Global Illumination

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).

Indirect Illumination

Color Bleeding

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

BB

B

1

2

n

EE

E

1 11 1 12 1 1

2 21 2 22

1

11

1

n

n n n nn

F F FF 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.

Intuition

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

BB

B

1

2

n

EE

E

1 11 1 12 1 1

2 21 2 22

1

11

1

n

n n n nn

F F FF F

F F

iBiB

Examples

Museum simulation. Program of Computer Graphics, Cornell University.50,000 patches. Note indirect lighting from ceiling.

Gauss-Siedel Iteration method1. For all i

Bi = Ei2. 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

Interpretation of Iteration Iteratively gather radiosity to elements

Progressive Radiosity

Progressive Radiosity Interpretation:

Iteratively shoot “unshot” radiosity from elements Select shooters in order of unshot radiosity

Progressive Radiosity

Adaptive Meshing Refine mesh in areas of

large errors

Adaptive Meshing

Uniform Meshing Adaptive Meshing

Hierarchical Radiosity Refine elements

hierarchically: Compute energy

exchange at different element granularity

satisfying a user-specified error tolerance

Hierarchical Radiosity

Hierarchical Radiosity

Displaying RadiosityUsually Gouraud Shading

Computed Rendered

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 Tracer1. Choose a ray (x, y), t; weight = 12. race ray to find intersection with nearest surface3. Randomly decide whether to compute emitted or

reflected light1. Step 3a: If emitted,

return weight * Le1. Step 3b: If reflected,

weight *= reflectanceGenerate ray in random directionGo to step 2

Bi-directional Path Tracing Role of source and receiver can be switched, flux

does not change

Bi-directional Path Tracing

Tracing from eye

Tracing from light

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 path tracing

1000 path / pixel

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.

Monte Carlo Integration

That’s why you always see “overcast” lighting like this.

Noise Filtering

van Jensen, Stanford

Unfiltered filtered

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

Photon Mapping

Direct illumination Photon Map

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.

Caustic illumination

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.

Indirect Illumination

Example

224,316 caustic photons, 3095 global photons

Example

Readings Textbook 16.13 Distribution Ray Tracing: Theory and Practice

 Shirley and  Wang. Proceedings of the 3rd Eurographics Rendering Workshop 1992

Global Illumination using Photon Maps Henrik Wann Jensen, Rendering Techniques '96