Last TimeTodayScatteringScattering is Visually ImportantOut-Scattering MathTransmittance PropertiesOptical ThicknessPhase FunctionIn-ScatteringSource TermIsotropic vs. Anisotropic MediaIsotropic vs. Anisotropic Phase FunctionsPhysically-Based Phase FunctionsHenyey-Greenstein FunctionAlternativesSampling Henyey-GreensteinPBRT Volume ModelsHomogeneous MediumHomogeneous with Varying DensityExponential Height FogComputing Optical ThicknessEquation of TransferSolving Transfer EquationEmission OnlySegment of InterestCumulative TransmittanceEmission ExampleSingle ScatteringSingle Scattering ExampleMultiple ScatteringNext Time02/25/05 © 2005 University of WisconsinLast Time•Meshing•Volume Scattering Radiometry (Adsorption and Emission)02/25/05 © 2005 University of WisconsinToday•Participating Media–Scattering theory–Integrating Participating Media02/25/05 © 2005 University of WisconsinScattering•Particles in the media act as little reflectors–They are too small to see, but they influence the light passing through•Scattering has two effects–Out-scattering: light along a line is scattered in a different direction–In-scattering: light from some other direction is scattered into the direction of interest•Out-scattering decreases radiance, in-scattering increases it02/25/05 © 2005 University of WisconsinScattering is Visually Important02/25/05 © 2005 University of WisconsinOut-Scattering Math•There is an out-scattering co-efficient s (p,) –The probability density that light is scattered per unit distance–Just like absorption coefficient, but it’s not being converted, it’s being redirected•Define attenuation coefficient: t=a+s•Define transmittance, Tr, between two points:       ,0,pppppppLTLeTrdttrdt02/25/05 © 2005 University of WisconsinTransmittance Properties•Transmittance from a point to itself is 1•Transmittance multiplies along a ray–In a voxel-based volume, we can compute transmittance through each voxel and multiply to get total through volume  100, dttrteTppp     pppppprrrTTT02/25/05 © 2005 University of WisconsinOptical Thickness•Define optical thickness, :•If the medium is homogeneous, t does not depend on p–Integration is easy and we get Beer’s law   dtdtt0,ppp drteT pp02/25/05 © 2005 University of WisconsinPhase Function•We need a function that tells us what directions light gets scattered in–The participating media equivalent of the BRDF•The phase function, p(’), gives the distribution of outgoing directions, ’, for an incoming direction, –A probability distribution, so it must be normalized over the hemisphere: 12Sdp02/25/05 © 2005 University of WisconsinIn-Scattering•The phase function tells us where light gets scattered•To find out how much light gets scattered into a direction, integrate over all the directions it could be scattered from     2,,SisdLppppIncoming radianceProportion scattered into direction Proportion scattered02/25/05 © 2005 University of WisconsinSource Term•Given the emission radiance and the phase function, we can define a source term, S–The total amount of radiance added per unit length–Note the resemblance to the surface scattering equation           2,,,,,,SisveodLpLSSdtdLppppppp02/25/05 © 2005 University of WisconsinIsotropic vs. Anisotropic Media•A medium is isotropic if the phase function depends only on the angle between the directions, –Write p(cos)•Most natural materials are like this, except crystal structures•Phase functions are also reciprocal: p(’)=p(’)02/25/05 © 2005 University of WisconsinIsotropic vs. Anisotropic Phase Functions•A phase function is isotropic if it scatters equally in all directions: pisotropic(’)=const•There is only one possible isotropic phase function–Why? What is the additional constraint on phase functions?•Homogeneous/inhomogeneous refers to spatial variation, isotropic/anisotropic refers to directional variation 41isotropicp02/25/05 © 2005 University of WisconsinPhysically-Based Phase Functions•Two common physically-based formulas•Air molecules are modeled by Rayleigh scattering–Optical extinction coefficient varies with -4–What phenomena does this explain?•Scattering due to larger particles (dust, water droplets) is modeled with Mie scattering–Scattering depends less on wavelength, so what color is haze?•Turbidity is a useful measurement: T=(tm+th)/tm–tm is vertical optical thickness of molecular atmosphere–th is vertical optical thickness of haze atmosphere02/25/05 © 2005 University of WisconsinHenyey-Greenstein Function•Single parameter, g, controls relative proportion of forward/backward scattering: g(0,1)   2322cos21141:cosggggpHG02/25/05 © 2005 University of WisconsinAlternatives•Linear combination of Henyey-Greenstein–Weights must sum to 1 to keep normalized•Schlick Approximation –Avoid 3/2 power computation–k roughly 1.55g-.55g3  22cos1141coskkpSchlick   niiHGigpwp1:coscos02/25/05 © 2005 University of WisconsinSampling Henyey-Greenstein•Because of the isotropic medium assumption, the distribution is separable into one for  and one for •Given 1 and 2:•Given an incoming direction, use these to generate a scattered direction22221211121cos2ggggg02/25/05 © 2005 University of WisconsinPBRT Volume Models•PBRT volumes must give–Extent (3D shape to intersect)–Functions to return scattering parameters–Function to return phase function at a point–Function to compute optical thickness between two points•Simplest is homogeneous volume–Everything is constant, and optical thickness comes from Beer’s law02/25/05 © 2005 University of WisconsinHomogeneous Medium02/25/05 © 2005 University of WisconsinHomogeneous with Varying Density•Assume that the same medium is present, but that the