Multiple Scattering in Vision and Graphics

Lecture #21

Thanks to Henrik Wann Jensen

Mist Fog

Glows of Light Sources

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

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

Recap

Different Orders of Scattering

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

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 )

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 )

Mountains

Distant objects appear Bright !

Combining 0th and 1st orders: Useful for Vision

Object

Observer

d

Attenuation

SunlightDiffuseSkylight

DiffuseGround Light

Airlight

Intensity

DistanceDistance

Intensity

Multiple Scattering : Higher orders of scattering

Incident Beam

Particle

Phase Function

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)

Plane Parallel and Spherical Radiative Transfer

Isotropic

Source

Homogeneous Medium

ScatteredLight Field

Plane Parallel Medium

Scattered Light Field

Distant SourceSun

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

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

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

Symbol Reference

Diffusion Approximation for Multiple Scattering

An incoming ray is transformed into a dipole source for the diffusion approximation

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

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

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)

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)

BRDF vs BSSRDF

BRDF vs BSSRDF

BRDF vs BSSRDF

Diffusion Approximation for Multiple Layers

Donner, Jensen, Siggraph 05

Plane Parallel and Spherical Radiative Transfer

Isotropic

Source

Homogeneous Medium

ScatteredLight Field

Plane Parallel Medium

Scattered Light Field

Distant SourceSun

Mist Fog

Glows of Light Sources

(Narasimhan & Nayar, CVPR 2003)

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

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 :

Radiant Intensity of Source

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 Accuracy

Low Milk Concentration High Milk Concentration

Model Fit Accuracy: Monte Carlo Simulations

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

Original Image

Rendering Glows using Convolution

Increasing Fog

Rendered Images

Joint work with Ramamoorthi

Original Image Single Scattering

Multiple Scattering (Mild Condition) Multiple Scattering (Dense Condition)

Single versus Multiple Scattering

Joint work with Ramamoorthi

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

Volume Rendering as Convolution

Analytic Multiple Scattering )())()((),( 10

mmm

m LTgTgTI

Shedding Light on the Weather

Model Validation using Milk

Summary

Next Class: Fluids

Lectures #22