Ph. Bekaert, M. Sbert, Y Willems Department of Computer Science, K.U.Leuven I.M.A., U.d.Girona

1. The Computation of Higher-Order Radiosity Approximations with a Stochastic Jacobi Iterative Method Ph. Bekaert, M. Sbert, Y Willems Department of Computer Science, K.U.Leuven I.M.A., U.d.Girona

2. The Radiosity Method Reflectivity Self-emitted radiosity Total radiosity Form factor radiative exchange factor

3. 4 Steps, 2 Problematic • Discretise the input scene Problem: discretisation artifacts • Compute form factors Problem: huge number of non-trivial integrals: 95% of the computing time, very large storage requirements, computational error. • Solve radiosity system • Tone mapping and display In practice intertwined!

4. Discretisation Artifacts Constant Approximation “true” solution Quadratic Approximation

5. Form Factor Singularities and Discontinuities

6. Higher-Order Approximations • “True” radiosity: • Find “best” polynomial approximation: Basis functions

8. Solution Methods: • By projecting the solution of the integral equation on the basis functions • By solving a discretised problem, e.g. Galerkin method: Dual basis function Generalised form factor

9. Random Walk Solution(Feda, Bouatouch and Pattanaik) • Trace “analog” light paths • Collisions are distributed with density proportional to “true” radiosity. • Basically average value of dual basis function at collision points: Note: no form factors! Feda: amount of work for K-th order approx is O(K2)

10. Jacobi Iterative Method • Power equations: • Deterministic Jacobi Algorithm: (quadratic cost)

11. Stochastic Jacobi iterations 1) Select patch j: (Neumann et al.) 2) Select i conditional on j: 3) Score (form factor cancels!!) VARIANCE: (log-linear cost)

12. Form Factor Sampling • Form factors Fij for fixed patch i form a probability distribution that can be sampled efficiently by tracing rays: Local Lines Global Lines (Sbert)

13. Higher order approximations • 1) Sample point y: • 2) Sample point x conditionally: • 3) Score:

14. Constant Linear Quadratic Cubic Stochastic Jacobi Random walk • As good as random walk • Variance proportional to number of basis functions Results

15. Results

16. Variance reduction methods • View-importance sampling: • Arbitrary variance reduction, high cost • Constant control variate (aka constant radiosity steps) • 5-50% variance reduction, low cost • Bidirectional energy transports • up to factor 2 variance reduction, very low additional cost

17. Conclusion • Basic method as good as (continuous random walks) • More easy variance reduction • Straightforward incorporation of hierarchical refinement • oracle needs to be cheap! • Needs discontinuity meshing to be perfect • Future work: discrete random walks