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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_o(p, \vec{v})} is the outgoing radiance from surface at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} in the direction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\ell}} and 0 otherwise.

Geometric computation of form factors

Project other polygons in the scene onto the hemisphere about point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{i,j}} is this form factor term.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.