Lec 15 Ray Tracing 3(Light Transport & Global Illumination)

Irradiance and Radiance

Irradiance is the power per (perpendicular / projected) unit area incident on a surfacee point.

\[ E(\mathbf{x})\equiv\frac{\mathrm{d}\Phi(\mathbf{x})}{\mathrm{d}A} \]

\[ \left[\frac{\text{W}}{\text{m}^2}\right]\,\left[\frac{\text{lm}}{\text{m}^2}=\text{lux}\right] \]

In Blinn-Phong model, "intensity falloff" should be corrected as "irradiance falloff".

Radiance is the fundamental field quantity that describes the distribution of light in an environment.

• Radiance is the quantity associated with a ray
• Rendering is all about computing radiance

Radiance is the power per unit solid angle, per projected unit area.

\[ L(\mathrm{p},\omega)\equiv\frac{\mathrm{d}\Phi(\mathrm{p},\omega)}{\mathrm{d}\mathrm{d}A\cos\theta} \]

\[ \left[\frac{\mathrm{W}}{\mathrm{sr}\,\mathrm{m}^2}\right]\,\left[\frac{\mathrm{cd}}{\mathrm{m}^2}=\frac{\mathrm{lm}}{\mathrm{sr}\,\mathrm{m}^2}=\mathrm{nit}\right] \]

• Radiance is irradiance per solid angle
• Radiance is intensity per projected area

Irradiance vs. radiance

Irradiance is total power received by area dA, from all angle.

\[ \begin{align*} dE(\mathrm{p},\omega)&=L_i(\mathrm{p},\omega)\cos\theta\mathrm{d}\omega \\ E(\mathrm{p})&=\int_{H^2}L_i(\mathrm{p},\omega)\cos\theta\mathrm{d}\omega \end{align*} \]

Bidirectional Reflectance Distribution Function (BRDF)

Radiance from direction \(\omega_i\) turns into the power E that dA receives.
Then power E will become the radiance to any other direction \(\omega\).

BRDF represents how much light is reflected into each outgoing direction \(\omega_r\) from each incoming direction.

\[ f_r(\omega_i\to\omega_r)=\frac{\mathrm{d}L_r(\omega_r)}{\mathrm{d}E_i(\omega_i)}=\frac{\mathrm{d}L_r(\omega_r)}{L_i(\omega_i)\cos\theta_i\mathrm{d}\omega_i} \quad\left[\frac{1}{\text{sr}}\right] \]

The reflection equation:

\[ L_r(\mathrm{p},\omega_r)=\int_{H^2}f_r(\mathrm{p},\omega_i\to\omega_r)L_i(\mathrm{p},\omega_i)\cos\theta_i\mathrm{d}\omega_i \]

The rendering equation:

Add an emission term to make it general.

\[ L_o(\mathrm{p},\omega_o)=L_e(\mathrm{p},\omega_o)+\int_{\Omega^+}L_i(\mathrm{p},\omega_i)f_r(\mathrm{p},\omega_i,\omega_o)(n\cdot\omega_i)\mathrm{d}\omega_i \]

(reflected light = emission + incident lignt * BRDF * incident angle)

One point light: no need of integral
Multiple point lights: sum over all light sources
Area light: replace sum with integal
Unknown reflection: regard reflection as light source

Simplify: L=E+KL

Approximate set of all paths of light in scene:

\[ \begin{align*} L&=E+KL \\ L&=(I-K)^{-1}E \\ &=(I+K+K^2+K^3+\cdots)E \\ &=E+KE+K^2E+K^3E+\cdots \end{align*} \]

• \(E\): emission directly from light sources
• \(KE\): direct illumination on surfaces
• \(K^2E\): indirect illumination (one bounce indirect)
• ...

Probability Review

\(X\): random variable
\(X\sim p(x)\): probability density function (PDF)

Requirements of a probability distrubution:

\[p_i\ge0\quad\sum_{i=1}^n p_i=1\]

Expected value:

\[E[X]=\sum_{i=1}^n x_i p_i\]

Continuous case: PDF

Conitions on p(x): \(p(x)\ge 0\,\text{and}\,\int p(x)dx=1\)
Expected value of X: \(E[X]=\int xp(x)dx\)

Function of a Random Variable

\[X\sim p(x)\quad Y=f(X)\]

Expected value of a function of a random varaible:

\[E[Y]=E[f(X)]=\int f(x)p(x)dx\]