Light Field = Radiance(Ray) - Computer Graphicsgraphics.stanford.edu/courses/cs348b-04/lectures/lecture... · 2004-04-12 · ∴ Radiance is a function on rays ∴ All other field

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CS348B Lecture 5 Pat Hanrahan, 2004

The Light Field

Light field = radiance function on rays

Surface and field radiance

Conservation of radiance

Measurement

Irradiance from area sources

Measuring rays

Form factors and throughput

Conservation of throughput

From London and Upton

Light Field = Radiance(Ray)

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Definition: The incoming surface radiance

(luminance) is the power per unit solid angle per

unit projected area arriving at a receiving surface

Incident Surface Radiance

ωω

ωΦ

≡i

2 ( , )( , ) i

i

d xL x

d dA

dAdω


Exitant Surface Radiance

Definition: The outgoing surface radiance (luminance) is the power per unit solid angle per unit projected area leaving at surface

Alternatively: the intensity per unit projected area leaving a surface

ωω

ωΦ

≡i

2 ( , )( , ) o

o

d xL x

d dA

dAdω

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Definition: The field radiance (luminance) at a

point in space in a given direction is the power

per unit solid angle per unit area perpendicular

to the direction

Field Radiance

dAdω

( , )L x ω


Environment Maps

Miller and Hoffman, 1984

( , )L θ ϕ

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Spherical Gantry ⇒ Light Field

Capture all the light leavingan object - like a hologram

( , , , )L x y θ ϕ( , )θ ϕ


Two-Plane Light Field

2D Array of Cameras 2D Array of Images

( , , , )L u v s t

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Multi-Camera Array ⇒ Light Field

Properties of Radiance

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Properties of Radiance

1. Fundamental field quantity that characterizes the distribution of light in an environment.

∴ Radiance is a function on rays

∴ All other field quantities are derived from it

2. Radiance invariant along a ray.

∴ 5D ray space reduces to 4D

3. Response of a sensor proportional to radiance.


1st Law: Conversation of Radiance

The radiance in the direction of a light ray remains

constant as the ray propagates

21 1 1 1d L d dAωΦ =

1 21 1 2 22

dA dAd dA d dA

rω ω= =

1 2L L∴ =

1dω

2dω

21d Φ 2

2d Φ Φ = Φ2 21 2d d

1dA

2dA

r

22 2 2 2d L d dAωΦ =

1L

2L

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Radiance: 2nd Law

The response of a sensor is proportional to theradiance of the surface visible to the sensor.

L is what should be computed and displayed.T quantifies the gathering power of the device.

A

R Ld dA LTωΩ

= =∫ ∫A

T d dAωΩ

= ∫ ∫

Aperture SensorΩ A


Quiz

Does the brightness that a wall appears to the

sensor depend on the distance?

No!!

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Irradiance from a Uniform Surface Source


Irradiance from the Environment

2

( ) ( , )cosi

H

E x L x dω θ ω= ∫

2 ( , ) ( , )cosi id x L x dAdω ω θ ωΦ =

( , ) ( , )cosidE x L x dω ω θ ω=ω( , )iL x

dA

θdω

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Irradiance from an Area Source

2

( ) cos

cos

H

E x L d

L d

L

θ ω

θ ωΩ

=

=

= Ω

∫

∫

A

Ω

Ω


Projected Solid Angle

cos dθ ω

θ

2

cosH

dθ ω π=∫

dω

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Uniform Disk Source

r

h

α

cos 2

1 0

cos2

1

2

2

2 2

cos cos

cos2

2

sin

d d

r

r h

α π

α

θ φ θ

θπ

π α

π

Ω =

=

=

=+

∫ ∫

Geometric Derivation Algebraic Derivation

2sinπ αΩ =

sinα


Spherical Source

2

2

2

cos

sin

d

r

R

θ ω

π α

π

Ω =

=

=

∫r

R

α

Geometric Derivation Algebraic Derivation

2sinπ αΩ =

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The Sun

Solar constant (normal incidence at zenith)

Irradiance 1353 W/m2

Illuminance 127,500 lm/m2 = 127.5 kilolux

Solar angle

= .25 degrees = .004 radians (half angle)

= 6 x 10-5 steradians

Solar radiance

3 27

5 2

1.353 10 / W2.25 10

6 10

E W mL

sr m sr−

×= = = ×Ω × ⋅

2 2sinπ α παΩ = ≈α


Polygonal Source

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Polygonal Source


Polygonal Source

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Lambert’s Formula

1 1

n n

i i ii i

A γ= =

= •∑ ∑ N Ni i iA γ= •N N

i iγ Niγ 3

1i

i

A=∑

1A

2A−

3A−


Penumbras and Umbras

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Measuring Rays = Throughput


Throughput Counts Rays

Define an infinitesimal beam as the set of raysintersecting two infinitesimal surface elements

Measure the number of rays in the beam

This quantity is called the throughput

2 1 22

1 2

dA dAd T

x x=

−

1 1 1( , )dA u v 2 2 2( , )dA u v

1 1 2 2( , , , )r u v u v

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Parameterizing Rays

Parameterize rays wrt to receiver

2 2 2( , )dA u vω θ φ2 2 2( , )d

2 12 2 22

1 2

dAd T dA d dA

x xω= =

−

2 2 2 2( , , , )r u v θ φ


Parameterizing Rays

Parameterize rays wrt to source

1 1 1( , )dA u v ω θ φ1 1 1( , )d

2 21 1 12

1 2

dAd T dA dA d

x xω= =

−

1 1 1 1( , , , )r u v θ φ

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Parameterizing Rays

Tilting the surfaces reparameterizes the rays

All these throughputs must be equal.

ii

2 1 21 22

1 2

1 1

2 2

cos cosd T dA dA

x x

d dA

d dA

θ θ

ω

ω

=−

=

=

1 1 1( , )dA u v 2 2 2( , )dA u v1 1 2 2( , , , )r u v u v


Parameterizing Rays: S2 × R2

Parameterize rays by

Measuring the number or rays that hit a shape

( , , , )r x y θ φ

ω θ ϕ

θ ϕ ω θ ϕ

π

=

=

=

∫ ∫

∫2 2

2

( , ) ( , )

( , ) ( , )

4

S R

S

T d dA x y

A d

A

( )A ωω

Sphere:2 24 4T A Rπ π= =

Projected area

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Parameterizing Rays: M2 × S2

Parameterize rays by ( , , , )r u v θ φ

2 2 ( )

( , ) cos ( , )M H

T dA u v d

S

θ ω θ ϕ

π

=

∫ ∫

N

Sphere: 2 24T S Rπ π= =

Crofton’s Theorem: 44

SA S Aπ π= ⇒ =

( , )u v

( , )θ φ

N

Form Factors

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Types of Throughput

1. Infinitesimal beam of rays

2. Infinitesimal-finite beam

3. Finite-finite beam

22

cos cos( , ) ( ) ( )d T dA dA dA x dA x

x x

θ θ′′ ′≡′−

2

cos cos( , ) ( ) ( )

A

dT dA A dA dA x dA xx x

θ θ

′

′′ ′≡

′− ∫

2

cos cos( , ) ( ) ( )

A A

T A A dA x dA xx x

θ θ

′

′′ ′≡

′−∫ ∫


Probability of Ray Intersection

Probability of a ray hitting A’ given it hits A

( , )Pr( | )

( )

( , )

T A AA A

T A

T A A

Aπ

′′ =

′=

A( )dA x

A′( )dA x′

π=( )T A A

2

cos cos( , ) ( ) ( )

A A

T A A dA x dA xx x

θ θ

′

′′ ′=

′−∫ ∫

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Probability of Ray Intersection

A( )dA x

A′

θ θπ

′≡′ ′

− ′2

cos cos( , ) ( , )G x x V x x

x x

0( , )

1

visibleV x x

visible

¬′ =

( )dA x′

2

cos cos( , ) ( ) ( )

( , ) ( ) ( )

A A

A A

T A A dA x dA xx x

G x x dA x dA x

θ θ

π

′

′

′′ ′=

′−

′ ′=

∫ ∫

∫ ∫


Form Factor

Probability of a ray hitting A’ given it hits A

Form factor definition

Form factor reciprocity

Pr( | ) ( ) Pr( | ) ( ) ( , )A A T A A A T A T A A′ ′ ′ ′= =

A( )dA x

A′

π=( )T A A

( )dA x′( , ) Pr( | )

( , ) Pr( | )

F A A A A

F A A A A

′ ′=′ ′=

( , ) ( , )F A A A F A A A′ ′ ′=

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Form Factor

Power transfer from a constant source

A( )dA x

A′

π=( )T A A

( )dA x′

( , ) ( , )

( , )( )

( )

( ) ( , )

A A LT A A

T A ALT A

T A

A F A A

′ ′Φ =′

′=′

′ ′=Φ


Differential Form Factor

Probability of a ray leaving dA(x) hitting A’

( )dA x

( , ) ( , ) ( , )Pr( | )

( )

( , ) ( )A

T dA A dA T dA A dA T dA AA dA

T dA dA

G x x dA x

π π

′

′ ′ ′′ = = =

′ ′= ∫ A′

( )dA x′

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Form Factor

Power transfer from a constant source

A( )dA x

A′( )dA x′

( , ) ( , )

( , )( )

( )

( ) ( , )

d dA A LT dA A

T dA ALT A

T A

A F dA A

′ ′Φ =′

′=′

′ ′=Φ


Radiant Exitance

Definition: The radiant (luminous) exitance is the energy per unit area leaving a surface.

In computer graphics, this quantity is oftenreferred to as the radiosity (B)

( ) odM x

dA

Φ≡

2 2

W lmlux

m m =

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Uniform Diffuse Emitter

ω( , )oL x

dA

θdω

θ ω

θ ω

π

=

=

=

∫

∫2

2

cos

cos

o

H

o

H

o

M L d

L d

L

π=o

ML


Form Factor

Irradiance from a constant source

A( )dA x

A′( )dA x′

( , ) ( ) ( , )

( )( , )

( ) ( , )

d dA A A F dA A

AA F dA A

AM A F A dA dA

′ ′ ′Φ =Φ′Φ ′ ′=′′ ′=

( ) ( ) ( , )E dA M A F A dA′ ′=

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Form Factors and Throughput

Throughput measures the number of rays in a set of rays

Form factors represent the probability of ray leaving a surface intersecting another surface

Only a function of surface geometry

Differential form factor

Irradiance calculations

Form factors

Radiosity calculations (energy balance)


Conservation of Throughput

Throughput conserved during propagation

Number of rays conserved

Assuming no attenuation or scattering

n2 (index of refraction) times throughput invariant under the laws of geometric optics

Reflection at an interface

Refraction at an interfaceCauses rays to bend (kink)

Continuously varying index of refractionCauses rays to curve; mirages

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Conservation of Radiance

Radiance is the ratio of two quantities:

1. Power

2. Throughput

Since power and throughput are conserved,

∴ Radiance conserved

∆ →

∆Φ ∆ Φ= =

∆0

( )( ) lim

T

T dL r

T dT


Quiz

Does radiance increase under a magnifying glass?

No!!

Documents

Light Field = Radiance(Ray) - Computer Graphicsgraphics.stanford.edu/courses/cs348b-04/lectures/lecture... · 2004-04-12 · ∴ Radiance is a function on rays ∴ All other field