Shedding Light on the Weather

Srinivasa Narasimhan and Shree NayarComputer Science Department

Columbia University

Sponsors : DARPA Human ID, NSF

IEEE CVPR ConferenceJune 2003, Madison, USA

Light Sources in Bad Weather

Mist Fog

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 :

TI

Radiance Rate of Change

dwTIP )',()',(41

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 :

'')',()',(41),(1 2

0

1

1

2

ddTIPTIITT

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

IncidentDirection

Exiting Direction

Light Source in a Spherical Medium

Isotropic

Source

Homogeneous Medium

Spherical Radiative Transfer Equation :

'')',()',(41),(1 2

0

1

1

2

ddTIPTIITT

I

Phase FunctionLight FieldCosine of Angle Optical Thickness

[ Chandrasekhar 1960 ]

ScatteredLight Field

])1(12)1([exp)( 10 Tq

mmTmITg 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

m160120 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 acquiredWith No Milk

Number of Milk Concentrations : 15Model 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

V 912.3

[ 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

Volume Rendering as Convolution

Analytic Multiple Scattering

)())()((),( 10

mmm

m LTgTgTI

Shedding Light on the Weather

Model Validation using Milk

Summary