Introduction to
Computer Vision
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Radiometry
Image: two-dimensional array of 'brightness' values.
Geometry: where in an image a point will project.
Radiometry: what the brightness of the point will be.
Brightness: informal notion used to describe both
scene and image brightness.
Image brightness: related to energy flux incident on
the image plane:
IRRADIANCE
Scene brightness: brightness related to energy flux
emitted (radiated) from a surface.
RADIANCE
Introduction to
Light
Computer Vision
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Electromagnetic energy
Wave model
Light sources typically radiate over a frequency spectrum
Φ watts radiated into 4π radians
Φ watts
Φ=
r
∫spheredΦ
dω
R = Radiant Intensity =
(of source)
dΦ
dω
Watts/unit solid
angle (steradian)
Introduction to
Irradiance
Computer Vision
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Light falling on a surface from all directions.
How much?
dΦ
dA
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Irradiance: power per unit area falling on a surface.
dΦ
Irradiance E = dΑ
watts/m2
Introduction to
Inverse Square Law
Computer Vision
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Relationship between radiance (radiant intensity) and irradiance
dω =
Φ watts
r
dω
dA
R: Radiant Intensity
E: Irradiance
R=
dΦ
dω
=
r2 dΦ
dΑ
Φ: Watts
ω : Steradians
E=
E=
R
r2
=
r2 E
dA
r2
dΦ
dΑ
Introduction to
Surface Radiance
Computer Vision
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Surface acts as light source
Radiates over a hemisphere
dω
Surface
Normal
R: Radiant Intensity
dAf
E: Irradiance
L: Scene radiance
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Radiance: power per unit foreshortened area emitted
into a solid angle
d 2Φ
L=
dA f dω
(watts/m2 - steradian)
Introduction to
Computer Vision
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Pseudo-Radiance
Consider two definitions:
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Radiance:
power per unit foreshortened area emitted into a solid angle
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Pseudo-radiance
power per unit area emitted into a solid angle
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Why should we work with radiance rather than pseudoradiance?
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Only reason: Radiance is more closely related to our
intuitive notion of “brightness”.
Introduction to
Computer Vision
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A particular point P on a Lambertian (perfectly matte)
surface appears to have the same brightness no
matter what angle it is viewed from.
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Lambertian Surfaces
Piece of paper
Matte paint
Doesn’t depend upon incident light angle.
What does this say about how they emit light?
Introduction to
Computer Vision
Lambertian Surfaces
Equal Amounts of Light Observed
From Each Vantage Point
Theta
Area of black box = 1
Area of orange box = 1/cos(Theta)
Foreshortening rule.
Introduction to
Computer Vision
Lambertian Surfaces
Relative magnitude of light scattered in each direction.
Proportional to cos (Theta).
Introduction to
Computer Vision
Lambertian Surfaces
Equal Amounts of Light Observed
From Each Vantage Point
Theta
Radiance
= 1/cos(Theta)*cos(Theta)
=1
Area of black box = 1
Area of orange box = 1/cos(Theta)
Foreshortening rule.
Introduction to
Geometry
Computer Vision
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Goal: Relate the radiance of a surface to the irradiance in
the image plane of a simple optical system.
α: Solid angle of patch
Φ
dAs: Area on surface
dAi: Area in image
dAs
Lens Diameter d
α
i
e
α
dAi
Introduction to
Light at the Surface
Computer Vision
dΦ
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E = flux incident on the surface (irradiance) = dΑ
i = incident angle
e = emittance angle
g = phase angle
ρ = surface reflectance
Φ watts
dω
r
Incident Ray
g
i
dAs
N (surface normal)
e
ρ
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We need to determine dΦ and dA
Emitted ray
Introduction to
Computer Vision
Reflections from a Surface I
dA
dA
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dA = dAscos i {foreshortening effect in direction
of light source}
dΦ
dΦ
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dΦ = flux intercepted by surface over area dA
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dA subtends solid angle dω = dAs cos i / r2
dΦ = R dω = R dAs cos i / r2
E = dΦ / dAs
Surface Irradiance: E = R cos i / r2
Introduction to
Reflections from a Surface II
Computer Vision
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Now treat small surface area as an emitter
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….because it is bouncing light into the world
How much light gets reflected?
Incident Ray
N (surface normal)
i
e
dA s
g
Emitted Ray
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E is the surface irradiance
L is the surface radiance = luminance
They are related through the surface reflectance function:
Ls
E
= ρ(i,e,g,λ)
May also be a function of the
wavelength of the light
Introduction to
Power Concentrated in Lens
Computer Vision
dA s
Lens Diameter d
Ima
ge P
α
θ
lane
α
dAi
Z
d2Φ
L s=
dA sdω
-f
Luminance of patch (known from previous step)
What is the power of the surface patch as a source in
the direction of the lens?
d2Φ = L s dA sdω
Introduction to
Through a Lens Darkly
Computer Vision
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In general:
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Ls is a function of the angles i and e.
Lens can be quite large
Hence, must integrate over the lens solid angle to get dΦ
dΦ = dA s
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Ω
L s dΩ
Introduction to
Simplifying Assumption
Computer Vision
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Lens diameter is small relative to distance from patch
Ls is a constant and can be
removed from the integral
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dΦ = dA s • L s dΩ
Ω
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dΦ = dA s L s • dΩ
Ω
Surface area of patch in
direction of lens
= dAs cos e
Solid angle subtended by
lens in direction of patch
=
Area of lens as seen from patch
(Distance from lens to patch)
2
=
π (d/2) cos α
(z / cos α) 2
2
Introduction to
Putting it Together
Computer Vision
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dΦ = dA s • L s dΩ
Ω
π (d/2) 2cos α
= dA scos e L s
(z / cos α)2
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Power concentrated in lens:
π
d 2
Ls dAs
cos e cos 3 α
dΦ =
Z
4
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Assuming a lossless lens, this is also the power
radiated by the lens as a source.
Introduction to
Through a Lens Darkly
Computer Vision
dA s
Lens Diameter d
Ima
ge P
α
θ
α
dAi
Z
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-f
dΦ
=E i
Image irradiance at dA i =
dA i
dAs π d
E i= Ls
dA i 4 Z
2
cos e cos 3 α
ratio of areas
lane
Introduction to
Patch ratio
Computer Vision
dA s
Lens Diameter d
Ima
ge P
α
θ
lane
α
dAi
Z
-f
The two solid angles are equal
dA s cos e
(Z / cos α) 2
=
dA i cos α
(-f / cos α) 2
dAs cos α Z
= cos e -f
dAi
2
Introduction to
The Fundamental Result
Computer Vision
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Source Radiance to Image Sensor Irradiance:
dA s cos α Z
= cos e -f
dA i
dA s π d
E i = Ls
dA i 4 Z
Z
E i = Ls cos α
cos e -f
2
2
2
cos e cos 3α
π d
4 Z
2
cos e cos 3α
π d 2
E i = Ls
cos 4 α
4 -f
Introduction to
Radiometry Final Result
Computer Vision
π d 2
E i = Ls
cos 4 α
4 -f
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Image irradiance is a function of:
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Scene radiance L s
Focal length of lens f
Diameter of lens d
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f/d is often called the 'effective focal length' of the lens
Off-axis angle α
Introduction to
44
Cos α Light Falloff
Computer Vision
Lens Center
Top view shaded by height
y
−π/2
π/2
−π/2
x