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

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

SCCG’ 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!

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

SCCG’20005 Form Factor Singularities and Discontinuities

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

SCCG’20007

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

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 )

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

SCCG’ 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)

SCCG’ 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:

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

SCCG’ 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

SCCG’ Results

SCCG’ 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

SCCG’ 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