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. SCCG’20002 Self-emitted radiosity Reflectivity Total radiosity The Radiosity Method Form factor radiative exchange factor

3. SCCG’20003 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 ¹Tone mapping and display In practice intertwined!

4. SCCG’20004 Discretisation Artifacts Constant Approximation “true” solution Quadratic Approximation

5. SCCG’20005 Form Factor Singularities and Discontinuities

6. SCCG’20006 Higher-Order Approximations l “True” radiosity: l Find “best” polynomial approximation: Basis functions

7. SCCG’20007

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

9. SCCG’20009 Random Walk Solution (Feda, Bouatouch and Pattanaik) l Trace “analog” light paths l Collisions are distributed with density proportional to “true” radiosity. l 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(K 2 )

10. SCCG’200010 l Power equations: l Deterministic Jacobi Algorithm: (quadratic cost) Jacobi Iterative Method

11. SCCG’200011 Stochastic Jacobi iterations (Neumann et al.) 1) Select patch j: 2) Select i conditional on j: 3) Score (form factor cancels!!) VARIANCE: (log-linear cost)

12. SCCG’200012 Form Factor Sampling Local Lines Global Lines (Sbert) l Form factors F ij for fixed patch i form a probability distribution that can be sampled efficiently by tracing rays:

13. SCCG’200013 Higher order approximations l 1) Sample point y: l 2) Sample point x conditionally: l 3) Score:

14. SCCG’200014 Results ConstantLinearQuadraticCubic Stochastic Jacobi Random walk As good as random walkAs good as random walk Variance proportional to number of basis functionsVariance proportional to number of basis functions

15. SCCG’200015 Results

16. SCCG’200016 Variance reduction methods l View-importance sampling: # Arbitrary variance reduction, high cost l Constant control variate (aka constant radiosity steps) # 5-50% variance reduction, low cost l Bidirectional energy transports # up to factor 2 variance reduction, very low additional cost

17. SCCG’200017 Conclusion l Basic method as good as (continuous random walks) l More easy variance reduction l Straightforward incorporation of hierarchical refinement # oracle needs to be cheap! l Needs discontinuity meshing to be perfect l Future work: discrete random walks