Nusselt Analog

CPSC 641 Computer Graphics:
Radiosity
Jinxiang Chai
Local Illumination
Ir = kaIa + Ii (kd (n.l) + ks(h.n)m )
ambient
diffuse
specular
Local Illumination
Ir = kaIa + Ii (kd (n.l) + ks(h.n)m )
ambient
diffuse
specular
Local Illumination
Ir = kaIa + Ii (kd (n.l) + ks(h.n)m )
ambient
diffuse
specular
if there are multiple lights there is a sum of the specular and
diffuse components for each light
Local Illumination
Ir = kaIa + Ii (kd (n.l) + ks(h.n)m )
ambient
diffuse
specular
if there are multiple lights there is a sum of the specular and
diffuse components for each light
What are limitations of local illumination?
Rendering: Illumination Computing
Direct (local) illumination

Light directly from light sources

No shadows
Direct and Indirect Light
Rendering: Illumination Computing
Direct (local) illumination

Light directly from light sources

No shadows
Indirect (global) illumination

Hard and soft shadows

Diffuse interreflections (radiosity)

Glossy interreflections (caustics)
Early Radiosity
Consolation Room
Challenge
To evaluate the reflection equation
the incoming radiance must be known
Lr ( x, r ) 
 f ( x,   )L ( x, )cos d
r
i
r
i
i
H2
To evaluate the incoming radiance
the reflected radiance must be known
i
i
Radiosity
Only consider inter-reflections between diffuse
surfaces!
Radiosity: Key Idea #1
Diffuse Surface
Radiosity: Key Idea #2
Constant Surface Approximation
Radiosity Equation
Radiosity Equation
Radiosity Algorithm
Radiosity Algorithm
Energy Conservation Equation
Energy Conservation Equation
Form factor
Compute Form Factors
Compute Form Factors
Radiant energy reaching Ay from Ax
Radiant energy leaving Ax in all directions
Form Factor: Reciprocity
Radiosity Equation
Linear System
Radiosity Algorithm
Form Factors
Form Factor: How to compute?
Closed Form
- anlytical
Hemicube
Monte Carlo
Form Factor: Analytical
Form Factor: How to compute?
Closed Form
- anlytical
Hemicube
Monte Carlo
Form Factor: Nusselt Analog
Form Factor: Nusselt Analog
Why is it true?
Form Factor: Nusselt Analog
Form Factor: Nusselt Analog
How can we use this property?
Form Factor: Nusselt Analog
How can we use this property?
- Speed up form-factor evaluation
Form Factor: HemiCube
Delta Form Factor: Top Face
Top of hemicube
Delta Form Factors: Side Faces
Side of hemicube
The Hemicube in Action
Form Factor: HemiCube
Form Factors
Radiosity Algorithm
How to Solve Linear System?
Matrix conversion
Iterative approaches
- Jacobian
- Gauss-Seidel
Matrix Conversion
Iterative Approaches
Jacobian
Successive Approximation
Le
K Le
Le
Le  K Le
K K Le
Le 
K 2 Le
K K K Le
Le 
K 3 Le
Gauss-Seidel
Gauss-Seidel (Cont.)
Radiosity Algorithm
Rendering
The final Bi's can be used in place of intensities in a
standard renderer (Gouraud)
Radiosities are constant over the extent of a patch
A standard renderer requires vertex intensities (or
radiosities)
If the radiosities of surrounding patches are know,
vertex radiosities can be estimated using bilinear
interpolation
Vertex Intensity: Bilinear Interpolation
Theatre
Steel Mills
Radiosity: Benefit
 Global illumination method: modeling diffuse interreflection
 Color bleeding: a red wall next to a white one casts a
reddish glow on the white wall near the corner
 Soft shadows – an “area” light source casts a soft
shadow from a polygon
 No ambient term hack, so when you want to look at
your object in low light, you don’t have to adjust
parameters of the objects – just the intensities of the
lights!
 View independent: it assigns a brightness to every
surface
Radiosity: Limitation
 Radiation is uniform in all directions
 Radiosity is piecewise constant
– usual renderings make this assumption, but then interpolate cheaply to
fake a nice-looking answer
– this introduces quantifiable errors
 No surface is transparent or translucent
 Reflectivity is independent of directions to source and
destination