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Multiple Scattering in Vision and Graphics
Lecture #21
Thanks to Henrik Wann Jensen
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Mist Fog
Glows of Light Sources
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Properties of Scattering Media
Scattering Coefficient: Fractional loss in intensity due to scattering per unit cross section
Absorption Coefficient: Fractional loss in intensity due to absorption per unit cross section
Extinction Coefficient: Scattering Coefficient
+ Absorption Coefficient
Scattering Albedo: Scat. Coeff. / Ext. Coeff.
0
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Phase Function
Incident
Direction
Exiting Direction
• Probability of light getting scattered in a single direction
• Phase function integrates to 1
• Light Scattered in any direction :
)(P
)(4
P
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Recap
Different Orders of Scattering
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Particle Scattering Mechanisms
( Mie 1908 )
Incident Beam
Size: 0.01μm Size: 0.1μm Size: 1μm
Single Scattering:
Independent Scattering:
Incident Beam
Distance of Separation >> Size of Particles
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Attenuation Model – Zeroth Order Scattering
ScatteringMedium
Unit Cross
Section
X = 0
dx
X = dIncident Light
AttenuatedExiting Light
0E
dE
Scattering Coefficient
Brightness at Distance d :
deEdE 0)( ( Bouguer’s Law, 1729 )
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Airlight Model – First Order (Single) Scattering
SunlightDiffuseSkylight
DiffuseGround Light
Object
Observer
d
dV
Brightness due to a Path of Length d :
)1()( deEdE
Horizon Brightness
( Koschmeider, 1924 )
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Mountains
Distant objects appear Bright !
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Combining 0th and 1st orders: Useful for Vision
Object
Observer
d
Attenuation
SunlightDiffuseSkylight
DiffuseGround Light
Airlight
Intensity
DistanceDistance
Intensity
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Multiple Scattering : Higher orders of scattering
Incident Beam
Particle
Phase Function
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Radiative TransferMathematical study of transport of radiation (in particular light).
Finite Difference method used to model the rate of change ofradiation along any direction in an infinitesimal volume.
Can model multiple scattering elegantly.
Solution to light transport gives the Light Field in the medium.
But, hard to solve analytically. Why?Depends on medium geometry and location of sources.
Only few special cases are known to have analytic solutions.(Plane Parallel, Spherical)
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Plane Parallel and Spherical Radiative Transfer
Isotropic
Source
Homogeneous Medium
ScatteredLight Field
Plane Parallel Medium
Scattered Light Field
Distant SourceSun
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Plane Parallel Medium
Radiative Transfer in Plane Parallel Media[ Chandrasekhar 1960 , Ishimaru 1997 ]
Scattered Light Field
Distant SourceSun
Collimated Source Outside Medium
Widely used in Atmospheric Optics, Remote Sensing
Popular configuration for Subsurface Scattering in Graphics
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Radiative Transfer in Plane Parallel Medium
Infinitesimal Scattering Volume :
),( TI
Extinction
Radiative Transfer Equation :
T
I
Radiance Rate of Change
dwTIP )',()',(4
1
Source Function
Phase Function Optical Thickness
dRdT
Incident BeamRadiance
Exiting BeamRadiance
dR
Direction
II
I
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BSSRDFs
• Bidirectional Surface Scattering Reflectance Distribution Function
• The BSSRDF relates the outgoing radiance to the incident flux
• The BRDF is an approximation of the BSSRDF for which it is assumes that light enters and leaves at the same point
• The outgoing radiance is computed by integrating the incident radiance over incoming directions and area, A
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Symbol Reference
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Diffusion Approximation for Multiple Scattering
An incoming ray is transformed into a dipole source for the diffusion approximation
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The Diffusion Approximation• The diffusion approximation is based on the observation that the light distribution in highly
scattering media tends to become isotropic
• The volumetric source distribution can be approximated using the dipole method
• The dipole method consists of positioning two point sources near the surface in such a way as to satisfy the required boundary condition
• The diffuse reflectance due to the dipole source can be computed as
• Taking into account the Fresnel reflection at the boundary for both the incoming light and the outgoing radiance
• Where Sd is the diffusion term of the BSSRDF, which represents multiple scattering
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Single Scattering Term• The total outgoing radiance, due to single scattering is computed by integrating the incident
radiance along the refracted outgoing ray
• The single scattering BSSRDF is defined implicitly by the second line of this equation
Single scattering occurs only when the refracted incoming and outgoing rays intersect, and is computed as an integral over path length s along the refracted outgoing ray
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The BSSRDF Model
• The complete BSSRDF model is a sum of the diffusion approximation and the single scattering term
• This model accounts for light transport between different locations on the surface, and it simulates both the directional component (due to single scattering) as well as the diffuse component (due to multiple scattering)
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Rendering Using the BSSRDF
• The BSSRDF model derived only applies to semi-infinite homogeneous media, for a practical model we must consider
– Efficient integration of the BSSRDF (importance sampling)
– Single scattering evaluation for arbitrary geometry
– Diffusion approximation for arbitrary geometry
– Texture (spatial variation on the object surface)
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BRDF vs BSSRDF
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BRDF vs BSSRDF
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BRDF vs BSSRDF
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Diffusion Approximation for Multiple Layers
Donner, Jensen, Siggraph 05
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Plane Parallel and Spherical Radiative Transfer
Isotropic
Source
Homogeneous Medium
ScatteredLight Field
Plane Parallel Medium
Scattered Light Field
Distant SourceSun
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Mist Fog
Glows of Light Sources
(Narasimhan & Nayar, CVPR 2003)
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Multiple Scattering in the Atmosphere
Incident Beam
Particle
Light Source
A T M O S P H E R E
Phase Function
Imaging Plane
GlowPinhole
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Light Source in a Spherical Medium
Isotropic
Source
Homogeneous Medium
Spherical Radiative Transfer Equation:
'')',()',(4
1),(
1 2
0
1
1
2
ddTIPTII
TT
I
Phase FunctionLight FieldCosine of Angle
Optical Thickness
[ Chandrasekhar 1960 ]
ScatteredLight Field
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Axially Symmetric Phase Functions
Legendre Polynomial Expansion: [ Ishimaru 1997 ] [ Henyey et al., 1941 ]
)(cos])12[()(cos0
mm
m LqmP
Legendre Polynomial
Forward Scattering Parameter
Incident
Direction
Exiting Direction
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Light Source in a Spherical Medium
Isotropic
Source
Homogeneous Medium
Spherical Radiative Transfer Equation:
'')',()',(4
1),(
1 2
0
1
1
2
ddTIPTII
TT
I
Phase FunctionLight FieldCosine of Angle
Optical Thickness
[ Chandrasekhar 1960 ]
ScatteredLight Field
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])1(12
)1([exp)( 10 Tq
m
mTmITg m
m
Phase Function ParameterOptical Thickness
Exponential Coefficients :
Radiant Intensity of Source
Legendre Polynomial
Analytic Multiple Scattering Solution
)())()((),( 10
mmm
m LTgTgTI
Scattered Light Field :
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Highlights of the Model
1.02 1.2 1.4 1.6 1.8 T
m160
120 60
30 10
• Small Number of Coefficients (m) :
• Absorbing and Purely Scattering Media
• Single and Multiple Scattering
• Isotropic and Anisotropic Phase Functions
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Scattered Light Field vs. Weather Condition
Mild Weather (T = 1.2) Dense Weather (T = 4)
Angular PSF : Scattered Light Field at a Point
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Validation: Multiple Scattering in Milk
Original Milk
Images
Increasing Milk Concentrations
Rendered Milk
Images
Image acquired
With No Milk
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Number of Milk Concentrations : 15
Model Fitting Error : [ 1 % to 3 % ]
Diffusion Fitting Error : [ 20 % to 50 % ]
Model Fit Accuracy
Low Milk Concentration High Milk Concentration
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Model Fit Accuracy: Monte Carlo Simulations
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Effect of Source Visibility
315 240 180 90 30o o o o o
Incr
easi
ng
M
ilk
Con
cen
trati
on
sObserved Milk
Images
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Original Image
Rendering Glows using Convolution
Increasing Fog
Rendered Images
Joint work with Ramamoorthi
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Original Image Single Scattering
Multiple Scattering (Mild Condition) Multiple Scattering (Dense Condition)
Single versus Multiple Scattering
Joint work with Ramamoorthi
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Inverse RTE : Weather from APSF
Measured APSF : ),( TI
Meteorological Visibility:
RT
V912.3
[ Middleton 1952]
Weather Condition:[ Van de Hulst 1957]
q0 1
PureAir
SmallAerosols Haze Mist Fog Rain
0.1 0.4 0.7 0.90.8
|||| )()),(),((),(minarg 10,
mmm
m LqTgqTgTIqT
Objective Function :
qT ,qT ,qT ,
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Computed Atmospheric Visibilities
A Camera-based Weather Station
45 images of a light source (WILD Database ECCV 02)
Computed Weather Conditions
Ground TruthEstimated
Ground TruthEstimated
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Volume Rendering as Convolution
Analytic Multiple Scattering )())()((),( 10
mmm
m LTgTgTI
Shedding Light on the Weather
Model Validation using Milk
Summary
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Next Class: Fluids
Lectures #22