CSCE 441 Lecture 26

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Radiosity

A non-real-time rendering method.

Allows for indirect illumination from reflected surfaces.

• Soft shadows
• Color bleeding

Assume perfectly diffuse surfaces (no specular highlights)

Not view-dependent; the camera can walk around the scene, and the computed lighting will be correct. Some games would precompute radiosity and then add dynamic lighting.

Rendering Equation

$L_{o}(p,{\vec {v}})=L_{e}(p,{\vec {v}})+\int _{\Omega }f(l,v)\,L_{o}(r(p,\ell ),-\ell )\,\cos {\theta _{i}}\,\mathrm {d} \omega _{i}$

• $L_{o}(p,{\vec {v}})$ is the outgoing radiance from surface at $p$ in the direction of ${\vec {v}}$.
• $L_{e}(p,{\vec {v}})$ is the emitted radiance from surface at $p$ in the direction of ${\vec {v}}$. This will be 0 for most surfaces (only light sources emit light)
• $f(l,{\vec {v}})$ is the BRDF of surface at $p$. governs material properties of our surface
• $r(p,{\vec {\ell }})$ is a ray cast from $p$ in the direction of $\ell$
  ($L(r(p,{\vec {\ell }}),-\ell )$ computes how much contribution is made from light reflected from other surfaces)
• $\theta _{i}$ is the angle between ${\vec {\ell }}$ and ${\vec {n}}$
• $\int _{\Omega }\ldots \,\mathrm {d} \omega _{i}$ is the integral about the hemisphere centered at $p$.

Bidirectional Reflectance Distribution Function

(See wikipedia:Bidirectional reflectance distribution function→)

If light comes in from direction $\ell$, how much light is scattered in the direction of ${\vec {v}}$. There are two parameters goverining the direction of ${\vec {\ell }}$ and ${\vec {v}}$, so $f$ is a four-dimensional function.

Governs all material properties and allows for all types of matte, glossy, etc. surfaces.

Discretizing the Rendering Equation

Assume perfectly diffuse surfaces, so $BRDF$ is constant: $f({\vec {\ell }},{\vec {v}})=c$

Break surfaces up into patches and assume color is constant per patch. The value from each patch will be summed to approximate the integral.

$L_{i}=L_{i,e}+\int _{\Omega }c_{i}\,L_{j}h_{i,j}\,\cos {\theta _{i}}\,\mathrm {d} \omega _{i}$

$h_{i,j}$ is 1 if patch $i$ is visible to patch $j$ along ${\vec {\ell }}$ and 0 otherwise.

Geometric computation of form factors

Project other polygons in the scene onto the hemisphere about point $p$. This represents the contribution of those polygons to this point's radiosity.

We can integrate over the angular coordinates at which the projections lie on the hemisphere:

$F_{i,j}$ is this form factor term.

$L_{i}=L_{i,e}+\sum _{j}c_{i}\,L_{j}\,F_{i,j}$

We can set up a large matrix of systems of equations

Advantages

• Global illumination methed: modeling diffuse inter-reflection
• Color Bleeding: red wall next to white wall casts reddish glow on white wall
• Soft shadows: area light source casts a soft shadow from a polygon
• No ambient hack
• View-independent: assigns a brightness to every surface. Not actually an advantage, but a side effect of the assumption that all surfaces are perfectly diffuse.

Disadvantages

• Assumption that BRDF is uniform in all directions
• Radiocity is piecewise constant
• No surface is transparent or translucent.
• Must determine how to subdivide shapes into small enough patches.

Photon Mapping: makes no simplifying assumptions; tries to modify individual photons of light.