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Shedding Light on the Weather
Srinivasa Narasimhan and Shree Nayar
Computer Science Department
Columbia University
Sponsors : DARPA Human ID, NSF
IEEE CVPR ConferenceJune 2003, Madison, USA
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
Radiative TransferInfinitesimal 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
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
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
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
])1(12
)1([exp)( 10 Tq
m
mTmITg m
m
Phase Function ParameterOptical Thickness
Exponential Coefficients :
Source Radiant Intensity
Legendre Polynomial
Analytic Multiple Scattering Solution
)())()((),( 10
mmm
m LTgTgTI
Scattered Light Field :
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
Scattered Light Field vs. Weather Condition
Mild Weather (T = 1.2) Dense Weather (T = 4)
Angular PSF : Scattered Light Field at a Point
Validation : Multiple Scattering in Milk
Original Milk
Images
Increasing Milk Concentrations
Rendered Milk
Images
Image acquired
With No Milk
Number of Milk Concentrations : 15
Model Fitting Error : [ 1 % to 3 % ]
Diffusion Fitting Error : [ 20 % to 50 % ]
Model Fit AccuracyLow Milk Concentration High Milk Concentration
Original Image
Rendering Glows using Convolution
Increasing Fog
Rendered Images
Joint work with Ramamoorthi (submitted to TOG)
Original Image Single Scattering
Multiple Scattering (Mild Condition)Multiple Scattering (Dense Condition)
Single versus Multiple Scattering
Joint work with Ramamoorthi (submitted to TOG)
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 ,
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