ボリュームレンダリング入門
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This document provides an introduction to volume rendering techniques. It discusses stochastic techniques for rendering participating media with constant refractive index. It also covers the radiance transfer equation and techniques for sampling distance through media, including ray marching, Woodcock tracking, and acceleration structures. Methods for importance sampling, density estimation using photon mapping and beams, and a unified framework are also summarized.
![[Křiánek et al. 2014]](https://image.slidesharecdn.com/201508rtcamp3-150830014521-lva1-app6892/85/-3-320.jpg)
![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]](https://image.slidesharecdn.com/201508rtcamp3-150830014521-lva1-app6892/85/-5-320.jpg)
![Interaction to volume
[Dutre et al. 2006]](https://image.slidesharecdn.com/201508rtcamp3-150830014521-lva1-app6892/85/-7-320.jpg)
![[Mitsuba renderer]](https://image.slidesharecdn.com/201508rtcamp3-150830014521-lva1-app6892/85/-17-320.jpg)
![[Mitsuba renderer]](https://image.slidesharecdn.com/201508rtcamp3-150830014521-lva1-app6892/85/-18-320.jpg)
![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]](https://image.slidesharecdn.com/201508rtcamp3-150830014521-lva1-app6892/85/-24-320.jpg)
![Beam formulation
• Jarosz et al., “A comprehensive theory of volumetric
radiance estimation using photon points and
beams”, TOG, 2011.
[Křiánek et al. 2014]](https://image.slidesharecdn.com/201508rtcamp3-150830014521-lva1-app6892/85/-47-320.jpg)
ボリュームレンダリング入門
- 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
- 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.
- 27. Extension to MLT • Pauly et al. “Metropolis light transport for participating Media”, EGSR, 1996.
- 32. Importance sampling: Equi-angular sampling • Kulla & Fajardo, “Importance sampling techniques for path tracing in participating media”, EGSR, 2012.
- 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.
- 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.
- 51. Density estimation: Unified framework • Extended MIS Regular sampling technique Extended sampling technique
- 52. Density estimation: Unified framework • Extended balance heuristics