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RadiometryRadiometry

Computer Vision ICSE252A

Lecture 5-Part II

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RadiometryRadiometry

• Read Chapter 4 of Ponce & Forsyth

• Solid Angle• Irradiance• Radiance• BRDF• Lambertian/Phong BRDF

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Measuring AngleMeasuring Angle

• The solid angle subtended by an object from a point P is the area of the projection of the object onto the unit sphere centered at P.

• Measured in steradians, sr• Definition is analogous to projected angle in 2D• If I’m at P, and I look out, solid angle tells me how

much of my view is filled with an object CSE 252A

Solid AngleSolid Angle

• By analogy with angle (in radians), the solid angle subtended by a region at a point is the area projected on a unit sphere centered at that point

• The solid angle subtended by a patch area dA is given by

dω =dAcosϑ

r2

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Radiance Radiance • Power is energy per unit time

• Radiance: Power traveling at some point in a specified direction, per unit area perpendicular to the direction of travel, per unit solid angle

• Symbol: L(x,θ,φ)

• Units: watts per square meter per steradian : w/(m2sr1)

x

dA

θ

dω

ωθ ddAP

L)cos(

=

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Radiance transfer Radiance transfer What is the power received by a small area dA2 at distance r from a small emitting area dA1?

From definition of radiance

ωθ ddAP

L)cos(

=

From definition of solid angle

2

cosr

dAd

θω =

21212

2111

coscos

cos

θθ

ωθ

dAdArL

dLdAP

=

= >−

2

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IrradianceIrradiance• How much light is arriving at a

surface?• Sensible unit is Irradiance• Incident power per unit area not

foreshortened• This is a function of incoming angle. • A surface experiencing radiance

L(x,θ,φ) coming in from solid angle dω experiences irradiance:

• Crucial property: Total power arriving at the surface is given by adding irradiance over all incoming angles Total power is

( ) ωφφθ dxL cos,,

( )∫Ω

φθθφφθ ddxL sincos,,

NφL(x,θ,φ)

x x

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Intermezzo:Intermezzo:Camera’s sensorCamera’s sensor

• Measured pixel intensity is a function of irradiance integrated over – pixel’s area– over a range of wavelengths– For some time

• Ideally, it’s linear to the radiance, but the camera response C(.) may not be linear

∫ ∫ ∫ ∫ =t x y

dtdydxdqyxstyxEIλ

λλλ )(),(),,,(

= ∫ ∫ ∫ ∫

t x y

dtdydxdqyxstyxECIλ

λλλ )(),(),,,(

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Image sensor

Two types :

è

1. CCD

2. CMOS

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CCD

separate photo sensor at regular positionsno scanning

charge-coupled devices (CCDs)

: interline transfer and frame transfer

photosensitive

storage

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CMOSCMOSEach photo sensor has its own amplifier

More noise (reduced by subtracting ‘black’ image)Lower sensitivity (lower fill rate)

Uses standard CMOS technologyAllows to put other components on chip‘Smart’ pixels

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CCD vs. CMOSCCD vs. CMOS

• Mature technology• Specific technology• High production cost• High power consumption• Higher fill rate• Blooming• Sequential readout

• Recent technology• Standard IC technology• Cheap• Low power• Less sensitive• Per pixel amplification• Random pixel access• Smart pixels• On chip integration

with other components

3

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Color CamerasColor Cameras

We consider 3 concepts:

1. Prism (with 3 sensors)2. Filter mosaic3. Filter wheel

… and X3

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Prism Prism colourcolour cameracameraSeparate light in 3 beams using dichroic prismRequires 3 sensors & precise alignmentGood color separation

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Filter mosaic Filter mosaic Coat filter directly on sensor

Demosaicing (obtain full colour & full resolution image)

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Filter wheelFilter wheelRotate multiple filters in front of lensAllows more than 3 colour bands

Only suitable for static scenes

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Prism vs. mosaic vs. wheelPrism vs. mosaic vs. wheelWheel

1GoodAverageLowMotion3 or more

approach# sensorsSeparationCostFramerateArtefactsBands

Prism3

HighHighHighLow

3

High-endcameras

Mosaic1

AverageLowHighAliasing

3

Low-endcameras

Scientific applications

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new color CMOS sensornew color CMOS sensorFoveon’sFoveon’s X3X3

better image qualitysmarter pixels

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Example: Radiometry of thin lensesExample: Radiometry of thin lenses

( ) ( )22 cos/cos

cos/'cos'

ββδ

ααδδω

zA

zA ==

2

'coscos

'

=

zz

AA

βα

δδ

( )3

2

2

2

cos4cos/

cos4

απ

ααπ

==Ω

zd

zd

βαδπ

βδδ coscos4

cos 32

ALzd

ALP

=Ω=

βαδδπ

δδ

coscos'4'

32

AA

Lzd

AP

E

==

Lzd

E

= α

π 42

cos4

E: Image irradianceL: emitted radianced : Lens diamterZ: depthα: Angle of patch from optical axis CSE 252A

Light at surfacesLight at surfaces

Many effects when light strikes a surface -- could be:

• transmitted– Skin, glass

• reflected– mirror

• scattered– milk

• travel along the surface and leave at some other point

• absorbed– sweaty skin

Assume that• surfaces don’t fluoresce

– e.g. scorpions, detergents • surfaces don’t emit light (i.e.

are cool)• all the light leaving a point is

due to that arriving at that point

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BRDFBRDFWith assumptions in previous slide• Bi-directional Reflectance Distribution

Function ρ(θin, φin ; θout, φout)

•• Ratio of incident irradiance to Ratio of incident irradiance to emitted radianceemitted radiance

• Function of– Incoming light direction:

θin , φin– Outgoing light direction:

θout , φout

n(θin,φin)

(θout,φout)

( ) ( )( ) ωφφθ

φθφθφθρ

dxLxL

xininini

outoutooutoutinin cos,;

,;,;,; =

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Surface Reflectance ModelsSurface Reflectance Models

• Lambertian• Phong• Physics-based

– Specular [Blinn 1977], [Cook-Torrance 1982], [Ward 1992]

– Diffuse [Hanrahan, Kreuger 1993]

– Generalized Lambertian [Oren, Nayar 1995]

– Thoroughly Pitted Surfaces [Koenderink et al 1999]

• Phenomenological[Koenderink, Van Doorn 1996]

Common Models Arbitrary Reflectance

• Non-parametric model• Anisotropic• Non-uniform over surface• BRDF Measurement [Dana et

al, 1999], [Marschner ]

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Lambertian (Diffuse) SurfaceLambertian (Diffuse) Surface

• BRDF is a constant called the albedo

• Emitted radiance is NOT a function of outgoing direction – i.e. constant in all directions.

• For lighting coming in single direction, emitted radiance is proportional to cosine of the angle between normal and light direction

Lr = N . ωι

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Lambertian reflectionLambertian reflection

• Lambertian• Matte• Diffuse

Light is radiated equally in all directions

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Specular Reflection: Smooth SurfaceSpecular Reflection: Smooth Surface

N

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Rough Specular SurfaceRough Specular Surface

Phong Lobe

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GonioreflectometersGonioreflectometers

• Three degrees of freedom spread among light source, detector, and/or sample

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GonioreflectometersGonioreflectometers

• Three degrees of freedom spread among light source, detector, and/or sample

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GonioreflectometersGonioreflectometers

• Can add fourth degree of freedom to measure anisotropic BRDFs

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Ward’s BRDF Measurement SetupWard’s BRDF Measurement Setup

• Collect reflected light with hemispherical (should be ellipsoidal) mirror [SIGGRAPH 92]

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Ward’s BRDF Measurement SetupWard’s BRDF Measurement Setup

• Result: each image captures light at all exitant angles

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Marschner’sMarschner’s ImageImage--BasedBasedBRDF MeasurementBRDF Measurement

• For uniform BRDF, capture 2-D slice corresponding to variations in normals

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BRDF Not Always AppropriateBRDF Not Always Appropriate

http://graphics.stanford.edu/papers/bssrdf/(Jensen, Marschner, Levoy, Hanrahan)

BRDF BSSRDF

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Light sources and shadingLight sources and shading

• How bright (or what colour) are objects?• One more definition: Exitance of a source is

– the internally generated power radiated per unit area on the radiating surface

• Similar to irradiance

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RadiosityRadiosity due to a point sourcesdue to a point sources

• small, distant sphere radius ε and exitance E, which is far away subtends solid angle of about

πεd

2

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Standard nearby point source modelStandard nearby point source model

• N is the surface normal• ρ is diffuse (Lambertian)

albedo• S is source vector - a

vector from x to the source, whose length is the intensity term– works because a dot-

product is basically a cosine

( ) ( ) ( )( )

•2xr

xSxNxdρ

NS

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Standard distant point source modelStandard distant point source model

• Issue: nearby point source gets bigger if one gets closer– the sun doesn’t for any reasonable binding of closer

• Assume that all points in the model are close to each other with respect to the distance to the source. Then the source vector doesn’t vary much, and the distance doesn’t vary much either, and we can roll the constants together to get:

( ) ( ) ( )( )xSxNx dd •ρN

S

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Line sources Line sources

radiosity due to line source varies with inverse distance, if the source is long enough

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Area sourcesArea sources• Examples: diffuser boxes,

white walls.• The radiosity at a point due to

an area source is obtained by adding up the contribution over the section of view hemisphere subtended by the source – change variables and add

up over the source

• See Forsyth & Ponce or a graphics text for details.