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ボリュームレンダリング入門

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レイトレ合宿3 (https://sites.google.com/site/raytracingcamp3) のセミナー資料です.

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ボリュームレンダリング入門

  1. 1. Introduction to volume rendering perim (@hi2p_perim)
  2. 2. http://nukeation.deviantart.com/art/Vue-7-5-Experiment-52-121086230
  3. 3. [Křiánek et al. 2014]
  4. 4. Participating media w/o volume w/ volume
  5. 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. 6. Early history • See – Cerezo et al., “A survey on participating media rendering techniques”, The Visual Computer, 2005. – PBRT
  7. 7. Interaction to volume [Dutre et al. 2006]
  8. 8. Absorption 𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥) Δ𝑥 𝐿 𝑥 + Δ𝑥 − 𝐿 𝑥 = −𝜎 𝑎 𝑥 𝐿 𝑥 Δ𝑥 Absorption coefficient
  9. 9. Out-scattering 𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥) Δ𝑥 𝐿 𝑥 + Δ𝑥 − 𝐿 𝑥 = −𝜎𝑠 𝑥 𝐿 𝑥 Δ𝑥 Scattering coefficient
  10. 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. 11. Transmittance • Solving the equation, we obtain 𝐿 𝑥 + 𝑡 = exp − 0 𝑡 𝜎𝑡 𝑥 + 𝑠 𝑑𝑠 𝐿(𝑥) Transmittance 𝑇𝑟
  12. 12. Emission 𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥) Δ𝑥 𝑑𝐿(𝑥) 𝑑𝑥 = 𝐿 𝑒 𝑥
  13. 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. 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. 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. 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. 17. [Mitsuba renderer]
  18. 18. [Mitsuba renderer]
  19. 19. Deterministic distance sampling: Ray marching • Perlin & Hoffert, “Hypertexture”, SIGGRAPH, 1989.
  20. 20. Deterministic distance sampling: Ray marching • Perlin & Hoffert, “Hypertexture”, SIGGRAPH, 1989.
  21. 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. 22. Unbiased distance sampling: Woodcock tracking Maximum extinction 𝜎𝑡 𝑡 = − log(𝑈) 𝜎𝑡 𝑥0 𝑥0 + 𝑡𝜔 𝜔
  23. 23. Unbiased distance sampling: Woodcock tracking Maximum extinction 𝜎𝑡 𝑥0 𝜔 𝜎𝑡 (𝑥0 + 𝑡𝜔) 𝜎𝑡 < 𝑈
  24. 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. 25. Extension to BDPT • Larfortune & Willems, “Rendering participating media with bidirectional path tracing”, EGSR, 1996.
  26. 26. Extension to BDPT
  27. 27. Extension to MLT • Pauly et al. “Metropolis light transport for participating Media”, EGSR, 1996.
  28. 28. Recall.. Perturbation in MLT
  29. 29. Extension to MLT: Perturbations
  30. 30. Extension to MLT: Perturbations 𝑝(𝑡 →⋅)
  31. 31. Extension to MLT: Perturbations
  32. 32. Importance sampling: Equi-angular sampling • Kulla & Fajardo, “Importance sampling techniques for path tracing in participating media”, EGSR, 2012.
  33. 33. Importance sampling: Equi-angular sampling
  34. 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. 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. 36. Importance sampling: Joint importance sampling • Georgiev et al., “Joint importance sampling of low- order volumetric scattering”, SIGGRAPH Asia, 2013.
  37. 37. Importance sampling: Joint importance sampling 𝐺 𝐺 𝐺
  38. 38. Importance sampling: Joint importance sampling
  39. 39. Density estimation • Jensen & Christensen, “Efficient simulation of light transport in scenes with participating media using photon maps”, SIGGRAPH, 1998.
  40. 40. Density estimation
  41. 41. Density estimation Eye ray Photons
  42. 42. Density estimation with beam: Beam radiance estimation • Jarosz et al., “The beam radiance estimate for volumetric photon mapping”, EG, 2008.
  43. 43. Density estimation with beam: Beam radiance estimation Photons Eye ray
  44. 44. Density estimation with beam: Photon beam • Jarosz et al., “Progressive photon beams”, SIGGRAPH Asia, 2011.
  45. 45. Density estimation with beam Eye ray Photon beams
  46. 46. Beam formulation
  47. 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. 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. 49. Density estimation: Unified framework • Křiánek et al., “Unifying Points, Beams, and Paths in Volumetric Light Transport Simulation”, SIGGRAPH, 2014.
  50. 50. Density estimation: Unified framework • Extended MIS Regular sampling technique Extended sampling technique
  51. 51. Density estimation: Unified framework • Extended balance heuristics

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