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Direct Illumination vs. Global Illumination
reflected, scattered and focused light (not discreet).
physical-based light transport calculations modeled around bidirectional reflective distribution functions (BRDFs).
discreet light source. efficient lighting calculations
based on light and surface vectors (i.e. fast cheats).
Contact Shadows
Notice the surface just under the sphere. The shadow gets much darker where the direct illumination as well as most of the indirect illumination is occluded. That dark contact shadow helps enormously in “sitting” the sphere in scene. Contact shadows are difficult to fake, even with area lights.
Caustics
Focused and reflected light, or caustics, are another feature of the real world that we lack in direct illumination.
Global illumination rendered images
Caustics are a striking and unique feature of Global Illumination.
BRDF
BRDF really just means “the way light bounces off of something.” Specular reflection: some of the light bounces right off of the surface along
the angle of reflection without really changing color much. Diffuse reflection: some of the light gets refracted into the plastic and
bounced around between red particles of pigment. Most of the green and the blue light is absorbed and only the red light makes it’s way back out of the surface. The red light bounced back is scattered every which way with fairly equal probability.
Rendering Equation
L is the radiance from a point on a surface in a given direction ω E is the emitted radiance from a point: E is non-zero only if x’ is emissive V is the visibility term: 1 when the surfaces are unobstructed along the direction ω, 0 otherwise G is the geometry term, which depends on the geometric relationship between the two surfaces x and x’ It includes contributions from light bounded many times off surfaces f is the BRDF
( , ) ( , ) ( , , ) ( , ) ( , ) ( , )rL E f G V L dA x x x x x x x x
Light Emitted from a Surface Radiance (L): Power per
unit area per unit solid angle Measured in W/m2sr dA is projected area –
perpendicular to given direction
Radiosity (B): Radiance integrated over all directions
Power from per unit area, measured in W/m2
dLB cos),(
Radiosity Concept Radiosity of each surface
depends on radiosity of all other surfaces Treat global illumination as
a linear system Need constant BRDF
(diffuse) Solve rendering equation
as a matrix problem Process
Mesh into patches Calculate form factors Solve radiosity Display patches
Cornell Program of Computer Graphics
Radiosity Equation
( , ) ( , ) ( , , ) ( , ) ( , ) ( , )rL E f G V L dA x x x x x x x x
( , , ) ( )r rf x f x
( , ) ( , ) ( ) ( , ) ( , ) ( , )rL x E x f x L x G x x V x x dA
Assume only diffuse reflection
Convert to radiosity
( , ) ( , ) ( ) ( , ) ( , ) ( , )e rB x B x f x B x G x x V x x dA
Radiosity Approximations
1
n
i i i ji jj
B E F B
2
cos cos1
i j
ij i jij j i
i A A
VF dA dA
A r
Discretize the surface into patches
The form factor
Fii = 0 (patches are flat)Fij = 0 if occludedFij is dimensionless
Radiosity Matrix
1
2
n
B
B
B
1
2
n
E
E
E
1 11 1 12 1 1
2 21 2 22
1
1
1
1
n
n n n nn
F F F
F F
F F
iBiB1
1
1
( )
n
i i i ji jj
n
i i i ji jj
n
i ij i ji jj
B E F B
E B F B
E F B
Such an equation exists for each patch, and in a closed environment, a set of n Simultaneous equations in n unknown Bi values is obtained:
A solution yields a single radiosity value Bi for each patch in the environment – a view-independent solution. The Bi values can be used in a standard renderer and a particular view of the environment constructed from the radiosity solution.
Form Factor Intuition
2
cos cosij i jdi dj i j
VF dAdA
r
2
cos cos1
i j
ij i jij j i
i A A
VF dA dA
A r
Hemicube
Compute form factor with image-space precision Render scene from centroid of Ai Use z-buffer to determine visibility of other surfaces Count “pixels” to determine projected areas
Monte Carlo Sampling
Compute form factor by random sampling Select random points on elements Intersect line segment to evaluate Vij
Evaluate Fij by Monte Carlo integration
Solving the Radiosity Equations
Solution methods: Invert the matrix – O(n3) Iterative methods – O(n2) Hierarchical methods – O(n)
1
2
n
B
B
B
1
2
n
E
E
E
1 11 1 12 1 1
2 21 2 22
1
1
1
1
n
n n n nn
F F F
F F
F F
iBiB
Examples
Museum simulation. Program of Computer Graphics, Cornell University.50,000 patches. Note indirect lighting from ceiling.
Gauss-Siedel Iteration method
1. For all i
Bi = Ei
2. While not converged
For each i in turn
3. Display the image using Bi as the intensity of patch i
i i i ij jj i
B E F B
Progressive Radiosity
Interpretation: Iteratively shoot “unshot” radiosity from elements Select shooters in order of unshot radiosity
Hierarchical Radiosity
Refine elements hierarchically: Compute energy
exchange at different element granularity
satisfying a user-specified error tolerance
Radiosity
Constrained by the resolution of your subdivision patches.
Have to calculate all of the geometry before you rendered an image.
No reflections or specular component.
Path Types
OpenGL L(D|S)E
Ray Tracing LDS*E
Radiosity LD*E
Path Tracing attempts to trace
“all rays” in a scene
Ray Tracing LDS*E Paths Rays cast from eye into scene
Why? Because most rays cast from light wouldn’t reach eye
Shadow rays cast at each intersectionWhy? Because most rays
wouldn’t reach the light source Workload badly distributed
Why? Because # of rays grows exponentially, and their result becomes less influential
objects lights
Tough Cases Caustics
Light focuses through a specular surface onto a diffuse surface
LSDE Which direction should
secondary rays be cast to detect caustic?
Bleeding Color of diffuse surface
reflected in another diffuse surface
LDDE Which direction should
secondary rays be cast to detect bleeding?
Path Tracing
Kajiya, SIGGRAPH 86 Diffuse reflection spawns
infinite rays Pick one ray at random Cuts a path through the
dense ray tree Still cast an extra shadow
ray toward light source at each step in path
Trace > 40 paths per pixel
objects lights
Monte Carlo Path Tracing
Integrate radiance for each pixel by sampling paths randomly
( , ) ( , ) ( , , ) ( , ) ( , ) ( , )rL E f G V L dA x x x x x x x x
Basic Monte Carlo Path Tracer
1. Choose a ray (x, y), t; weight = 1
2. race ray to find intersection with nearest surface
3. Randomly decide whether to compute emitted or reflected light
1. Step 3a: If emitted,
return weight * Le
1. Step 3b: If reflected,
weight *= reflectance
Generate ray in random direction
Go to step 2
Monte Carlo Path Tracing Advantages
Any type of geometry (procedural, curved, ...)
Any type of BRDF (specular, glossy, diffuse, ...)
Samples all types of paths (L(SD)*E)
Accurate control at pixel level
Low memory consumption Disadvantages
Slow convergence Noise in the final image
Monte Carlo Ray Tracing
It’s worse when you have small light sources (e.g. the sun) or lots of light and dark variation (i.e. high-frequency) in your environment.
Photon Mapping
Monte Carlo path tracing relies on lots of camera rays to “find” the bright areas in a scene. Small bright areas can be a real problem. (Hence the typical “overcast” lighting).
Why not start from the light sources themselves, scatter light into the environment, and keep track of where the light goes?
Photon Mapping
Jensen EGRW 95, 96 Simulates the transport of individual photons Photons emitted from source Photons deposited on diffuse surfaces Photons reflected from surfaces to other
surfaces Photons collected by rendering
What is a Photon? A photon p is a particle of
light that carries flux p(xp, p) Power: p – magnitude
(in Watts) and color of the flux it carries, stored as an RGB triple
Position: xp – location of the photon
Direction: p – the incident direction i used to compute irradiance
Photons vs. rays Photons propagate flux Rays gather radiance
p
p
xp
Sources Point source
Photons emitted uniformly in all directions
Power of source (W) distributed evenly among photons
Flux of each photon equal to source power divided by total # of photons
For example, a 60W light bulb would send out a total of 100K photons, each carrying a flux of 0.6 mW
Photons sent out once per simulation, not continuously as in radiosity
Russian Roulette Arvo & Kirk, Particle Transport and Image
Synthesis, SIGGRAPH 90, pp. 63-66. Reflected flux only a fraction of incident
flux After several reflections, spending a lot of
time keeping track of very little flux Instead, absorb some photons and reflect
the rest at full power Spend time tracing fewer full power
photons Probability of reflectance is the
reflectance Probability of absorption is 1 – .
?
Mixed Surfaces
Surfaces have specular and diffuse components d – diffuse reflectance
s – specular reflectance
d + s < 1 (conservation of energy)
Let be a uniform random value from 0 to 1 If < d then reflect diffuse
Else if < d + s then reflect specular Otherwise absorb
Rendering
Photons in photon map are collected by eye rays cast by a distributed ray tracer
Multiple photon maps Indirect irradiance map Caustic map
Rays use the radiance constructed from reflected flux density from nearest neighbor photons
Photon Mapping Rendering
Ray Tracing At each hit:
Ray trace further if the contribution > threshold (more accurate)
Use photon map approximation otherwise
Caustics rendered directly
A = r2
Caustic photon map
The caustics photon map is used only to store photons corresponding to caustics.
It is created by emitting photons towards the specular objects in the scene and storing these as they hit diffuse surfaces.
Global Photon Map
The global photon map is used as a rough approximation of the light/flux within the scene
It is created by emitting photons towards all objects.
It is not visualized directly and therefore it does not require the same precision as the caustics photon map.