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LATTICE-BOLTZMANN LIGHTING BY ROBERT GEIST, KARL RASCHE, JAMES WESTALL AND ROBERT SCHALKOFF
William MossAdvanced Image Synthesis, Fall 2008
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MOTIVATION
September 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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wxLxwwxLwwxfxwxLxx
wx,Ltsea
,'',,',,
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Radiance =
PARTICIPATING MEDIA BACKGROUND
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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 METHODS
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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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j
i
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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 SYSTEM
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Radiance =
PARTICIPATING MEDIA BACKGROUND
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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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RESULTS
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Forward Scattering
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RESULTS
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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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