For a 3D surface assumed that its surface appearance follows the Lambertian reflectance i.e. $I = \rho (\mathbb{l} \cdot \mathbb{n})$, I found that it also can be expressed by using following second-order spherical harmonics[1] :
$I=\rho \mathbb{y}^T\phi(\mathbb{n})$
where $\mathbb{y}=[y^r, y^g, y^b]$ is weight vector
and $\phi(\mathbb{n})=[1, n_x, n_y,n_z, n_xn_y, n_xn_z, n_yn_z, n_x^2-n_y^2, 3n_z^2-1]$.
I tried to google to find a theoretical base of this relationship. But no articles which are about spherical harmonics lighting have the expressions like $\phi(\mathbb{n})$. Can someone recommend a document or article about this relationship?
Thanks in advance.
[1] Ichim, Alexandru Eugen, Sofien Bouaziz, and Mark Pauly. "Dynamic 3D avatar creation from hand-held video input." ACM Transactions on Graphics (ToG) 34.4 (2015): 45.