Introduction to
volume rendering
perim
(@hi2p_perim)

http://nukeation.deviantart.com/art/Vue-7-5-Experiment-52-121086230

[Křiánek et al. 2014]

Participating media
w/o volume w/ volume

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]

Early history
• See
– Cerezo et al., “A survey on participating media
rendering techniques”, The Visual Computer,
2005.
– PBRT

Interaction to volume
[Dutre et al. 2006]

Absorption
𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥)
Δ𝑥
𝐿 𝑥 + Δ𝑥 − 𝐿 𝑥 = −𝜎 𝑎 𝑥 𝐿 𝑥 Δ𝑥
Absorption coefficient

Out-scattering
𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥)
Δ𝑥
𝐿 𝑥 + Δ𝑥 − 𝐿 𝑥 = −𝜎𝑠 𝑥 𝐿 𝑥 Δ𝑥
Scattering coefficient

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

Transmittance
• Solving the equation, we obtain
𝐿 𝑥 + 𝑡 = exp −
0
𝑡
𝜎𝑡 𝑥 + 𝑠 𝑑𝑠 𝐿(𝑥)
Transmittance 𝑇𝑟

Emission
𝑥 𝐿(𝑥 + Δ𝑥)𝐿(𝑥)
Δ𝑥
𝑑𝐿(𝑥)
𝑑𝑥
= 𝐿 𝑒 𝑥

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

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

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

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

[Mitsuba renderer]

[Mitsuba renderer]

Deterministic distance sampling:
Ray marching
• Perlin & Hoffert, “Hypertexture”, SIGGRAPH, 1989.

Deterministic distance sampling:
Ray marching
• Perlin & Hoffert, “Hypertexture”, SIGGRAPH, 1989.

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.

Unbiased distance sampling:
Woodcock tracking
Maximum extinction 𝜎𝑡
𝑡 = −
log(𝑈)
𝜎𝑡
𝑥0 𝑥0 + 𝑡𝜔
𝜔

Unbiased distance sampling:
Woodcock tracking
Maximum extinction 𝜎𝑡
𝑥0
𝜔
𝜎𝑡 (𝑥0 + 𝑡𝜔)
𝜎𝑡
< 𝑈

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]

Extension to BDPT
• Larfortune & Willems, “Rendering participating
media with bidirectional path tracing”, EGSR, 1996.

Extension to BDPT

Extension to MLT
• Pauly et al. “Metropolis light transport for
participating Media”, EGSR, 1996.

Recall..
Perturbation in MLT

Extension to MLT:
Perturbations

Extension to MLT:
Perturbations
𝑝(𝑡 →⋅)

Extension to MLT:
Perturbations

Importance sampling:
Equi-angular sampling
• Kulla & Fajardo, “Importance sampling techniques
for path tracing in participating media”, EGSR, 2012.

Importance sampling:
Equi-angular sampling

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

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

Importance sampling:
Joint importance sampling
• Georgiev et al., “Joint importance sampling of low-
order volumetric scattering”, SIGGRAPH Asia, 2013.

Importance sampling:
Joint importance sampling
𝐺
𝐺
𝐺

Importance sampling:
Joint importance sampling

Density estimation
• Jensen & Christensen, “Efficient simulation of light
transport in scenes with participating media using
photon maps”, SIGGRAPH, 1998.

Density estimation

Density estimation
Eye ray
Photons

Density estimation with beam:
Beam radiance estimation
• Jarosz et al., “The beam radiance estimate for
volumetric photon mapping”, EG, 2008.

Density estimation with beam:
Beam radiance estimation
Photons
Eye ray

Density estimation with beam:
Photon beam
• Jarosz et al., “Progressive photon beams”, SIGGRAPH
Asia, 2011.

Density estimation with beam
Eye ray
Photon beams

Beam formulation

Beam formulation
• Jarosz et al., “A comprehensive theory of volumetric
radiance estimation using photon points and
beams”, TOG, 2011.
[Křiánek et al. 2014]

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.

Density estimation:
Unified framework
• Křiánek et al., “Unifying Points, Beams, and Paths in
Volumetric Light Transport Simulation”, SIGGRAPH,
2014.

Density estimation:
Unified framework
• Extended MIS
Regular sampling
technique
Extended sampling
technique

Density estimation:
Unified framework
• Extended balance heuristics