LATTICE-BOLTZMANN LIGHTING BY ROBERT GEIST, KARL RASCHE, JAMES WESTALL AND ROBERT SCHALKOFF

William MossAdvanced Image Synthesis, Fall 2008

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MOTIVATIONSeptember 25, 2008

Quickly!

Visual simulation of smoke, Fedkiw et al., 2001

Metropolis Light Transport for Participating Media, Pauly et al, 2000

Efficient simulation of light transport in scenes with participating media using photon maps, Jensen and Christensen, 1998

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OVERVIEW Background on participating media Participating media as diffusion Lattice-Boltzmann methods

Background Solving the diffusion equation Results

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PARTICIPATING MEDIA BACKGROUND Interactions take place at all points in the

medium, not just the boundaries Solve the volume radiative transfer equation:

where f is the phase, σa is the absorption coefficient, σs is the scattering coefficient, σt = σa + σs and Le is the emissive field

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Radiance =

PARTICIPATING MEDIA BACKGROUNDSeptember 25, 2008

Emission − Out-scattering− Absorption + In-scattering

Diffuse Intensity (Id)

Reduced Incident Intensity (Iri)

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PARTICIPATING MEDIA BACKGROUND

Ray xu = x0 - us

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REDUCED INCIDENT INTENSITY

Where I is the incident intensity, σt = σa + σs and ρ is the density of the medium

Trace a ray through the medium, integrating:

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DIFFUSE INTENSITY As the medium becomes thick

Number of scattering events increases Directional dependence decreases Light distribution tends towards uniformity

Approximate the diffuse intensity First two Taylor expansion terms in the directional

component Results in a diffusion equation for average Id

See Stam, ‘95 for full derivation

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LATTICE-BOLTZMANN METHODS First introduced in as the Lattice-Gas Automaton

(1987) Lattice-Boltzmann model (1988)

A system specified by interactions with neighbors Simple local interaction functions can model complex

macroscopic phenomena Dates back to 1940s with cellular automata

Game of life

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LATTICE-BOLTZMANN METHODS Heart of the method is the lattice

Some have used hexagonal, they choose a grid Each point connected to the surrounding points

Stores “directional density,” density flowing in that direction

For a 3D grid, 18 directional densities per node

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LATTICE-BOLTZMANN METHODSSeptember 25, 2008

2 distanceat

shown) (8 neighbors 121 distanceat neighbors 6

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LATTICE-BOLTZMANN METHODS Flow is represented by a matrix, Θ

Θij – fraction of flow in direction j that will be diverted to direction i

Updates performed synchronously

ΘI = O I is the vector of current densities O is the vector of densities flowing out of that site

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ji

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LATTICE-BOLTZMANN METHODS An alterative to FEM for solving coupled PDEs

Comparable speed, stability, accuracy and storage Widely used for solving fluid flow in physics

Multiple methods for simulating the incompressible, time-dependant Navier-Stokes

Used in graphics for modeling gases Also Navier-Stokes

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LATTICE-BOLTZMANN METHODS Advantages

Easy to implement Easy to parallelize Easy to handle complex boundary conditions

Disadvantages Specified by microscopic particle density interactions Difficult to deduce rules given a macroscopic system

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LATTICE-BOLTZMANN SOLUTION Choose Θ such that in the limit, we get the

diffusion equations Start simple, isotropic scattering

σa, is absorption at each lattice point

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LATTICE-BOLTZMANN SOLUTION For axial rows (i = 1…6):

For non-axial rows (i = 7…18):

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LATTICE-BOLTZMANN SOLUTION Show this simulates a diffusion process

Put light into the system, let it “settle” and render

Start with:

Where fi(r, t) is the density at site r at time t in direction ci, λ Is the lattice spacing, τ is the time step and Θi is the ith row of Θ

The ci directions are all 18 previous flow directions

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LATTICE-BOLTZMANN SOLUTION Let λ and τ go to 0 (see paper for full, 1 page, proof): Result is a diffusion equation (phew):

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ANISOTROPIC SCATTERING Modify Θ

Remember Θij is the fraction of flow in direction cj that will be diverted to direction ci

Weight values unevenly For forward-scattering, weight values where cj ·ci < 0 more

heavily For back-scattering, do the reverse

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ANISOTROPIC SCATTERING Scale σs in Θij by

Where pi,j is a discrete version of Henyey-Greenstein phase function

Where ni = ci / |ci| and g defines the scattering g > 0 provides forward scattering, g < 0 back scattering

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INCIDENT LIGHT Add light at the boundaries of the lattice

Choose the lattice direction with the largest dot product with the light direction

Fix the inflow in that direction to the dot product Reduce the remaining incident light by that amount Repeat for remain directions Apply the inflow to boundary nodes

Only handles directional light Fine for clouds

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SOLVING THE SYSTEM Inject the light at the boundaries For each node

Distribute the incoming density according to the collision rules (i.e. ΘI = O)

Flow the distributed density to the neighboring nodes Repeat

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SOLVING THE SYSTEMSeptember 25, 2008

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Radiance =

PARTICIPATING MEDIA BACKGROUNDSeptember 25, 2008

Emission − Out-scattering− Absorption + In-scattering

Diffuse Intensity (Iri)

Reduced Incident Intensity (Id)

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RENDERING THE VOLUME DENSITIES Have the outward flowing density at every point

This represents the number of photons Sum all the directions to represent the illuminate

Could use viewer location, if desired Shoot rays into the volume

Attenuate the value due to the reduced indecent intensity

Increase the value due to the illuminate at intersected lattice cells

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RESULTS

Isotropic Scattering

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RESULTSSeptember 25, 2008

Forward Scattering

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RESULTSSeptember 25, 2008

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REFERENCES Robert Geist, Karl Rasche, James Westall and

Robert Schalkoff, Lattice-Boltzmann Lighting, Proc. Eurographics Symposium on Rendering, June, 2004

Jos Stam, Multiple scattering as a diffusion process, Eurographics Rendering Workshop, 1995

Eva Cerezo, Frederic Pérez, Xavier Pueyo, Francisco J. Seron and François X. Sillion, A survey on participating media rendering techniques, The Visual Computer, June 2005

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