[PPT] - - Volume Rendering - Pascal Grittmann, Jaroslav K ivnek Using PowerPoint Presentation

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Computer Graphics

Volume Rendering -

Pascal Grittmann, Jaroslav Křivánek

Using pictures from: Monte Carlo Methods for Physically Based Volume Rendering; SIGGRAPH 2018 Course; Jan Novák, Iliyan Georgiev, Johannes Hanika, Jaroslav Křivánek, Wojciech Jarosz

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Overview

So far:
Light interactions with surfaces
Assume vacuum in and around objects
This lecture:
Participating media
How to represent volumetric data
How to compute volumetric lighting effects
How to implement a very basic volume renderer

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Fundamentals

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Volumetric Effects

Light interacts not only with surfaces but everywhere inside!
Volumes scatter, emit, or absorb light

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http://coclouds.com http://wikipedia.org http://commons.wikimedia.org

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Approximation: Model Particle Density

Modeling individual particles of a volume is, of course, not practical
Instead, represent statistically using the average density
(Same idea as, e.g., microfacet BSDFs)

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Volume Representation

Many possibilities (particles, voxel octrees, procedural,…)
A common approach: Scene objects can “contain” a volume

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Volume Representation

Homogeneous:
Constant density
Constant absorption, scattering, emission,
Constant phase function (later)
Heterogeneous:
Coefficients and/or phase function vary across the volume
Can be represented using 3D textures
(e.g., voxel grid, procedural)

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http://wikipedia.org

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Data Acquisition

Real-world measurements via tomography
Simulation, e.g.,
Fluids,
Fire and smoke,
Fog

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https://docs.blender.org

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Simulating Volumes

Mathematical Formulation of Volumetric Light Transport

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Compute 𝑀𝑝(𝑦, 𝜕𝑝) using the rendering equation

So far: Assume Vacuum

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𝑦

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Compute 𝑀𝑝(𝑦, 𝜕𝑝) using the rendering equation
Only a fraction 𝑈 𝑏, 𝑐 𝑀𝑝 𝑦, 𝜕𝑝 arrives at the eye

Volume Absorbs and Scatters Light

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𝑦 𝑏 𝑐

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Compute 𝑀𝑝(𝑦, 𝜕𝑝) using the rendering equation
Only a fraction 𝑈 𝑏, 𝑐 𝑀𝑝 𝑦, 𝜕𝑝 arrives at the eye
Every point 𝑨 between 𝑏 and 𝑐 might emit light

Volume Emits Light

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𝑦 𝑏 𝑐 𝑨

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Compute 𝑀𝑝(𝑦, 𝜕𝑝) using the rendering equation
Only a fraction 𝑈 𝑏, 𝑐 𝑀𝑝 𝑦, 𝜕𝑝 arrives at the eye
Every point 𝑨 between 𝑏 and 𝑐 might emit light
Every point 𝑨 might be illuminated through the volume

Volume Scatters Light

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𝑦 𝑏 𝑐 𝑨

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Attenuation

Computing Absorption and Out-Scattering

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𝑦 𝑏 𝑐 𝑀𝑗 = ? 𝑀0 𝑦

http://commons.wikimedia.org

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Attenuation = Absorption + Out-Scattering

Every point in the volume might absorb light or scatter it in other directions
Modeled by absorption and scattering coefficients: 𝜈𝑏 𝑨 and 𝜈𝑡 𝑨

(both in [m−1])

Might depend on position, direction, time, wavelength,…
For simplicity: we assume only positional dependence

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𝑦 𝑏 𝑐 𝑨

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Computing Absorption – Intuition

Consider a small segment Δ𝑨
Along that segment, radiance is reduced from 𝑀 to 𝑀′
𝑀′ = 𝑀 − 𝑀(𝜈𝑏Δ𝑨)
Where 𝜈𝑏 is the percentage of radiance that is absorbed (per unit distance)

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𝑏 𝑐 Δ𝑨 𝑀 𝑀′

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Computing Absorption – Intuition

Consider a small segment Δ𝑨
Along that segment, radiance is reduced from 𝑀 to 𝑀′
𝑀′ = 𝑀 − 𝑀(𝜈𝑏Δ𝑨)
Where 𝜈𝑏 is the percentage of radiance that is absorbed (per unit distance)
Lets rewrite this:
Δ𝑀 = 𝑀′ − 𝑀 = −𝑀𝜈𝑏Δ𝑨

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𝑏 𝑐 Δ𝑨 𝑀 𝑀′

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Computing Absorption – Exponential Decay

Δ𝑀 = −𝜈𝑏 𝑀 Δ𝑨
For infinitely small Δ𝑨, this becomes
𝑒𝑀 = −𝜈𝑏 𝑀 𝑒𝑨
A differential equation that models exponential decay!
Solution: 𝑀 𝑏 = 𝑀𝑝 𝑦 𝑓− ׬

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𝑦 𝑀 𝑐 = 𝑀𝑝 𝑦 𝑀 𝑏 𝑏 𝑐 Δ𝑨 𝑀 𝑀′

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Computing Out-Scattering

Same as absorption, only different factor!
𝑀 𝑏 = 𝑀𝑝 𝑦 𝑓− ׬

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𝑦 𝑏 𝑐 𝑨

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Computing Attenuation

Fraction of light that is either absorbed or out-scattered (per unit distance)
𝜈𝑢 = 𝜈𝑏 + 𝜈𝑡
Many different names: extinction / attenuation / transport coefficient
𝑀 𝑏 = 𝑀𝑝 𝑦 𝑓− ׬

𝑨(𝜈𝑏 𝑢 +𝜈𝑡 𝑢 ) 𝑒𝑢

Attenuation: 𝑈 𝑏, 𝑐 = 𝑓− ׬

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𝑦 𝑏 𝑐 𝑨

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Estimating Attenuation

We need to evaluate another integral:
𝑈 𝑏, 𝑐 = 𝑓− ׬

𝑐 𝑏 𝜈𝑢 𝑢 𝑒𝑢

Many approaches, e.g., Monte Carlo integration or deterministic quadrature

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Estimating Attenuation – Ray Marching

We need to evaluate another integral:
𝑈 𝑏, 𝑐 = 𝑓− ׬

𝑐 𝑏 𝜈𝑢 𝑢 𝑒𝑢

Many approaches, e.g., Monte Carlo integration or deterministic quadrature
Ray marching: evaluate at discrete positions (fixed stepsize Δ𝑨)
׬

𝑐 𝑏 𝜈𝑢 𝑢 𝑒𝑢 ≈ σ𝑗 𝜈𝑢(𝑨𝑗 + 𝜁) Δ𝑨

Randomized offset 𝜻 for each ray to avoid aliasing problems

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𝑦 𝑏 𝑐 𝑨1 𝑨2 𝑨3 Δ𝑨 Δ𝑨 Δ𝑨 Δ𝑨

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Emission

Explosions!

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𝑏 𝑐 𝑨

http://wikipedia.org

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Every Point Might Emit Light

Assume 𝑨 emits 𝑀𝑓 𝑨 towards 𝑏
Some of that light might be absorbed or out-scattered: It is attenuated
𝑀 𝑏 = 𝑀𝑓 𝑨 𝑈 𝑨, 𝑏

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𝑏 𝑐 𝑨

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Every Point Might Emit Light

Assume 𝑨 emits 𝑀𝑓 𝑨 towards 𝑏
Some of that light might be absorbed or out-scattered: It is attenuated
𝑀 𝑏 = 𝑀𝑓 𝑨 𝑈 𝑨, 𝑏
Happens at every point along the ray!
𝑀 𝑏 = ׬

𝑏 𝑐 𝑀𝑓 𝑨 𝑈 𝑨, 𝑏 𝑒𝑨

Another integral…

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𝑏 𝑐 𝑨

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Ray Marching for Emission

Same as before: integrate via quadrature
׬

𝑏 𝑐 𝑀𝑓 𝑨 𝑈 𝑨, 𝑏 𝑒𝑨 ≈ σ𝑗 𝑀𝑓 𝑨𝑗 𝑈 𝑨𝑗, 𝑏 Δ𝑨

Attenuation 𝑈 𝑨𝑗, 𝑏 estimated as before

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𝑦 𝑏 𝑐 𝑨1 𝑨2 𝑨3 Δ𝑨 Δ𝑨 Δ𝑨 Δ𝑨

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Ray Marching for Emission

Same as before: integrate via quadrature
׬

𝑏 𝑐 𝑀𝑓 𝑨 𝑈 𝑨, 𝑏 𝑒𝑨 ≈ σ𝑗 𝑀𝑓 𝑨𝑗 𝑈 𝑨𝑗, 𝑏 Δ𝑨

Attenuation 𝑈 𝑨𝑗, 𝑏 estimated as before
Attenuation can be incrementally updated:
𝑈 𝑨𝑗, 𝑏 = 𝑈 𝑨𝑗−1, 𝑏 𝑈 𝑨𝑗, 𝑨𝑗−1
(because it is an exponential function)

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𝑦 𝑏 𝑐 𝑨1 𝑨2 𝑨3 Δ𝑨 Δ𝑨 Δ𝑨 Δ𝑨

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In-Scattering

Accounting for “Reflections” Inside the Volume

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𝑏 𝑐 𝑨

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Direct Illumination (Single-scattering)

Account for the (attenuated) direct illumination at every point 𝑨
Similar to the rendering equation:
𝑀𝑝 𝑨, 𝜕𝑝 = ׬

Ω 𝑀𝑗 𝑦, 𝜕𝑗 𝑔 𝑞 𝜕𝑗, 𝜕𝑝 𝑒𝜕𝑗

Integration over the whole sphere Ω
The phase function 𝑔

𝑞 takes on the role of the BSDF

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𝑏 𝑐 𝑨 𝜕𝑗 𝜕𝑝

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Phase Functions

𝑔

𝑞 𝜕𝑗, 𝜕𝑝

Describe what fraction of light is reflected from 𝜕𝑗 to 𝜕𝑝
Similar to BSDF for surface scattering
Simplest example: isotropic phase function
𝑔

𝑞 𝜕𝑗, 𝜕𝑝 = 1 4𝜌

(energy conservation: ׬

Ω 1 4𝜌 𝑒𝜕 = 1)

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Phase Functions: Henyey-Greenstein

Widely used
Easy to fit to measured data
𝑔

𝑞 𝜕𝑗, 𝜕𝑝 = 1 4𝜌 1−𝑕2 1+𝑕2+2𝑕 cos 𝜕𝑗,𝜕𝑝

3 2

𝑕: asymmetry (scalar)
cos 𝜕𝑗, 𝜕𝑝 : cosine of angle between the directions

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Henyey-Greenstein: Asymmetry Parameter

𝑕 = 0: isotropic
Negative 𝑕: back scattering
Positive 𝑕: forward scattering

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http://coclouds.com

Forward Scattering

http://commons.wikimedia.org

Back Scattering

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How to Estimate Volumetric Direct Illumination

Reflected radiance at a point 𝑨: 𝑀𝑝 𝑨, 𝜕𝑝 = ׬

Ω 𝑀𝑗 𝑦, 𝜕𝑗 𝑔 𝑞 𝜕𝑗, 𝜕𝑝 𝑒𝜕𝑗

In our framework:
Sum over all point lights (as for surfaces)
Trace shadow ray (as for surfaces)
Estimate attenuation along the shadow ray (as for surfaces)

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𝑏 𝑐 𝑨 𝜕𝑗 𝜕𝑝

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Ray Marching to Compute In-Scattering

Same as for emission
Goal: estimate the integral ׬

𝑏 𝑐 𝑈 𝑨, 𝑏 𝑀𝑗 𝑨 𝑔 𝑞 𝑒𝑨

Quadrature:
׬

𝑏 𝑐 𝑈 𝑨, 𝑏 𝑀𝑗 𝑨 𝑔 𝑞 𝑒𝑨 ≈ σ𝑗 𝑈 𝑨𝑗, 𝑏 𝑀𝑗 𝑨𝑗 𝑔 𝑞 Δ𝑨

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𝑏 𝑐 𝑨1 𝑨2 𝑨3 Δ𝑨 Δ𝑨 Δ𝑨 Δ𝑨

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Ray Marching to Compute In-Scattering

Same as for emission
Goal: estimate the integral ׬

𝑏 𝑐 𝑈 𝑨, 𝑏 𝑀𝑗 𝑨 𝑔 𝑞 𝑒𝑨

Quadrature:
׬

𝑏 𝑐 𝑈 𝑨, 𝑏 𝑀𝑗 𝑨 𝑔 𝑞 𝑒𝑨 ≈ σ𝑗 𝑈 𝑨𝑗, 𝑏 𝑀𝑗 𝑨𝑗 𝑔 𝑞 Δ𝑨

Attenuation 𝑈 𝑨𝑗, 𝑏 estimated as before
Attenuation can be incrementally updated:
𝑈 𝑨𝑗, 𝑏 = 𝑈 𝑨𝑗−1, 𝑏 𝑈 𝑨𝑗, 𝑨𝑗−1
(because it is an exponential function)

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𝑏 𝑐 𝑨1 𝑨2 𝑨3 Δ𝑨 Δ𝑨 Δ𝑨 Δ𝑨

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Estimate direct illumination at x (as before)
If volume: continue straight ahead until no volume (yields intersections y, z)
Ray marching to estimate attenuation, emission, and in-scattering
Shadow rays to the lights + ray marching to compute attenuation
Compute illumination at z (as before)
Add together:
Attenuated illumination from z
Volumetric emission along 𝑦𝑧
In-scattering along 𝑦𝑧
Direct illumination at 𝑦

𝑦 𝑧 𝑨

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