CSCE 641 Computer Graphics: Radiosity Jinxiang Chai.
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Transcript of CSCE 641 Computer Graphics: Radiosity Jinxiang Chai.
CSCE 641 Computer Graphics: Radiosity
Jinxiang Chai
Rendering: Illumination Computing
Direct (local) illumination
Light directly from light sources
No shadows
Indirect (global) illumination
Transparent, reflective surfaces, and hard shadows (Ray tracing)
Diffuse interreflections, color bleeding, and soft shadow (radiosity)
Rendering: Illumination Computing
Direct (local) illumination
Light directly from light sources
No shadows
Indirect (global) illumination
Transparent, reflective surfaces, and hard shadows (Ray tracing)
Diffuse interreflections, color bleeding, and soft shadow (radiosity)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light sources)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light sources)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light sources)
- reflection ray (light reflected by an object)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light sources)
- reflection ray (light reflected by an object)
- transparent ray (light passing through an object)
Review: Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light sources)
- reflection ray (light reflected by an object)
- transparent ray (light passing through an object)
Ray Tracing Assumption
The illumination of a point is determined by
- illumination/shadow ray (direct lighting from light sources)
- reflection ray (light reflected by an object)
- transparent ray (light passing through an object)
Pros and Cons of Ray Tracing
Advantages of ray tracing
All the advantages of the local illumination model
Also handles shadows, reflection, and refraction
Disadvantages of ray tracing
Computational expense
No diffuse inter-reflection between surfaces (i.e., color bleeding)
Not physically accurate
Radiosity handles these shortcomings for diffuse surfaces!
Radiosity vs. Local Illumination
Radiosity
Physical Image vs. Radiosity Rendering
Radiostiy
Definition: The radiant (luminous) exitance is the radiant flux/power per unit area leaving a surface.
( ) odM x
dA
2 2
W lmlux
m m
Radiosity
Model light effects by considering the physical laws governing the radiant energy transfer;
The radiosity model computes radiant-energy interactions between all the surfaces in a scene
Radiosity: Key Idea #1
Diffuse Surface
Radiosity: Key Idea #2
Constant Surface Approximation
Radiosity Equation
N
jjijiselfi RadiosityaRadiosityRadiosity
1,
Radiosity Equation
N
jjjself RadiosityaRadiosityRadiosity
111,1
N
jjjself RadiosityaRadiosityRadiosity
122,2
N
jjNjNselfN RadiosityaRadiosityRadiosity
1,
Radiosity Algorithm
Energy Conservation Equation
N
jjiiei ijF
1, )(
N
jjiiei ijF
1, )(
Energy Conservation Equation
The total rate of radiant energy leaving surface i per unit square
N
jjiiei ijF
1, )(
Energy Conservation Equation
The rate of energy emitted from surface i per unit area
- zero if surface i is not a light source
N
jjiiei ijF
1, )(
Energy Conservation Equation
Reflectivity factorPercent of incident light that is reflected in all
directions
N
jjiiei ijF
1, )(
Energy Conservation Equation
Form factor
Fractional amount of radiant energy from surface j that reaches surface i
Compute Form Factors
ji
AA xy
yx
j
dAdAyxVrA
ijFji
),(coscos1
)(,
2
The form factor specifies the fraction of the energy leaving one patch and arriving at the other. In other words, it is an expression of radiant exchange between two surface patches!
Compute Form Factors
ji
AA xy
yx
j
dAdAyxVrA
ijFji
),(coscos1
)(,
2
Radiant energy reaching Ax from Ay
Radiant energy leaving Ay in all directions
The form factor specifies the fraction of the energy leaving one patch and arriving at the other. In other words, it is an expression of radiant exchange between two surface patches!
Form Factor: Reciprocity
yx
AA xy
yx
j
dAdAyxVrA
ijFji
),(coscos1
)(,
2
xy
AA xy
yx
i
dAdAyxVrA
jiFji
),(coscos1
)(,
2
ji AijFAjiF )()(
Radiosity Equation
Radiosity for each polygon
Linear system:
- : radiosity of patch I (unknown)
- : emission of patch I (known)
- : reflectivity of patch I (known)
- : form-factor (known)
N
jjiiei ijFi
1, )(:
i
i
jiF
ie,
Linear System
3,
2,
1,
2
1
21
22222122
11211111
......
1...
............
...1
...1
e
e
e
NNNNNNNN
N
N
FFF
FFF
FFF
A X = B
Radiosity Algorithm
Form Factors for Infinitesimal Surfaces
),(coscos
)(2
yxVdAr
ijF ixy
yx
• Visibility
- if not visible, receive zero power
Form Factors for Subdivided Patches
ji
AA xy
yx
j
dAdAyxVrA
ijFji
),(coscos1
)(,
2• Visibility
- if not visible, receive zero power
Form Factor: How to compute?
Closed Form
- analytical
Hemicube
Form Factor: Analytical
yx
AA xy
yx
j
dAdAyxVrA
ijFji
),(coscos1
)(,
2
Form Factor: How to compute?
Closed Form
- analytical
Hemicube
Form Factor: Nusselt Analog
Nusselt developed a geometric analog which allows the simple and accurate calculation of the form factor between a surface and a point on a second surface.
3D diagram
Form Factor: Nusselt Analog
The form factor is, then, the area projected on the base of the hemisphere divided by the area of the base of the hemisphere, or (A/B)
AB
2D diagram
Form Factor: Nusselt Analog
Form Factor: Nusselt Analog
So how can we use Nusselt Analog to compute the form factor?
Form Factor: Nusselt Analog
So how can we use Nusselt Analog to compute the form factor?
- answer: precomputing
Form Factor: HemiCube
Form Factor: HemiCube
• Project path on hemicube
• Add hemicube cells to compute form factors
AB
2D diagram
Precomputing Form Factor
How to calculate the form factor for each cell?
Delta Form Factor: Top Face
Top of hemicube
Delta Form Factors: Side Faces
Side of hemicube
The Hemicube in Action
Form Factors: HemiCube
Form Factors
Radiosity Algorithm
How to Solve Linear System?
Matrix conversion
Iterative approaches
- Jacobian (gathering)
- Gauss-Seidel (gathering)
- progressive refinement (shooting)
Matrix Conversion
3,
2,
1,
1
21
22222122
11211111
2
1
...
1...
............
...1
...1
...
e
e
e
NNNNNNN
N
N
N FFF
FFF
FFF
- Computational cost: O(N3)
- Very slow for a large set of polygons
Iterative Approaches
N
jjiiei ijF
1, )(
Jacobian Iterations
For all patches i, i=1,…,N,
While not converged:
for all patches i=1,…,N
iei ,0
N
j
iterjiie
iteri ijF
1
1, )(
Jacobian Iterations
For all patches i, i=1,…,N,
While not converged:
for all patches i=1,…,N
iei ,0
N
j
iterjiie
iteri ijF
1
1, )(
Update of one patch requires evaluation of N Form Factors
What’s the computational cost?
Successive Approximation
eL
e eL K L eL
eK L eK K L eK K K L
2e eL K L 3
e eL K L
Rendering
- The final Φi's can be used in place of intensities in a standard renderer (Gouraud, per-veretx shading)
- 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
Consolation Room
Theatre
Steel Mills
Radiosity: Benefit
Global illumination method: modeling diffuse inter-reflection
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