Optics

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When light hits a material:n Reflectionn Absorptionn Transmission

Opaque objects - majority of incident light is reflected/absorbed (little/no transmission)

Translucent objects - significant light transmission

Light Physics

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Incoming Light

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Incoming Light§ Light doesn’t just come from light sources§ Light also comes from all visible objects in the world

§ The tree’s reflection on the car is created by light from the tree (with the color/brightness of the tree) hitting the car and reflecting towards the camera

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Color Bleeding• Using incoming light from all directions makes an image

appear more realistic, and also captures various physical phenomena such as color bleeding

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Incoming Light

using light from all directionsusing only the light source

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Measuring Incoming Light§ Place and photograph a small chrome sphere (light

probe) to collect and record the intensities of the light incoming from all directions

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Measuring Incoming Light§ Technically this should be done with a very tiny

sphere and at every point on the object of interest§ But one often approximates with a single finite size

sphere for the entire object of interest§ The incoming light information can be used to render

a new synthetic object in the original scene

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Object Interaction

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Modeling Approximations§ BRDF (Bidirectional

Reflectance Distribution Function) models how much light is reflected

§ BTDF (Bidirectional Transmittance Distribution Function) models how much light is transmitted

§ BSSRDF (Bidirectional Surface Scattering Reflectance Distribution Function) is a combinedreflection/transmission model

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Opaque (BRDF) vs. Translucent (BSSRDF)

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Paint (BRDF) vs. Milk (BSSRDF)

BRDF BSSRDF

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Reflection Only (BRDF)

Modeled by Stephen Stahlberg

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Subsurface Scattering (BSSRDF)

Modeled by Stephen Stahlberg

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BRDF

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BRDF( , , , , )i o u vl w w

Dependencies

How much light is reflected depends on:• wavelength of the light• 2D incoming light direction • 2D outgoing reflected direction • spatial position on the object surface

• many real world materials (e.g. wood) consist of sub-materials with spatially varying characteristics

),( iii fqw =),( ooo fqw =

l

),( vu

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Simplifying AssumptionsSpatial invariance

n Modulate the BRDF by a texture to approximation spatial variance

Three channels: R, G, Bn Variation throughout all visible wavelengths is

approximated by three separate BRDFs

Thus, 3 separate BRDFs (1 Red, 1 Green, 1 Blue), each of the form:

This is still a function of 4 variables, i.e. a 4D function!

),BRDF( oi ww

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Measuring/Approximating

• 4D BRDF data can be acquired with a gonioreflectometer to obtain a 4D table of values

• Alternatively, some simpleanalytical models: • Blinn-Phong Model – simplest and

general purpose (plastic)• Cook-Torrance Model – better specular

(metal)• Ward Model – anisotropic (brushed

metal, hair)• Oren-Nayar Model – non-Lambertian

(concrete, plaster, the moon)

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Lighting Equation

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Summary (Road Map)• Given a point on an object:

• Light from every incoming direction hits that point• For each incoming direction , light is reflected

outwards in every direction • BRDF indicates how much light is reflected in

each of the outgoing directions given an incoming direction

• Since light is reflected in all outgoing directions, we can all see that point on the object

• But it looks different to each of us, since we all see different light

• To render a synthetic scene, figure out what light each pixel of the camera’s film sees

iw

iw

ow

owiw

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The Lighting Equation

The amount of light reflected in an outgoing direction is the integral of the amount of light reflected in that direction due to light from every incoming direction:

n The BRDF gives

åÎ

=in

todue ),(i

oiioo LL ww

),( todue oiioL ww

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For each pixel, integrate the BRDF across all incoming directions for every point in the projected area

iw

Outgoin

g dire

ction

All incoming directions

Image plane

pixel

in

( , ) ( )o i o i ii

L BRDF dEw w wÎ

= ò

iEoL

The Lighting Equation

iwd

Vanishingly small surface elementwithin the projected area

ow

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For each pixel, integrate the BRDF across all incoming directions for every point in the projected area

iw

Outgoin

g dire

ction

All incoming directions

Image plane

pixel

in

( , ) ( )o i o i ii

L BRDF dEw w wÎ

= ò

iEoL

The Lighting Equation

iwd

Vanishingly small surface elementwithin the projected area

ow

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Math L

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Lights

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Solid Angle

Two-dimensional angle in 3D defined by a point and a surface patch, measured by a dimensionless unit called a steradian (sr)

Angle: (a circle has 2π radians)

Solid angle: (a sphere has 4π steradians)

rlarc=q

2sphereon

rA

=w 24A rp=

2C rp=

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Radiant Intensity of a Light

Power per unit solid angle:

where is the total power of the light

Note: is a function of for anisotropic light emission

The relation between power and radiant intensity for an isotropicpoint light is:

ww

dd)( F

ºI

F

I w

II pw 4dsphere

==F ò

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Irradiance on a Surface

Power per unit surface area:

Irradiance decreases as you tilt the object, since less light hits it

AE

ddF

º

q

q

q

cos

cos

)( cos

tilted

EA

AE A

A

=

F=

F=

F

qcosA

F

A

Aq

qcosA

N¢ˆq

qp -2

qp -2

A

A

q

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The relationship between solid angle and area is

where is the area of the orthogonal cross section

Solid Angle vs. Area

http://users.eecs.northwestern.edu/~yingwu/teaching/EECS432/Notes/lighting.pdf

2

cosddr

A qw =

qcosd A

unitsphere

surfacepatch

q

Ad

wd

0S

r N̂

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No energy can flow through a point; it must go through either a solid angle or an area (interchangeable)

Total light power per unit solid angle à Radiant Intensity• Measure of how strong a (point) light source is

Total light power per unit area à Irradiance• Measure of how much light is hitting a surface• Varies based on the distance to the light and the angle of

the surface (see last slide)• The farther away and more tilted a surface area patch is,

the smaller the solid angle of light that hits that surface area patch

Summary

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More Math L L

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Area Lights

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• Light power is emitted per unit area (not from a single point)

• This light goes in various directions, or solid angles

• Can approximate an area light by breaking it up into small area chunks, where each area chunk emits light into each of the solid angle directions • i.e. radiant intensity per area chunk

• Each direction has a cosine term similar to irradiance

• Radiance – radiant intensity per area chunk

Area Lights

2d d dcos d cos d cosI EL

dA dAq w q w qæ öF

= = =ç ÷è ø

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ow

BRDFRelates the incoming light that hits a surface patch

(irradiance ) to the outgoing light emitted from the surface patch acting as if it were an area light(radiance )

We care about the outwards radiance of a surface patch in the direction of a pixel, given an incoming irradiance from a direction onto that surface patch

(An “area light” patch has radiant intensity in every outgoing direction, but we only care about the direction of our pixel)

)(d)(d),BRDF(

ii

oooi E

Lwwww =

iE

oL

iw

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The Lighting Equation

Multiplying the BRDF by an incoming irradiance gives the outgoing radiance

For more realistic lighting, we bounce light all around the scene, and it is tedious to convert between irradiance and radiance --- so we use

(from two slides earlier) to obtain:

Finally, the outgoing radiance considering the light coming from all incoming directions is

qw cosdd LE =

òÎ

=in

dcos),(i

iiioio LBRDFL wqww

ioioiio EBRDFL d),(),( todue wwww =

iiioioiio LBRDFL qwwwww cosd),(),( todue =

completing the derivation

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Pixels

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Pixel Color

(integrating an equation from 2 slides ago)

(so we can store L instead of E, and scale later)

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Summary

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Question 1 (short/long form)Summarize… (use class time for short; long form 1000 words)

?