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Transcript of RI 6 1DUDVLPKDQ &08...,PDJH )RUPDWLRQ 5DGLRPHWU\:KDW GHWHUPLQHV WKH EULJKWQHVV RI DQ LPDJH SL[HO"...
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Introduction to computer vision VI
Instructors: Jean Ponce and Matthew [email protected], [email protected]
TAs: Jiachen (Jason) Zhu and Sahar [email protected], [email protected]
Slides will be available after class at: https://mtrager.github.io/introCV-fall2019/
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• Radiometry• Radiance• Irradiance• BRDF• Photometric stereo
• Color• Spectral radiometry• Source and surface colors• Trichromacies• The eye• Color spaces• The bichromatic model
Light, shading, color and shodows
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Light and shadows
[These slides courtesy of S. Narasimhan, CMU.]
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Reflections
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Reflections and refractions
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Refraction
Caustics
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Is glass really transparent?
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Interreflections
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Scattering
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The two main types of reflection: diffuse and specular
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Shading and orientation (distant light sources)Interreflections and soft shadows
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Area sources and soft shadows
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Indoor scene with some directional componenentbut almost no shadows: small occluders and largesources
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Image Formation: Radiometry
What determines the brightness of an image pixel?
The lightsource(s)
The surfacenormal
The surfacepropertiesThe optics
The sensorcharacteristics
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d2P = L( P, v ) dA dw
DEFINITION: The radiance is the power traveling at some pointin a given direction per unit area perpendicular to this direction,per unit solid angle.
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d2P = L( P, v ) dA dw d2P = L( P, v ) cosq dA dw
DEFINITION: The radiance is the power traveling at some pointin a given direction per unit area perpendicular to this direction,per unit solid angle.
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DEFINITION: Angles and Solid Angles
R
Ll θ (radians)
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DEFINITION: Angles and Solid Angles
R
Ll θ
2R
Aa
(radians)
(steradians)
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d2P = L( P, v ) dA dw d2P = L( P, v ) cosq dA dw
PROPERTY: Radiance is constant along straight lines (in vacuum).
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DEFINITION: Irradiance
The irradiance is the power perunit area incident on a surface.
d2P=dE dA=Li( P , vi ) cosqidwi dA
dE = Li( P , vi ) cosqi dwi
E=sH Li (P, vi) cosqi dwi
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Lz
dE
4
2
cos'4
Photometry
• L is the radiance.
• E is the irradiance.
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DEFINITION: The Bidirectional Reflectance Distribution Function (BRDF)
The BRDF is the ratio ofthe radiance in the outgoingdirection to the incident irradiance (sr-1).
dLo ( P, vo ) = rBD ( P, vi , vo ) d Ei ( P, vi )
= rBD ( P, vi , vo ) Li ( P, vi ) cos qi dwi
Helmoltz reciprocity law: rBD ( P, vi , vo ) = rBD ( P, vo , vi )
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DEFINITION: Specular Surfaces as Perfect or Rough Mirrors
qi qsqi qs
Perfect mirror Rough mirror
Phong (non-physical model): Lo (P,vo)= rsLi(P,vi) cosn d
d
Hybrid model: Lo (P,vo)= rdsH Li(P, vi) cosqi dwi+rsLs(P,vi) cosn d
Perfect mirror: Lo (P,vs) = Li (P,vi) [BRDF=double delta function]
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DEFINITION: Radiosity
The radiosity is the total powerLeaving a point on a surface per unit area (W x m-2).
B(P) = sH Lo ( P , vo ) cosqo dw
Important case: Lo is independent of vo.
B(P) = π Lo(P)
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DEFINITION: Lambertian (or Matte) Surfaces
A Lambertian surface is a surface whose BRDF is independentof the outgoing direction (and by reciprocity of the incomingdirection as well).
rBD( vi , vo ) = rBD = constant.
The albedo is rd = P rBD.Note: 1 ¸ P rBD
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DEFINITION: Point Light Sources
A point light source is an idealization of an emitting sphere with radius ε at distance R, with ε << R and uniform radianceLe emitted in every direction.
For a Lambertian surface, the corresponding radiosity is
ied PRLPPB qr cos
)()()(
2
2
P
qi qi
N
S
2)(
)()()(
PR
PPPd
SN r
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Lambertian sphere with the sun behind its observer
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Is the moon Lambertian?
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The moon “looks” matte but is not Lambertian
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Local Shading Model
• Assume that the radiosity at a patch is the sum of theradiosities due to point light sources alone.
No interreflections.
• For point sources:
• For point sources at infinity:
2 visible )(
)()()()(
PR
PPPPB
s
sd
s
SN r
s
sd PPPPB visible
)()()()( SNr
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Photometric Stereo (Woodham, 1979)
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Problem: Given n images of an object, taken by a fixed camera underdifferent (known) light sources, reconstruct the object shape.
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I(P) = kB(P) = kr N(P)¢ S
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Photometric Stereo: Example (1)
• Assume a Lambertian surface and distant point light sources.
I(P) = kB(P) = kr N(P)¢ S = g(P)¢Vwith g(P)=r N(P) and V= k S
• Given n images, we obtain n linear equations in g:
g = Vy i
I1 V1.g V1T
I2 V2.g V2T
i= … = … = g i = V gIn Vn.g Vn
T
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Photometric Stereo: Example (2)
• What about shadows?
• Just skip the equations corresponding to zero-intensitypixels.
• Only works when there is no ambient illumination.
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Photometric Stereo: Example (3)
g(P)=r(P)N(P)
r(P) = | g(P) |
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Reflectance map: known reflectance and light source
[Here: Lambertian case.]
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Using more images: photometric stereo
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Example-based photometric stereo
[Silver, 1980; Woodham, 1982, Seitz & Hertzmann, 2003]
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Problem: Interreflections
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COLOR: Light energy usually depends on wavelength
P (W)
l (nm)400 700
What causes the apparent color of a surface?• the color and geometry of the light sources,• the color and geometry of the surface,• the spectral characteristics and geometry of the sensor.
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Spectral Units
• Radiance L(P,v) – Spectral Radiance Lλ(P,v)
• The spectral radiance is the power traveling at somepoint in a given direction per unit area perpendicularto the direction of travel, per unit angle and per unitWavelength (W m-2sr-1).
• The power emitted in the range [l,l+dl] is
• Irradiance -> Spectral Irradiance• BRDF -> Spectral BRDF• Etc.
dP = Ll(P,v) cosq dw dA dl.
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Sources
• Black body (Planck’s Law):
( ) ( ) 1/exp
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• Sun (daylight) and sky (airlight)[sky brighter at horizon]
• Incandescent lights• Fluorescent lights• LEDs• Sodium or mercury arc lamps
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Spectral distribution of day light
The sun looks yellow because shorter-wavelength frequencies (blue)are scattered out of sun rays faster than longer-wavelength frequencies (red) -> yellow sun and blue sky.
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Day light vs Tungsten filament lightbulb
More or lesslike a black body
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High-speed electronsstrike gas in a bulb,releasing UV rays thatcause phospors coatingthe bulb to fluoresce
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Spectral albedoesof flowers
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Spectral albedoesof leaves
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Red flowers
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Green leaves
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Color Matching Experiments
Adjust the knobs on the primaries until the split field looks uniform.
T = w1P1+w2P2+ … +wkPk Notation only!
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Trichromacy
Three primaries are sufficient to match any test light!
Note:
• This requires using independent primaries andallowing subtractive matching.
• For a given set of primaries and test light, peopletend to pick the same weights.
• This is related to our three types of cones.
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Color Matching Experiments
Adjust the knobs on the primaries until the split field looks uniform.
T = w1P1+w2P2+ … +wkPk Notation only!
T+w1P1+ … +wnPn = wn+1Pn+1+ … +wkPkSubtractive matching
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Linearity of Colour Matching (Grassman’s Laws)
Ta = wa1P1+wa2P2+wa3P3 and Tb = wb1P1+wb2P2+wb3P3
l Ta+m Tb = (l wa1+m wb1)P1+(l wa2+m wb2)P2 +(l wa3+m wb3)P3
Ta = w1P1+w2P2+w3P3
and
Tb = w1P1+w2P2+w3P3
Ta = Tb
Note: this does not mean that Ta and Tb have the same spectrum.
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Principle of Univariance
• The eye photoreceptors do not directly measurespectral radiance.
• Two lights will match if they produce the same photoreceptor responses even if they have quitedifferent spectral radiances.
• Because of linearity, the response of each photoreceptor can be modeled as
Pk=sL sk(l)E(l)dl.
• The sensitivity sk can be measured experimentally.
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Cones in the fovea
Rods and cones in the periphery
The humaneye..
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Spectral sensitivity of thethree types of cones
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Color Matching Functions
L(l) = f1(l)P1+f2(l)P2+f3(l)P3
• Problem: given a set of primaries, what are the weightsmatching a given spectral radiance?
l
1
• Experiments:
Color matchingfunctions
• To match S use linearity:S=(sL f1(l)S(l)dl)P1+(sL f2(l)S(l)dl)P2+(sL f3(l)S(l)dl)P3
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RGB Color Matching Functions
R=645.16nmG=526.32nmB=444.44nm
Negative weights
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A rather poor reproduction ofthe RGB colorcube..
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CIE XYZ Color Matching Functions
Note: there are no physicalXYZ primaries!
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CIE XYZ and xy spaces
All visibleXYZ colors
Boundary: Single wavelength colors