Lattice-Boltzmann Lighting By Robert Geist , Karl Rasche , James Westall and Robert Schalkoff

1. William Moss Advanced Image Synthesis, Fall 2008 Lattice-Boltzmann LightingBy Robert Geist, Karl Rasche, James Westall and Robert Schalkoff

2. Quickly! Motivation Visual simulation of smoke, Fedkiw et al., 2001 Efficient simulation of light transport in scenes with participating media using photon maps, Jensen and Christensen, 1998 Metropolis Light Transport for Participating Media, Pauly et al, 2000

3. Background on participating media • Participating media as diffusion • Lattice-Boltzmann methods • Background • Solving the diffusion equation • Results Overview

4. 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+ σsand Le is the emissive field Participating Media Background

5. Radiance = Participating Media Background Emission • + In-scattering • − Absorption • − Out-scattering • Diffuse Intensity (Id) • Reduced Incident Intensity (Iri)

6. Ray xu = x0 - us Participating Media Background

7. Where I is the incident intensity,σt = σa + σsand ρ is the density of the medium • Trace a ray through the medium, integrating: Reduced Incident Intensity

8. 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 Diffuse Intensity

9. 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 Lattice-Boltzmann Methods

10. 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 Lattice-Boltzmann Methods

11. Lattice-Boltzmann Methods

12. Flow is represented by a matrix, Θ • Θij– fraction of flow in direction jthat 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 Lattice-Boltzmann Methods j i

13. 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 Lattice-Boltzmann Methods

14. 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 Lattice-Boltzmann Methods

15. Choose Θ such that in the limit, we get the diffusion equations • Start simple, isotropic scattering • σa, is absorption at each lattice point Lattice-Boltzmann solution

16. For axial rows (i = 1…6): For non-axial rows (i = 7…18): Lattice-Boltzmann solution

17. 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 Θiis the ith row of Θ • The ci directions are all 18 previous flow directions Lattice-Boltzmann solution

18. Let λand τ go to 0 (see paper for full, 1 page, proof): • Result is a diffusion equation (phew): Lattice-Boltzmann solution

19. Modify Θ • Remember Θijis the fraction of flow in direction cj that will be diverted to direction ci • Weight values unevenly • For forward-scattering, weight values where ĉj·ĉi< 0 more heavily • For back-scattering, do the reverse Anisotropic Scattering

20. Scale σs in Θijby • 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 Anisotropic Scattering

21. 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 Incident Light

22. 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 Solving the System

23. Solving the System

24. Radiance = Participating Media Background Emission • + In-scattering • − Absorption • − Out-scattering • Diffuse Intensity (Iri) • Reduced Incident Intensity (Id)

25. 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 Rendering The Volume Densities

26. Isotropic Scattering Results

27. Results • Forward Scattering

28. Results

29. Robert Geist, Karl Rasche, James Westall and Robert Schalkoff, Lattice-Boltzmann Lighting, Proc. Eurographics Symposium on Rendering, June, 2004 JosStam, 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 References