1. The Natural-Constraint Representation of the Path Space for Efficient Light Transport Simulation Anton S. Kaplanyan 1,2 and Johannes Hanika 1 and Carsten Dachsbacher 1 1 Karlsruhe Institute of Technology, 2 Lightrig

2. Motivation 2 [Wikipedia Commons] Rendered with manifold exploration [JakobMarschner12] Rendered with out method Jewelry

3. Intuition: Constraints in a Lens System Two endpoints with specular constraints 3

4. Idea: Deviate from Specular Path Rough scattering: use soft constraints 4

5. Outline Introduction Prior work New generalized coordinates Properties of the domain Overview: Metropolis light transport New half-vector mutation Results Conclusion 5 Theory Practice

6. Rendering with Light Transport Generate image by computing Flux incident on the sensor Sample all possible paths Stochastic integration 6

7. Emission BSDFs Absorption Geometric terms Path Integral Framework 7

8. Prior Work: Specular Paths Specular paths are hard constraints Obey Fermat principle [Alhazen1021] Ray transfer matrices used in optics [Gauss1840] Pencil tracing in graphics [Shinya87] 8 Start point End point Optical system

9. Prior Work: Specular Paths Rendering specular paths with geometric knowledge Solving for constraints [MitchellHanrahan92] First and second order analysis [ChenArvo00] Predictor-corrector perturbations [JakobMarschner12] Our work is inspired by manifold exploration 9 [Mitchell and Hanrahan 1992] [Chen and Arvo 2000] [Jakob and Marschner 2012]

10. THEORY The Domain of Halfway Vectors for Light Paths

11. Generalized Coordinates 11

12. Deviating the Half Vectors A path in generalized coordinates 12

13. Path Contribution with New Formulation 13 Emission Scattering distributions Camera responsivity Transfer matrix + Geometric term = Generalized G Area measure Half vector domain

14. Simplified Measurement 14

15. Decomposition of Path Integral Decorrelated islands Set of 2D integrals Easy-to-predict spectrum Mostly changes local BSDF Well-studied sampling 15

16. PRACTICAL RENDERING Mutation Strategy for Metropolis Light Transport

17. Metropolis Light Transport Take a path and perturb it [VeachGuibas97] Specialized mutation strategies Manifold exploration (ME) [JakobMarschner12] 17

18. Metropolis Light Transport 18

19. Half-vector Space Mutation Mutation 1. Perturb half-vectors 2. Find a new path Similar machinery to ME (see ME paper) Specular chains: fall back to ME Jump over geometric parts Take prediction as a proposal 19

20. Importance-Sample All BSDFs Query avg. BSDF roughness Approximate with Beckmann lobe Sample as ~2D Gaussian Known optimal step sizes from MCMC 20

21. Results: Kitchen 21 HSLT PSSMLT VMLT MEMLT HSLT

22. Results: Necklace 22 HSLT VMLT MEPT MEMLT HSLT

23. Conclusion Convenient domain for paths on surfaces Generalized coordinates Beneficial properties of path integral Sampling in generalized coordinates Robust light transport (especially glossy and specular) Importance sampling of all BSDFs along a path Practical stratification for MLT 23

24. Limitations and Future Work Geometric smoothness Level of detail for displaced geometry Rare events: needle in a haystack Probability-1 search Participating media More dimensions, new soft constraints 24

25. Thank You Questions?

26. Backup: Stratification for MCMC Expected change on the image from changes in half vectors 26 Without stratificationWith stratification

27. Backup: Spectral Sampling How to distribute step sizes among half vectors? Spectral sampling for MC [SubrKautz2013] Convex combination based on bandwidth (see paper) 27 Without redistributionWith redistribution