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### ボリュームレンダリング入門

1. Introduction to volume rendering perim (@hi2p_perim)
2. http://nukeation.deviantart.com/art/Vue-7-5-Experiment-52-121086230
3. [Křiánek et al. 2014]
4. Participating media w/o volume w/ volume
5. Focuses • Stochastic techniques • General techniques – i.e., not focus on solving specific phenomina, e.g., subsurface scattering • Participating media with constant IOR – For non-constant cases, e.g., see [Ament et al. 2014] • Rendering of steady state – For rendering of transient state, see [Jarabo et al. 2014] [Frisvad et al. 2015] [Ament et al. 2014] [Jarabo et al. 2014]
6. Early history • See – Cerezo et al., “A survey on participating media rendering techniques”, The Visual Computer, 2005. – PBRT
7. Interaction to volume [Dutre et al. 2006]
8. Absorption 𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥) Δ𝑥 𝐿 𝑥 + Δ𝑥 − 𝐿 𝑥 = −𝜎 𝑎 𝑥 𝐿 𝑥 Δ𝑥 Absorption coefficient
9. Out-scattering 𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥) Δ𝑥 𝐿 𝑥 + Δ𝑥 − 𝐿 𝑥 = −𝜎𝑠 𝑥 𝐿 𝑥 Δ𝑥 Scattering coefficient
10. Attenuation coefficient • Absorption + out-scattering • Taking Δ𝑥 → 0, we obtain 𝐿 𝑥 + Δ𝑥 − 𝐿 𝑥 = −𝜎𝑡(𝑥)𝐿 𝑥 Δ𝑥 𝜎𝑡 𝑥 ≡ 𝜎 𝑎 𝑥 + 𝜎𝑠 𝑥 Attenuation coefficient 𝑑𝐿 𝑥 𝑑𝑥 = −𝜎𝑡(𝑥)𝐿 𝑥
11. Transmittance • Solving the equation, we obtain 𝐿 𝑥 + 𝑡 = exp − 0 𝑡 𝜎𝑡 𝑥 + 𝑠 𝑑𝑠 𝐿(𝑥) Transmittance 𝑇𝑟
12. Emission 𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥) Δ𝑥 𝑑𝐿(𝑥) 𝑑𝑥 = 𝐿 𝑒 𝑥
13. In-scattering 𝑥 𝐿(𝑥 + Δ𝑥, 𝜔𝑖)𝐿(𝑥, 𝜔𝑖) Δ𝑥 𝑑𝐿(𝑥) 𝑑𝑥 = 𝜎𝑠(𝑥) 𝒮 𝑝(𝜔𝑖, 𝜔) 𝐿 𝑥, 𝜔 𝑑𝜔 𝜔 Phase function
14. Radiance transfer equation a.k.a. Volume rendering equation 𝐿 𝑥, 𝜔𝑖 = 𝑇𝑟 𝑥, 𝑥 + 𝑡𝜔𝑖 𝐿 𝑥 + 𝑡𝜔𝑖, 𝜔𝑖 + 0 𝑡 𝑇𝑟 𝑥, 𝑥 + 𝑠𝜔𝑖 𝜎𝑠 𝑥 + 𝑠𝜔𝑖 𝐿 𝑠 𝑥 + 𝑠𝜔𝑖, 𝜔𝑖 𝑑𝑠 𝐿 𝑠 𝑥, 𝜔𝑖 ≡ 𝐿 𝑒 𝑥, 𝜔𝑖 + 𝒮 𝑝(𝜔𝑖, 𝜔) 𝐿 𝑥, 𝜔 𝑑𝜔 Absorption + Out-scattering Emission In-scattering
15. Rendering participating media 𝐿 𝑥, 𝜔𝑖 = 𝑇𝑟 𝑥, 𝑥 + 𝑡𝜔𝑖 𝐿 𝑥 + 𝑡𝜔𝑖, 𝜔𝑖 + 0 𝑡 𝑇𝑟 𝑥, 𝑥 + 𝑠𝜔𝑖 𝜎𝑠 𝑥 + 𝑠𝜔𝑖 𝐿 𝑠 𝑥 + 𝑠𝜔𝑖, 𝜔𝑖 𝑑𝑠 𝐿 𝑠 𝑥, 𝜔𝑖 ≡ 𝐿 𝑒 𝑥, 𝜔𝑖 + 𝒮 𝑝(𝜔𝑖, 𝜔) 𝐿 𝑥, 𝜔 𝑑𝜔 Integration w.r.t. distance Integration w.r.t. solid angle
16. Sampling distance: Homogeneous media • Attenuation coefficient is constant • Transmittance • Sample distance from exp. dist 𝐿 𝑥 + 𝑡 = 𝑒−𝜎𝑡 𝑡 𝐿(𝑥) 𝜎𝑡 𝑥 ≡ 𝜎𝑡 𝑝 𝑡 = 𝜎𝑡 𝑒−𝜎𝑡 𝑡
17. [Mitsuba renderer]
18. [Mitsuba renderer]
19. Deterministic distance sampling: Ray marching • Perlin & Hoffert, “Hypertexture”, SIGGRAPH, 1989.
20. Deterministic distance sampling: Ray marching • Perlin & Hoffert, “Hypertexture”, SIGGRAPH, 1989.
21. Unbiased distance sampling: Woodcock tracking • Woodcock et al., “Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry”, 1965. • Raab et al., “Unbiased global illumination with participating media”, MCQMC, 2006.
22. Unbiased distance sampling: Woodcock tracking Maximum extinction 𝜎𝑡 𝑡 = − log(𝑈) 𝜎𝑡 𝑥0 𝑥0 + 𝑡𝜔 𝜔
23. Unbiased distance sampling: Woodcock tracking Maximum extinction 𝜎𝑡 𝑥0 𝜔 𝜎𝑡 (𝑥0 + 𝑡𝜔) 𝜎𝑡 < 𝑈
24. Unbiased distance sampling: Acceleration • Kd-tree – Yue et al., “Unbiased, adaptive stochastic sampling for rendering inhomogeneous participating media”, SIGGRAPH Asia, 2010. • Uniform grid – Szirmay-Kalos et al., “Free path sampling in high resolution inhomogeneous participating media”, CGF, 2011. [Yue et al. 2010]
25. Extension to BDPT • Larfortune & Willems, “Rendering participating media with bidirectional path tracing”, EGSR, 1996.
26. Extension to BDPT
27. Extension to MLT • Pauly et al. “Metropolis light transport for participating Media”, EGSR, 1996.
28. Recall.. Perturbation in MLT
29. Extension to MLT: Perturbations
30. Extension to MLT: Perturbations 𝑝(𝑡 →⋅)
31. Extension to MLT: Perturbations
32. Importance sampling: Equi-angular sampling • Kulla & Fajardo, “Importance sampling techniques for path tracing in participating media”, EGSR, 2012.
33. Importance sampling: Equi-angular sampling
34. Importance sampling: Equi-angular sampling • Assumption: single scattering, point light source Point light source 𝐿 𝑥, 𝜔 = 𝑎 𝑏 𝜎𝑠 𝑒−𝜎𝑡(𝑡+Δ+ 𝐷2+𝑡2) Φ 𝐷2 + 𝑡2 𝑑𝑡 Introduces weak singurality
35. Importance sampling: Equi-angular sampling • Sample a distance 𝑡 from 𝑝 𝑡 ≡ 𝐷 (𝜃 𝑏 − 𝜃 𝑎)(𝐷2 + 𝑡2) 𝑡 𝑈 = 𝐷 tan 1 − 𝑈 𝜃 𝑎 + 𝑈𝜃 𝑏 Sample angle uniformly between 𝜃 𝑎 and 𝜃 𝑏
36. Importance sampling: Joint importance sampling • Georgiev et al., “Joint importance sampling of low- order volumetric scattering”, SIGGRAPH Asia, 2013.
37. Importance sampling: Joint importance sampling 𝐺 𝐺 𝐺
38. Importance sampling: Joint importance sampling
39. Density estimation • Jensen & Christensen, “Efficient simulation of light transport in scenes with participating media using photon maps”, SIGGRAPH, 1998.
40. Density estimation
41. Density estimation Eye ray Photons
42. Density estimation with beam: Beam radiance estimation • Jarosz et al., “The beam radiance estimate for volumetric photon mapping”, EG, 2008.
43. Density estimation with beam: Beam radiance estimation Photons Eye ray
44. Density estimation with beam: Photon beam • Jarosz et al., “Progressive photon beams”, SIGGRAPH Asia, 2011.
45. Density estimation with beam Eye ray Photon beams
46. Beam formulation
47. Beam formulation • Jarosz et al., “A comprehensive theory of volumetric radiance estimation using photon points and beams”, TOG, 2011. [Křiánek et al. 2014]
48. Beam formulation: many-light technique • Novak et al., “Virtual ray lights for rendering scenes with participating media”, SIGGRAPH, 2012. • Novak et al., “Progressive virtual beam lights”, EGSR, 2012.
49. Density estimation: Unified framework • Křiánek et al., “Unifying Points, Beams, and Paths in Volumetric Light Transport Simulation”, SIGGRAPH, 2014.
50. Density estimation: Unified framework • Extended MIS Regular sampling technique Extended sampling technique
51. Density estimation: Unified framework • Extended balance heuristics

### Editor's Notes

1. Efficiency heavily depends on maximum extension
2. And, compute transition probability with correct measure conversion
3. Importance sampling, distance sampling