PRT

Spherical Harmonics (SH)

• Basis Functions: A set of orthonormal functions used to represent other functions via linear combination (like Polynomial series and Fourier series.).

• SH is A set of 2D basis functions \(B_i(\omega) = Y_l^m(\theta,\phi)\) defined on the sphere. (as an analogue to 1D Fourier series.)

  the exact form is complicated: $$ {\displaystyle Y_{\ell }^{m}(\theta ,\ \varphi )=(-1)^{m}{\sqrt {{(2\ell +1) \over 4\pi }{(\ell -|m|)! \over (\ell +|m|)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }\,!} \ P_{\ell }^{m}(x)=(1-x^{2})^{{|m|/2}}\ {\frac {d^{{|m|}}}{dx^{{|m|}}}}P_{\ell }(x)\, \ P_{\ell }(x)={1 \over 2^{\ell }\ell !}{d^{\ell } \over dx^{\ell }}(x^{2}-1)^{l} $$

  

• For any function \(f\) defined on the sphere, we can represent it with a linear combination of SHs (usually truncated):

\[ \displaylines{ f(\omega) = \sum_ic_iB_i(\omega) } \]

SH property (orthonormal):

\[ \displaylines{ \int_\Omega B_i(\omega)B_j(\omega) = 0, i\ne j \\ \int_\Omega B_i(\omega)B_j(\omega) = 1, i= j \\ } \]

SH property (projection): the coefficient can be obtained with:

\[ \displaylines{ c_i = \int_\Omega f(\omega)B_i(\omega)d\omega } \]

• 9 Parameter Approximation:

  Use the first 3 orders of SH (9 parameters) is enough for approximating diffuse BRDF.

  

Precomputed Radiance Transfer

• Basic idea:

  • Assume the scene is static, so only light conditions can vary.
  • view the rendering function as lighting and light transport (or bounce, in fact the visibility & BRDF).

\[ \displaylines{ L(o) = \int_\Omega \underbrace{L(i)}_{\text{lighting}}\underbrace{V(i)\rho(i, o)\max(0, n \cdot i)}_{\text{light transport}}di } \]

* approximate lighting with SH.

* For the diffuse case, BRDF is constant so $\rho(i,o) = \rho$ :
  $$
  L = L(o) = \rho\int_\Omega {L(i)} V(i)\max(0, n \cdot i) di \\
        \approx \rho\int_\Omega \sum c_jB_j(i)V(i)\max(0, n \cdot i) di \\
        = \rho\sum c_j\underbrace{\int_\Omega B_j(i)V(i)\max(0, n \cdot i) di}_{T_j \text{ (pre-compute!)}} \\
        =\rho\sum c_j T_j
  $$
    We need to save a vector $T_j$ for each point.

    Another way to calculate it (with a more formal rendering function):
  $$
  L_o(p,o)=\int_\Omega L_i(p,i)f_r(p,i,o)\cos\theta_iV(p,i)di \\
        \approx \int_\Omega \sum_pc_pB_p(i)\sum_qd_qB_q(i)di \\
        = \sum_p\sum_qc_qd_p\int_\Omega B_p(i)B_q(i)di \\
        = \sum_jc_jd_j
  $$

* For the glossy case, BRDF is not constant and is dependent on $(i,o)$.
  $$
  L(o) = \int_\Omega {L(i)} V(i)\rho(i,o)\max(0, n \cdot i) di \\
        \approx \sum_j c_jT_j(o) \\
        \approx \sum_jc_j\sum_kt_{j,k}B_k(o)
  $$
    Now we need to save a matrix for each point.

• SH property (rotation): any rotated SH basis function, can be represented with a linear combination of basis functions of the same level.

  So to rotate the lighting, we don't need to re-compute \(T_j\), but apply the rotation on \(T_j\).

• With PRT, rendering at each point is reduced to a dot product between a vector and another vector (diffuse) or matrix (glossy), and can be easily implemented in real-time.