SphericalHarmonics

P31

Spherical Harmonics

$$Y_ {lm}(\theta ,\phi )= N_ {lm}P_ {lm}(\cos \theta )e^ {Im \phi }$$

$$\begin{align*} x& = \sin \theta \cos \phi \\\\ y & = \sin \theta \sin \phi\\\\ z & = \cos\theta \end{align*}$$

Complex sphere integration can be approximated by quadratic polynomial：

$$\int\limits_{\theta =0}^{\pi } \int\limits_{\phi =0}^{2\pi } L(\theta,\phi )Y_{lm}(\theta ,\phi )\sin \theta d\theta d\phi \approx \begin{bmatrix} x \\\\ y \\\\ z\\\\ 1 \end{bmatrix}^TM\begin{bmatrix} x \\\\ y \\\\ z \\\\ 1 \end{bmatrix}$$

利用球谐函数定义了一组基，通过对球谐基的加权平均，可以组合出任意复杂的球面。

P32

Spherical Harmonics 基

Spherical Harmonics, a mathematical system analogous to the Fourier transform but defined across the surface of a sphere. The SH functions in general are defined on imaginary numbers

绿色表示正值，红色表示负值。 每一个维度的所有基都是正交的。 二阶导永远 0（光滑）。

P33

Spherical Harmonics Encoding