1. Path Differentials for MC Rendering Frank Suykens Department of Computer Science K.U.Leuven, Belgium Dagstuhl 2001: Stochastic methods in Rendering

2. Path differentials2 Monte Carlo Rendering Few samples  noise –Supersampling / Variance reduction –Exploit coherence in the ‘neighborhood’ of a path

3. Path differentials3 Neighborhood of a path Region of influence Distance to neighboring paths Density ‘Footprint’ Applications: Texture filtering Splatting Hierarchical refinement … Noise vs. Bias V

4. Path differentials4 Related Work (Non MC) Extend path to finite width: beam, cone, pencil (Heckbert, Amanatides, Shinia et.al.) –difficult intersections, reflections and refractions Maintain connectivity (Collins) –diverging ray trees Ray differentials (Igehy) –Derivatives : based on point samples! Specular path perturbations (Chen, Arvo)

5. Path differentials5 Overview Path sampling Path derivatives, differentials, footprint Path gradient Applications & results Conclusions Compute derivatives of directions and vertices in a path to estimate a ‘footprint’, the region of influence of the path Overview Generalization of Ray differentials

6. Path differentials6 Path Sampling D V D’ V’  D’ = h( D, V, x, y ) Random variables Sampling of new directions and vertices in a path: Importance Sampling: BRDF Light sources …

7. Path differentials7 Path Sampling D V D’ V’  D’ = g( x 1, x 2, …, x k ) Unit random variables Sampling of new directions and vertices in a path: ‘Path generation function’ DomainPath g

8. Path differentials8 Path Derivatives D V D’ V’ Partial derivatives for all random variables: Sensitivity in terms of x j

9. Path differentials9 Path Derivatives Vertex derivatives : V’ N

10. Path differentials10 Path Perturbation Vertex perturbation : Differential Vector

11. Path differentials11 Footprint approximation Path Footprint Footprint : All points reachable given perturbation intervals [ -  x j,  x j ] (for each j) Other footprint definitions possible (e.g. convolution)

12. Path differentials12 Derivative Computation Derive all sampling procedures: D’ = h( D, V, x, y ) Transfer: V, D’  V’ = r ( V, D’ ) ( Trace a ray, cfr. Ray differentials) Relatively simple!

13. Path differentials13 Determine footprint size Based on number of samples Many samples: smaller delta’s: Based on path gradient Choose Intervals 0 1 1 DomainPath

14. Path differentials14 Path Gradient Partial derivatives of path evaluation Error threshold   x j

15. Path differentials15 Summary Partial differentials Perturbation intervals  Differential vectors  Footprint

16. Path differentials16 Application: Texture Filtering Classical ray tracing, but: Glossy reflection/refraction (mod. Phong) Texture filtering over footprint (Box filter) Glossy V

17. Path differentials17 Application: Texture Filtering Filtering, no gradient Filtering, gradient No filtering Reference Samples: 1/pixel, 4/scatter

18. Path differentials18 Application: Radiosity Hierarchical particle tracing radiosity Footprint size  subdivision level

19. Path differentials19 Application: Radiosity 400.000 paths, with gradient

20. Path differentials20 Application: Radiosity Hierarchical (footprint) Non hierarchical (pre-meshed)

21. Path differentials21 Conclusions Derivatives of vertices, directions, path evaluation useful for: –Footprint estimates, perturbations, … Exploit coherence over footprint –Applicable to many MC methods based on path sampling –Bias vs. Noise See EGWR 2001 paper

22. Path differentials22 Future Work Other footprint estimates –use convolution Other derivative uses –Use path perturbations directly Apply to other global illumination algorithms (Importance maps) Include visibility

23. Path differentials23 Example Path Diffuse Glossy

24. Path differentials24 Derivative Computation …