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1 Computer Graphics - Volume Rendering - Pascal Grittmann Using pictures from: Monte Carlo Methods for Physically Based Volume Rendering; SIGGRAPH 2018 Course; Jan Novák, Iliyan Georgiev, Johannes Hanika, Jaroslav Křivánek, Wojciech Jarosz
2 Overview So far: Light interactions with surfaces Assume vacuum in and around ojects This lecture: Participating media How to represent volumetric data How to compute volumetric lighting effects How to implement a very asic volume renderer CG - Volume Rendering - Pascal Grittmann 2
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8 Fundamentals CG - Volume Rendering - Pascal Grittmann 8
9 Volumetric Effects Light interacts not only with surfaces ut everywhere inside! Volumes scatter, emit, or asor light CG - Volume Rendering - Pascal Grittmann 9
10 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) CG - Volume Rendering - Pascal Grittmann 10
11 Volume Representation Many possiilities (particles, voxel octrees, procedural, ) A common approach: Scene ojects can contain a volume CG - Volume Rendering - Pascal Grittmann 11
12 Volume Representation Homogeneous: Constant density Constant asorption, scattering, emission, Constant phase function (later) Heterogeneous: Coefficients and/or phase function vary across the volume Can e represented using 3D textures (e.g., voxel grid, procedural) CG - Volume Rendering - Pascal Grittmann 12
13 Data Acquisition Real-world measurements via tomography Simulation, e.g., Fluids, Fire and smoke, Fog CG - Volume Rendering - Pascal Grittmann 13
14 Simulating Volumes Mathematical Formulation of Volumetric Light Transport CG - Volume Rendering - Pascal Grittmann 14
15 So far: Assume Vacuum Compute L o (x, ω o ) using the rendering equation x CG - Volume Rendering - Pascal Grittmann 15
16 Volume Asors and Scatters Light Compute L o (x, ω o ) using the rendering equation Only a fraction T a, L o x, ω o arrives at the eye a x CG - Volume Rendering - Pascal Grittmann 16
17 Volume Emits Light Compute L o (x, ω o ) using the rendering equation Only a fraction T a, L o x, ω o arrives at the eye Every point z etween a and might emit light a z x CG - Volume Rendering - Pascal Grittmann 17
18 Volume Scatters Light Compute L o (x, ω o ) using the rendering equation Only a fraction T a, L o x, ω o arrives at the eye Every point z etween a and might emit light Every point z might e illuminated through the volume a z x CG - Volume Rendering - Pascal Grittmann 18
19 L i =? L 0 x a x Attenuation Computing Asorption and Out-Scattering CG - Volume Rendering - Pascal Grittmann 19
20 Attenuation = Asorption + Out-Scattering Every point in the volume might asor light or scatter it in other directions Modeled y asorption and scattering densities: μ a z and μ s z Might depend on position, direction, time, wavelength, For simplicity: we assume only positional dependence a z x CG - Volume Rendering - Pascal Grittmann 20
21 Computing Asorption Intuition Consider a small segment Δz Along that segment, radiance is reduced from L to L L = L μ a L Δz Where μ a is the percentage of radiance that is asored (per unit distance) a L Δz L CG - Volume Rendering - Pascal Grittmann 21
22 Computing Asorption Intuition Consider a small segment Δz Along that segment, radiance is reduced from L to L L = L μ a L Δz Where μ a is the percentage of radiance that is asored (per unit distance) Lets rewrite this: ΔL = L L = μ a L Δz a L Δz L CG - Volume Rendering - Pascal Grittmann 22
23 Computing Asorption Exponential Decay ΔL = μ a L Δz For infinitely small Δz, this ecomes dl = μ a L dz A differential equation that models exponential decay! Solution: L a = L o x e a μa t dt L = L o x L a a L Δz L x CG - Volume Rendering - Pascal Grittmann 23
24 Computing Out-Scattering Same as asorption, only different factor! μ s z : percentage of light scattered at point z L a = L o x e a μs t dt a z x CG - Volume Rendering - Pascal Grittmann 24
25 Computing Attenuation Fraction of light that is neither asored nor out-scattered μ t = μ a + μ s L a = L o x e a (μa t +μ s t ) dt Attenuation: T a, = e a μt t dt a z x CG - Volume Rendering - Pascal Grittmann 25
26 Computing Attenuation Homogeneous Simple case: constant density / attenuation μ t z = μ t z T a, = e a μt t dt = e (a )μ t a z x CG - Volume Rendering - Pascal Grittmann 26
27 Estimating Attenuation We need to solve another integral: T a, = e a μt t dt Many solutions, e.g., Monte Carlo integration (next semester) Simple solution: Quadrature CG - Volume Rendering - Pascal Grittmann 27
28 Estimating Attenuation Ray Marching We need to solve another integral: T a, = e a μt t dt Simple solution: Quadrature Ray marching: evaluate at discrete positions (fixed stepsize Δz) a μt t dt σ i μ t (z i ) Δz a Δz Δz Δz Δz z 1 z 2 z 3 x CG - Volume Rendering - Pascal Grittmann 28
29 a z Emission Explosions! CG - Volume Rendering - Pascal Grittmann 29
30 Every Point Might Emit Light Assume z emits L e z towards a Some of that light might e asored or out-scattered: It is attenuated L a = L e z T z, a a z CG - Volume Rendering - Pascal Grittmann 30
31 Every Point Might Emit Light Assume z emits L e z towards a Some of that light might e asored or out-scattered: It is attenuated L a = L e z T z, a Happens at every point along the ray! L a = a Le z T z, a dz Another integral a z CG - Volume Rendering - Pascal Grittmann 31
32 Ray Marching for Emission Same as efore: integrate via quadrature a Le z T z, a dz σ i L e z i T z i, a Δz Attenuation T z i, a estimated as efore a Δz Δz Δz Δz z 1 z 2 z 3 x CG - Volume Rendering - Pascal Grittmann 32
33 Ray Marching for Emission Same as efore: integrate via quadrature a Le z T z, a dz σ i L e z i T z i, a Δz Attenuation T z i, a estimated as efore Attenuation can e incrementally updated: T z i, a = T z i 1, a T z i, z i 1 (ecause it is an exponential function) a Δz Δz Δz Δz z 1 z 2 z 3 x CG - Volume Rendering - Pascal Grittmann 33
34 a z In-Scattering Accounting for Reflections Inside the Volume CG - Volume Rendering - Pascal Grittmann 34
35 Direct Illumination Account for the (attenuated) direct illumination at every point z Similar to the rendering equation: L o z, ω o = Ω L i x, ω i f p ω i, ω o dω i Integration over the whole sphere Ω The phase function f p takes on the role of the BSDF a ω o z ω i CG - Volume Rendering - Pascal Grittmann 35
36 Phase Functions L o z, ω o = Ω L i x, ω i f p ω i, ω o dω i Descrie what fraction of light is reflected from ω i to ω o Similar to BSDF for surface scattering Simplest example: isotropic phase function f p ω i, ω o = 1 4π (energy conservation: Ω 1 4π dω = 1) CG - Volume Rendering - Pascal Grittmann 36
37 Phase Functions: Henyey-Greenstein Widely used Easy to fit to measured data f p ω i, ω o = 1 4π 1 g 2 1+g 2 +2g cos ω i,ω o 3 2 g: asymmetry (scalar) cos ω i, ω o : cosine of the angle formed y ω i and ω o CG - Volume Rendering - Pascal Grittmann 37
38 Henyey-Greenstein: Asymmetry Parameter g = 0: isotropic Negative g: ack scattering Positive g: forward scattering Back Scattering Forward Scattering CG - Volume Rendering - Pascal Grittmann 38
39 How to Estimate Volume Direct Illumination Reflected radiance at a point z: L o z, ω o = Ω L i x, ω i f p ω i, ω o dω i 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) a ω o z ω i CG - Volume Rendering - Pascal Grittmann 39
40 Ray Marching to Compute In-Scattering Same as for emission Goal: estimate the integral a T z, a μs z L i z f p dz Quadrature: a T z, a μs z L i z f p dz σ i T z i, a μ s z L i z i f p Δz a Δz Δz Δz Δz z 1 z 2 z CG - Volume Rendering - Pascal Grittmann 40
41 Ray Marching to Compute In-Scattering Same as for emission Goal: estimate the integral a T z, a Li z f p dz Quadrature: a T z, a Li z f p dz σ i T z i, a L i z i f p Δz Attenuation T z i, a estimated as efore Attenuation can e incrementally updated: T z i, a = T z i 1, a T z i, z i 1 (ecause it is an exponential function) a Δz Δz Δz Δz z 1 z 2 z CG - Volume Rendering - Pascal Grittmann 41
42 Putting it all Together A Simple Volume Integrator CG - Volume Rendering - Pascal Grittmann 42
43 A Simple Volume Integrator Estimate direct illumination at x (as efore) If volume: continue straight ahead until no volume (yields intersections y, z) x y z CG - Volume Rendering - Pascal Grittmann 43
44 A Simple Volume Integrator Estimate direct illumination at x (as efore) If volume: continue straight ahead until no volume (yields intersections y, z) Ray marching to estimate attenuation, emission, x y z CG - Volume Rendering - Pascal Grittmann 44
45 A Simple Volume Integrator Estimate direct illumination at x (as efore) 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 x y z CG - Volume Rendering - Pascal Grittmann 45
46 A Simple Volume Integrator Estimate direct illumination at x (as efore) 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 efore) x y z CG - Volume Rendering - Pascal Grittmann 46
47 A Simple Volume Integrator Estimate direct illumination at x (as efore) 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 efore) Add together: Attenuated illumination from z Volumetric emission along xy In-scattering along xy Direct illumination at x x y z CG - Volume Rendering - Pascal Grittmann 47
48 A Simple Volume Integrator Estimate direct illumination at x (as efore) 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 efore) Add together: { Attenuated illumination from z Multiply Volumetric emission along xy y BTDF! In-scattering along xy Direct illumination at x x y z CG - Volume Rendering - Pascal Grittmann 48
- Volume Rendering -
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,
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