.

\begin{eqnarray}

Y_0^0 &=& \frac{1}{2} \sqrt{\frac{1}{\pi}} &=& \frac{1}{2} \sqrt{\frac{1}{\pi}} \cr

Y_1^{-1} &=& \frac{-1}{2} \sqrt{\frac{3}{\pi}} \sin(\theta) \sin(\phi) &=& \frac{-1}{2} \sqrt{\frac{3}{\pi}} y \cr

Y_1^0 &=& \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos(\theta) &=& \frac{1}{2} \sqrt{\frac{3}{\pi}} z \cr

Y_1^1 &=& \frac{-1}{2} \sqrt{\frac{3}{\pi}} \sin(\theta) \cos(\phi) &=& \frac{-1}{2} \sqrt{\frac{3}{\pi}} x \cr

Y_2^{-2} &=& \frac{1}{2} \sqrt{\frac{15}{\pi}} \sin^2(\theta) \sin(2\phi) &=& \frac{1}{2} \sqrt{\frac{15}{\pi}} xy \cr

Y_2^{-1} &=& \frac{-1}{2} \sqrt{\frac{15}{\pi}} \sin(\theta) \cos(\theta) \sin(\phi) &=& \frac{-1}{2} \sqrt{\frac{15}{\pi}} yz \cr

Y_2^0 &=& \frac{1}{4} \sqrt{\frac{5}{\pi}} (3 \cos^2(\theta) - 1) &=& \frac{1}{4} \sqrt{\frac{5}{\pi}} (3z^2 - 1) \cr

Y_2^1 &=& \frac{-1}{2} \sqrt{\frac{15}{\pi}} \sin(\theta) \cos(\theta) \cos(\phi) &=& \frac{-1}{2} \sqrt{\frac{15}{\pi}} zx \cr

Y_2^2 &=& \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin^2(\theta) \cos(2\phi) &=& \frac{1}{4} \sqrt{\frac{15}{\pi}} (x^2 - y^2) \cr

\end{eqnarray}

We'll work only with the first three bands as this is enough for accurately representing directional diffuse (Lambertian BRDF) lighting.

##
Projecting Light and Transfer Functions into SH

Remember we're dealing with directional lights acting on Lambertian surfaces. The light function $L$ is simply constant (light) color ($lc$), projected into SH in the direction of the light ($ld$). Thus, $L$ projected into SH gives $L_l^m$:

\begin{eqnarray}

L_0^0 &=& \frac{1}{2} \sqrt{\frac{1}{\pi}} lc \cr

L_1^{-1}
&=& \frac{-1}{2} \sqrt{\frac{3}{\pi}} ld_y lc \cr

L_1^0 &=& \frac{1}{2} \sqrt{\frac{3}{\pi}} ld_z lc \cr

L_1^1 &=& \frac{-1}{2} \sqrt{\frac{3}{\pi}} ld_x lc \cr

L_2^{-2} &=& \frac{1}{2} \sqrt{\frac{15}{\pi}} ld_x ld_y lc \cr

L_2^{-1}
&=& \frac{-1}{2} \sqrt{\frac{15}{\pi}}
ld_y ld_z lc \cr

L_2^0 &=& \frac{1}{4} \sqrt{\frac{5}{\pi}} (3ld_z^2 -
1) lc \cr

L_2^1 &=& \frac{-1}{2}
\sqrt{\frac{15}{\pi}} ld_z ld_x lc \cr

L_2^2 &=& \frac{1}{4}
\sqrt{\frac{15}{\pi}} (ld_x^2 - ld_y^2) lc \cr

\end{eqnarray}

This is just light. We also need to project into SH the transfer function $A$, which in case of our Lambertian BRDF function is simply $\cos(\theta)$. The formula for this function projected into SH is derived

here; have a look at equations (19) (this equation has two typos which are fixed here, thanks to

KriS) and (26) in particular. Formulas:

\begin{eqnarray}

l = 1: & A_l &=& \frac{\sqrt{\pi}}{3} N_l \cr

l > 1, \mbox{odd}: & A_l &=& 0 \cr

l, \mbox{even}: & A_l &=& 2 \pi \sqrt{\frac{2l+1}{4 \pi}} \frac{(-1)^{l/2-1}}{(l+2)(l-1)} \frac{l!}{2^l ((n/2)!)^2} N_l \cr

\mbox{where} & N_l &=& \sqrt{\frac{4 \pi}{2l + 1}} \cr

\end{eqnarray}

Note that $A$ varies per SH band index. $A$ values for the first three SH bands are:

\begin{eqnarray}

A_0 &=& \pi \cr

A_1 &=& \pi \frac{2}{3} \cr

A_2 &=& \frac{\pi}{4} \cr

\end{eqnarray}

The final formula for each SH coefficient of light and transfer functions, projected into SH and convolved is:

\begin{eqnarray}

E_l^m = L_l^m A_l

\end{eqnarray}

##
Final Vertex Color

To finally calculate the lighting for a vertex, given its normal $n$, we need to project the vertex's color ($vc$) in the direction of the normal into SH. This is pretty much the same what we did for the light function:

\begin{eqnarray}

V_0^0 &=& \frac{1}{2} \sqrt{\frac{1}{\pi}} vc \cr

V_1^{-1}
&=& \frac{-1}{2} \sqrt{\frac{3}{\pi}} n_y vc \cr

V_1^0 &=& \frac{1}{2} \sqrt{\frac{3}{\pi}} n_z vc \cr

V_1^1 &=& \frac{-1}{2} \sqrt{\frac{3}{\pi}} n_x vc \cr

V_2^{-2} &=& \frac{1}{2} \sqrt{\frac{15}{\pi}} n_x n_y vc \cr

V_2^{-1}
&=& \frac{-1}{2} \sqrt{\frac{15}{\pi}} n_y n_z vc \cr

V_2^0 &=& \frac{1}{4} \sqrt{\frac{5}{\pi}} (3n_z^2 -
1) vc \cr

V_2^1 &=& \frac{-1}{2}
\sqrt{\frac{15}{\pi}} n_z n_x vc \cr

V_2^2 &=& \frac{1}{4}
\sqrt{\frac{15}{\pi}} (n_x^2 - n_y^2) vc \cr

\end{eqnarray}

Convolved with $E_l^m$ it gives the final vertex color $C_l^m$ for each band:

\begin{eqnarray}

C_l^m &=& V_l^m E_l^m

\end{eqnarray}

To retrieve the final vertex color $C$ all we need is to sum all SH components:

\begin{eqnarray}

C = \sum\limits_{l=0}^2 \sum\limits_{m=-l}^l C_l^m

\end{eqnarray}