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):

SH property (orthonormal):

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

• 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).

• 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.