Theory

Read Chapters 1 and 2 in Cohen and Wallace(CW)[10].

Overview

• Goals
• Structure

Integral Equation for Global Illumination

$$L(x, \omega) = L_e(x, \omega) + \int_{H^{in}} \rho(x;\omega \rightarrow \omega') L(x',\omega') \cos(\theta_x) d\omega'$$

where $L(x, \omega)$ is the radiance at x in the direction $\omega$ (watts per projected meter squared per steradian), $L_e(x, \omega)$ is the emitted radiance, Hⁱⁿ is the hemisphere at x, and $\rho(x;\omega \rightarrow \omega')$ is the reflectance function (inverse steradians).

By changing variables using

$$d\omega = {\rm vis}(x,x') \frac{\cos(\theta_{x'})}{ \left\vert \left\vert x - x' \right\vert \right\vert ^2}dA$$

we can express integration over surfaces rather than direction

$$L(x, \omega) = L_e(x, \omega) + \int_{\cal E} \rho(x,\omega \... ...{ \left\vert \left\vert x - x' \right\vert \right\vert ^2}dA.$$

By assuming a diffuse environment we have $\rho(x,\omega \rightarrow \omega') = k < 1 / \pi$ and

$$L(x) = L_e(x) + \rho(x) \int_{\cal E} L(x') {\rm vis}(x,x')\f... ...{ \left\vert \left\vert x - x' \right\vert \right\vert ^2}dA.$$

Discretization and Basis functions

• What are basis functions?
• Why are they good?
• Examples

Duals

• How do we determine the coefficients for the basis functions?
• Choices of duals.

More info

Andrew Glassner has a fairly nice discussion of basis function in Principles of Digital Image Synthesis [12] pages 175-183. Duals are mentioned on page 182, but stop reading on page 183 at sentence ``We can construct...'' as this no longer describes the duals he defines on page 182.

Jim Arvo formalized finite element methods for global illumination in a Siggraph 94 paper [3].

Many math texts deal with basis function, look at linear algebra texts, numerical analysis texts, anything with fourier analysis in it.