02/25/05© 2005 University of Wisconsin Last Time Meshing Volume Scattering Radiometry (Adsorption and Emission)

02/25/05 © 2005 University of Wisconsin

Last Time

• Meshing

• Volume Scattering Radiometry (Adsorption and Emission)


Today

• Participating Media– Scattering theory

– Integrating Participating Media


Scattering

• 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 it


Scattering is Visually Important


Out-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,

pppp

ppp

LTL

eT

r

dtt

r

d

t


Transmittance 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

1

0

0,

dtt

r

teT

ppp

pppppp rrr TTT


Optical Thickness

• Define optical thickness, :

• If the medium is homogeneous, t does not depend on p– Integration is easy and we get Beer’s law

dt dtt0 , ppp

dr

teT pp


Phase 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

S dp


In-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

,,S is dLp ppp

Incoming radiance

Proportion scattered into direction

Proportion scattered


Source 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,,,,

,,

S isve

o

dLpLS

Sdt

dL

ppppp

pp


Isotropic 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(’)


Isotropic 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

4

1isotropicp


Physically-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 atmosphere


Henyey-Greenstein Function

• Single parameter, g, controls relative proportion of forward/backward scattering: g(0,1)

232

2

cos21

1

4

1:cos

gg

ggpHG


Alternatives

• 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

2

2

cos1

1

4

1cos

k

kpSchlick

n

iiHGi gpwp

1

:coscos


Sampling 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

2

2

22

1

21

11

2

1cos

2

gg

gg

g


PBRT 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 law


Homogeneous Medium


Homogeneous with Varying Density

• Assume that the same medium is present, but that the density varies

• All parameters are scaled by density– Except optical thickness, which may be hard to compute

• Options:– 3D Grids – give sampled density on grid and interpolate

– Exponential density from some ground plane:

– Aggregates of volumes

bhaehd


Exponential Height Fog


Computing Optical Thickness

• Recall:

• Obviously we can use:

• The best way to get the T(j) is to use stratified sampling with a fixed offset– The offset is different for each query, but fixed among the T(j)

dt dtt0 , ppp

N

jj

jt

Tp

T

N 1)(

)( ,1 p

t0 t1u

T(5)


Equation of Transfer

• Radiance arriving is radiance leaving a surface that is attenuated plus radiance that gets in-scattered and emitted on the way from the surface– The transmittance describes the out-scattering and adsorption

– The source term describes the emission and in-scattering

t

t

tdSTLTLt

rori

pp

pp

ppppppp

0

0

00 ,,,


Solving Transfer Equation

• The hard part is the integral– Transmittance is simple – it depends only on optical thickness,

which we just saw how to compute

– Implementation increases step for transfer that is NOT to the camera

• Several possible assumptions in the integral– Emission only – simple because radiance from other directions is

not required

– Single-scattering only – simple because only radiance from light sources is considered

– Multiple – hard because you have to account for radiance from all directions, including other scattering events, so it blows up


Emission Only

• Choose points through the volume to evaluate emission

• Attenuate via transmittance

• Sum over points in Monte Carlo:

– Point are chosen using uniform offset stratified sampling (a few slides back) within the part of the ray that the volume occupies

– The transmittance can be computed cumulatively as we step along the ray

N

j

jve

jr LT

N

tt

1

)()(01 , ppp


Segment of Interest

• Viewer could be inside

• Visible surface could be inside

• Could pass right through


Cumulative Transmittance


Emission Example


Single Scattering

• Evaluates

• Very similar to previous slide, except:– At each point, sample light sources and

push through phase function to get and estimate of the inner integral

– Have to account for transmittance between light and sample point

– Actually, only sample one light for each sample point along the ray

t

S isver tddLpLT0

2,,, pppppp


Single Scattering Example


Multiple Scattering

• Can do it like path sampling– At each point along ray, sample multiple outgoing directions

– For each sampled direction, find first hit surface• Add in outgoing radiance from that surface – itself expensive to

compute

– For ray to first hit surface, recursively apply the algorithm• Account for scattering within the volume into this dircection

• Very computationally inefficient

• Speedups: Bi-Directional, Volumetric Photon Mapping


Next Time

• Sky models

• Sub-surface scattering

Documents

02/25/05© 2005 University of Wisconsin Last Time Meshing Volume Scattering Radiometry (Adsorption and Emission)