Volume and Participating Media

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Jerome Cox
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1 Volume and Participating Media Digital Image Synthei Yung-Yu Chuang 12/31/2008 with lide by Pat Hanrahan and Torten Moller

Participating media We have by far aumed that the

2 Participating media We have by far aumed that the cene i in a vacuum. Hence radiance i contant along the ray. However ome real-world ituation uch a fog and moke attenuate and catter light. They participate in rendering. Natural phenomena Fog moke fire Atmophere haze Beam of light through cloud Suburface cattering

3 Volume cattering procee Aborption converion from light to other form Emiion i contribution from luminou particle Scattering direction change of particle Out-cattering In-cattering Single cattering v.. multiple cattering Homogeneou v.. inhomogeneouheterogeneou emiion in-cattering out-cattering aborption

4 Single cattering and multiple cattering attenuation ingle cattering multiple cattering

5 Aborption The reduction of energy due to converion of light to another form of energy e.g. heat + d σ a d d σ a d Aborption cro-ection: σ Probability of being aborbed per unit length a

6 Tranmittance d σ a d d + d ' ' σ a d σ a + ' ln + ln σ a + ' d' τ Optical ditance or depth τ σ + ' d' 0 a Homogenou media: contant σ τ σ a a 0 σ a 0 + ' ' d' ' 0

7 Tranmittance and opacity d d a σ d + d ' d d a σ + + a d d 0 ' ' ' ' σ + + a d 0 ' ' ln ln τ σ τ T τ T e + Tranmittance T e T τ Opacity + e T 1 T + 1 T α

8 Aborption

9 Emiion Energy that i added to the environment from luminou particle due to chemical thermal or nuclear procee that convert energy to viible light. ve : emitted radiance added to a ray per unit ditance at a point in direction d ve d + d ve d

10 Emiion

11 Out-cattering ight heading in one direction i cattered to other direction due to colliion with particle + d σ d d σ d Scattering cro-ection: σ Probability of being cattered per unit length σ

12 Etinction + d σ t d d σ t d Total cro-ection σ t σa + σ Albedo W σ σ σ σ + σ t a Attenuation due to both aborption and cattering τ σ + ' d' τ 0 t

13 Etinction Beam tranmittance 0 Tr ' e σ + ' d' : ditance between and t ' Propertie of Tr: In vaccum Tr ' 1 Multiplicative Tr '' Tr ' Tr ' '' Beer law in homogeneou medium Tr ' e σ t

14 In-cattering Increaed radiance due to cattering from other direction d ' σ + d d σ p ' ' d' d Ω Phae function p Reciprocity p p Energy conerving p d 1 S 2

15 Source term S + σ p ' ' d' S i determined by ve Volume emiion Ω d S d Phae function which decribe the angular ditribution of cattered radiation volume analog of BSDF for urface

16 Phae function Phae angle coθ Phae function from the phae of the moon 1 1. Iotropic pco θ 4 π -imple co θ pco θ 2. Rayleigh 4 4 λ - Molecule ueful for very mall particle whoe radii maller than wavelength of light 3. Mie cattering - mall phere baed on Mawell equation; good model for cattering in the atmophere due to water droplet and fog θ

17 Henyey-Greentein phae function Empirical phae function g -0.3 pco θ g 4π 1+ g 2gcoθ g 0.6 π 2 π pco θcoθ dθ g 0 g: average phae angle

18 Henyey-Greentein approimation Any phae function can be written in term of a erie of egendre polynomial typically n<4 p co θ 1 2 n + 1 b n P n co θ 4π n 0 b n 1 < 1 p co θ P co θ > p co θ P n n co θ d co θ P0 1 P P P

19 Schlick approimation Approimation to Henyey-Greentein k p Schlick coθ 4 π 1 k co θ K play a imilar role like g 0: iotropic -1: back cattering 2 Could ue k 1.55g 0.55g 2

20 Importance ampling for HG co g p θ co co p g g θ π θ + φ 2πξ 0 if ξ g + + otherwie co ξ θ g g g g g ξ g g g

21 Pbrt implementation core/volume.* volume/* t 1 t cla VolumeRegion { 0 public: bool InterectPRay &ray float *t0 float *t1; Spectrum igma _ apoint & Vector &; Spectrum igma_point & Vector &; Spectrum vepoint & Vector &; // phae function: pbrt ha iotropic Rayleigh // Mie HG Schlick virtual float ppoint & Vector & Vector &; // attenuation coefficient; _a+_ Spectrum igma_tpoint & Vector &; // calculate optical thickne by Monte Carlo or // cloed-form olution Spectrum TauRay &ray float tep1. t float offet0.5; 0 tep t 1 }; offet

22 Homogenou volume Determined by contant σ and σ g in phae function Emiion i Spatial etent a ve Spectrum TauRay &ray float float{ float t0 t1; if!interectpray&t0&t1 return 0.; return Ditancerayt0rayt1 * ig_a + ig_; }

23 Homogenou volume

24 Varying-denity volume Denity i varying in the medium and the volume cattering propertie at a point i the product of the denity at that point and ome baeline value. DenityRegion 3D grid VolumeGrid Eponential denity EponentialDenity

25 DenityRegion cla DenityRegion : public VolumeRegion { public: DenityRegionSpectrum &ig_a Spectrum &ig_ float g Spectrum &e Tranform &VolumeToWorld; float DenityPoint &Pobj cont 0; igma_apoint &p Vector & { return DenityWorldToVolumep * ig_a; } Spectrum igma _ Point &p Vector & { return DenityWorldToVolumep * ig_; } Spectrum igma_tpoint &p Vector & { return DenityWorldToVolumep*ig ig_a+ig_;} ;} Spectrum vepoint &p Vector & { return DenityWorldToVolumep * le; }... protected: Tranform WorldToVolume; Spectrum ig_a ig_ le; float g; };

26 VolumeGrid Standard form of given data Tili Tri-linear interpolation i of data to give continuou volume Often ued in volume rendering

27 VolumeGrid VolumeGridSpectrum &a Spectrum & float gg Spectrum &emit BBo &e Tranform &v2w int n int ny int nz cont float *d; float VolumeGrid::Denitycont Point &Pobj cont { if!etent.inidepobj return 0; // Compute voel coordinate and offet float vo Pobj. - etent.pmin. / etent.pma. - etent.pmin. * n -.5f; float voy Pobj.y - etent.pmin.y / etent.pma.y - etent.pmin.y * ny -.5f; float voz Pobj.z - etent.pmin.z / etent.pma.z - etent.pmin.z * nz -.5f;

28 VolumeGrid int v Floor2Intvo; int vy Floor2Intvoy; int vz Floor2Intvoz; float d vo - v dy voy - vy dz voz - vz; // Trilinearly interpolate denity value float d00 erpd Dv vy vz Dv+1 vy vz; float d10 erpd Dv vy+1 vz Dv+1 vy+1 vz; float d01 erpd Dv vy vz+1 Dv+1 vy vz+1; float d11 erpd Dv vy+1vz+1dv+1vy+1vz+1; float d0 erpdy d00 d10; float d1 erpdy d01 d11; return erpdz d0 d1; float Dint int y int z { } Clamp 0 n-1; y Clampy 0 ny-1; z Clampz 0 nz-1; return denity[z*n*ny+y*n+]; * * }

29 Eponential denity Given by bh d h ae EponentialDenity Where h i the height in the direction of the up-vector

30 EponentialDenity cla EponentialDenity : public DenityRegion { public: EponentialDenitySpectrum &a Spectrum & float g Spectrum &emit BBo &e Tranform &v2w float aa float bb Vector &up... float Denitycont Point &Pobj cont { if!etent.inidepobj return 0; float height DotPobj - etent.pmin updir; return a * epf-b * height; } private: h }; BBo etent; float a b; Vector updir; etent.pmin updir Pobj

31 ight tranport Emiion + in-cattering ource term ' ' ' σ d p S ve Ω + Aborption + out-cattering etinction d S d Aborption + out cattering etinction d d t σ Combined σ S d d t +

32 Infinite length no urface Aume that there i no urface and we have an infinite length we have the olution Tr ' S ' d 0 0 Tr ' e σ t + ' d' S ' ' Tr ' '

33 With urface The olution Tr Tr ' S ' d d 0 from the urface point Tr 0 d

34 With urface The olution Tr Tr ' S ' d d 0 from the urface point 0 from the participating media 0 0 Tr 0 ' d '

35 Simple atmophere model Aumption Homogenou media Contant ource term airlight σ + S t σt 1 σt e S + e C S C Fog Haze

36 OpenG fog model C fc + 1 f in C fog G_EXP f z e d denity z G_ EXP2 f z e denity z 2 G_INEAR f z end end z tart From

37 VolumeIntegrator cla VolumeIntegrator : public Integrator { Beam tranmittance for a given public: ray from mint to mat virtual Spectrum Tranmittance }; cont Scene *cene cont Ray &ray cont Sample *ample float *alpha cont 0; Pick up function Preproce RequetSample and i from Integrator.

38 Emiion only Solution for the emiion-only implification ' ' S ev d d Tr Tr ev d ' ' Monte Carlo etimator N i i ev i N i i ev i Tr N t t p Tr N i i i N p N 1 1

39 Emiion only Ue multiplicativity of Tr Tr i Tr i i Tr i 1 1 Break up integral and compute it incrementally by ray marching Tr can get mall in a long ray Early ray termination Either ue Ruian Roulette or determinitically

40 EmiionIntegrator cla EmiionIntegrator : public VolumeIntegrator { public: EmiionIntegratorfloat { tepsize ; } void RequetSampleSample *ample cont Scene *cene; Spectrum Tranmittancecont Scene * cont Ray &ray cont Sample *ample float *alpha cont; Spectrum icont Scene * cont RayDifferential &ray cont Sample *ample float *alpha cont; private: float tepsize; int tausampleoffet cattersampleoffet; }; ingle 1D ample for each

41 EmiionIntegrator::Tranmittance if!cene->volumeregion return Spectrum1; float tep ample? tepsize : 4.f * tepsize; ue larger tep for hadow and float offet indirect ray for efficiency ample? ample->oned[tausampleoffet][0] : RandomFloat; Spectrum tau cene->volumeregion->tauraytepoffet; return Ep-tau; τ σ + ' d' 0 a T e τ

42 EmiionIntegrator::i VolumeRegion *vr cene->volumeregion; float t0 t1; if!vr!vr->interectpray &t0 &t1 return 0; // Do emiion-only volume integration in vr Spectrum v0.; // Prepare for volume integration tepping int N Ceil2Intt1-t0 / tepsize; float tep t1 - t0 / N; Spectrum Tr1.f; Point p rayt0 pprev; Vector w -ray.d; if ample t0 + ample->oned[cattersampleoffet][0]*tep; ele t0 + RandomFloat * tep;

43 EmiionIntegrator::i for int i 0; i < N; ++i t0 + tep { // Advance to ample at t0 and update T pprev p; p rayt0; Tr i Tr i i Tr i 1 Spectrum teptau vr->tauraypprevp-pprev01.5f * tepsize RandomFloat; Tr * Ep-tepTau; // Poibly terminate if tranmittance i mall if Tr.y < 1e-3 { cont float continueprob.5f; if RandomFloat > continueprob break; Tr / continueprob; } // Compute emiion-only only ource term at _p_ v + Tr * vr->vep w; } N t t return v * tep; 1 0 Tr i ev i N i 1 1

44 Emiion only eponential denity

45 Single cattering Conider incidence radiance due to direct illumination illumination d S Tr Tr d ' ' ' ' ' σ d p S d ve Ω + Ω 0 '

46 Single cattering Conider incidence radiance due to direct illumination illumination d S Tr Tr d ' ' ' ' ' σ d p S d ve Ω + 0 Ω 0

47 Single cattering d may be attenuated by participating media At each point of the integral we could ue multiple importance ampling to get σ p ' ' d' Ω But in practice we can jut pick up light ource randomly. d

48 Single cattering

Radiometric properties of media. Mathieu Hébert

Radiometric properties of media. Mathieu Hébert 1 Radiometric propertie of media Mathieu Hébert Aborbing media 2 Seawater i blue due to wavelength-dependent aborption http://www.webexhibit.org/caueofcolor/5b.html Almot all media are aborbing. The trength