P31
Spherical Harmonics
Y l m ( θ , ϕ ) = N l m P l m ( cos θ ) e I m ϕ Y_ {lm}(\theta ,\phi )= N_ {lm}P_ {lm}(\cos \theta )e^ {Im \phi } Y l m ( θ , ϕ ) = N l m P l m ( cos θ ) e I m ϕ x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ \begin{align*}
x& = \sin \theta \cos \phi \\\\
y & = \sin \theta \sin \phi\\\\
z & = \cos\theta
\end{align*} x y z = sin θ cos ϕ = sin θ sin ϕ = cos θ Complex sphere integration can be approximated by quadratic polynomial：
∫ θ = 0 π ∫ ϕ = 0 2 π L ( θ , ϕ ) Y l m ( θ , ϕ ) sin θ d θ d ϕ ≈ [ x y z 1 ] T M [ x y z 1 ] \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} θ = 0 ∫ π ϕ = 0 ∫ 2 π L ( θ , ϕ ) Y l m ( θ , ϕ ) sin θ d θ d ϕ ≈ x y z 1 T M x y z 1 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
Last updated 2 months ago
