Post on 14-Dec-2015
Computer Vision
Radiometry
Bahadir K. Gunturk 2
Radiometry
Radiometry is the part of image formation concerned with the relation among the amounts of
light energy emitted from light sources, reflected from surfaces, and registered by sensors.
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Foreshortening
A big source, viewed at a glancing angle, must produce the same effect as a small source viewed frontally.
This phenomenon is known as foreshortening.
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Solid Angle
Solid angle is defined by the projected area of a surface patch onto a unit sphere of a point.
(Solid angle is subtended by a point and a surface patch.)
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Solid Angle
Arc length
r
d r d
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Solid Angle
Solid angle is defined by the projected area of a surface patch onto a unit sphere of a point.
2sin sindA rd r d r d d
22 2
0 0
sin 4TotalArea r d d r
dA
2sin
dAdw d d
r
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Solid Angle
Similarly, solid angle due to a line segment is
r
dl
d
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Radiance
The distribution of light in space is a function of position and direction.
The appropriate unit for measuring the distribution of light in space is radiance, which is defined as the power (the amount of energy per unit time) traveling at some point in a specified direction, per unit area perpendicular to the direction of travel, per unit solid angle.
In short, radiance is the amount of light radiated from a point… (into a unit solid angle, from a unit area).
Radiance = Power / (solid angle x foreshortened area)
W/sr/m2W is Watt, sr is steradian, m2 is meter-squared
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Radiance
Radiance from dS to dR
Radiance = Power / (solid angle x foreshortened area)
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Radiance
Example: Infinitesimal source and surface patches
Source
Illuminated surface
2
1 1 21 1 2 2 1 1
( , )cos cos cos
d r dL
dw dA dA dA
x x x
Radiance at x1 leaving to x2
2 22
cosdAdw
r
Radiance = Power / (solid angle x foreshortened area)
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Radiance
Source
Illuminated surface
Power at x1 leaving to x2
2 22
cosdAdw
r
1 1 2 1 1( , ) cosd L dw dA x x x
1 1 2 2 2 1 12
( , ) cos cosL dA dA
r
x x x
Radiance = Power / (solid angle x foreshortened area)
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Radiance
The medium is vacuum, that is, it does not absorb energy. Therefore, the power reaching point x2 is equal to the power leaving for x2 from x1.
Power at x2 from direction x1 is
1 1 2 2 2 1 12
( , ) cos cosL dA dAd
r
x x x
Let the radiance arriving at x2 from the direction of x1 is
Source
Illuminated surface
2
2 1 22 2 1 1 2 2
( , )cos cos cos
d r dL
dw dA dA dA
x x x
1 12
cosdAdw
r
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Radiance
Radiance is constant along a straight line.
1 1 2 2 1 2( , ) ( , )L L x x x x x x
Source
Illuminated surface
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Point Source
Many light sources are physically small compared with the environment in which they stand.
Such a light source is approximated as an extremely small sphere, in fact, a point.
Such a light source is known as a point source.
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Radiance Intensity
If the source is a point source, we use radiance intensity.
2
2 2cos
d r dI
dw dA
2 22
cosdAdw
r
Radiance intensity = Power / (solid angle)
Illuminated surface
Source
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Light at Surfaces
When light strikes a surface, it may be absorbed, transmitted, or scattered; usually, combination of these effects occur.
It is common to assume that all effects are local and can be explained with a local interaction model. In this model:
The radiance leaving a point on a surface is due only to radiance arriving at this point.
Surfaces do not generate light internally and treat sources separately.
Light leaving a surface at a given wavelength is due to light arriving at that wavelength.
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Light at Surfaces
In the local interaction model, fluorescence, [absorb light at one wavelength and then radiate light at a different wavelength], and emission [e.g., warm surfaces emits light in the visible range] are neglected.
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Irradiance
Irradiance is the total incident power per unit area.
Irradiance = Power / Area
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Irradiance
What is the irradiance due to source from angle ?
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Irradiance
( , , ) cos( , , ) cosi i i ii i i i
L x dw dAdIrradiance L x dw
dA dA
dA
What is the irradiance due to source from angle ?
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Irradiance
What is the total irradiance?
Integrate over the whole hemisphere.
Exercise: Suppose the radiance is constant from all directions. Calculate the irradiance.
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Irradiance
Exercise: Calculate the irradiance at O due to a plate source at O’.
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Irradiance due to a Point Source
For a point source,
2
cosi i
d r dI
dw dA
2
cosi idAdw
r
2
cosi idAd I
r
2
cos i
i
dIrradiance I
dA r
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The Relationship Between Image Intensity and Object Radiance
We assume that there is no power loss in the lens.
The power emitted to the lens is
0 0cosobjectd L dA dw
Diameter of lens
Radiance of object
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The Relationship Between Image Intensity and Object Radiance
The solid angle for the entire lens is
The power emitted to the lens is
0 0cosobjectd L dA dw
Diameter of lens 2
0 2
/ 4 cosddw
r
2
0 2
coscos
4object
dL dA
r
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The Relationship Between Image Intensity and Object Radiance Diameter of
lensThe solid angle at O can be written in two ways.
022
coscos
'
pdAdA
r OA
' / cosOA f
Note that
3
02 2
coscos pdAdA
r f
Therefore
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The Relationship Between Image Intensity and Object Radiance Diameter of
lensCombine
3
02 2
coscos pdAdA
r f
to get
2
0 2
coscos
4object
dd L dA
r
2
4cos4 object p
dd L dA
f
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The Relationship Between Image Intensity and Object Radiance Diameter of
lensTherefore the irradiance on the image plane is
2
4cos4 object
p
d dIrradiance L
dA f
The irradiance is converted to pixel intensities, which is directly proportional to the radiance of the object.
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Surface Characteristics
We want to describe the relationship between incoming light and reflected light.
This is a function of both the direction in which light arrives at a surface and the direction in which it leaves.
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Bidirectional Reflectance Distribution Function (BRDF)
BRDF is defined as the ratio of the radiance in the outgoing direction to the incident irradiance.
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Bidirectional Reflectance Distribution Function (BRDF)
The radiance leaving a surface due to irradiance in a particular direction is easily obtained from the definition of BRDF:
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Bidirectional Reflectance Distribution Function (BRDF)
The radiance leaving a surface due to irradiance in all incoming directions is
where Omega is the incoming hemisphere.
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Lambertian Surface
A Lambertian surface has constant BRDF.
constant
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Lambertian Surface
A Lambertian surface looks equally bright from any view direction.
The image intensities of the surface only changes with the illumination directions.
constant
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Lambertian Surface
For a Lambertian surface, the outgoing radiance is proportional to the incident radiance.
If the light source is a point source, a pixel intensity will only be a function of
constant
2
cos i
i
dIrradiance I
dA r
Remember, for a point source
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Specular Surface
The glossy or mirror like surfaces are called specular surfaces.
Radiation arriving along a particular direction can only leave along the specular direction, obtained from the surface normal.
*The term Specular comes from the Latin word speculum, meaning mirror.
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Specular Surface
Few surfaces are ideally specular. Specular surfaces commonly reflect light into a lobe of directions around the specular direction.
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Lambertian + Specular Model
Relatively few surfaces are either ideal diffuse or perfectly specular.
The BRDF of many surfaces can be approximated as a combination of a Lambertian component and a specular component.
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Lambertian + Specular Model
Lambertian Lambertian + Specular
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Radiosity
Radiosity, defined as the total power leaving a point. To obtain the radiosity of a surface at a point, we can
sum the radiance leaving the surface at that point over the whole hemisphere.
Part II
Shading
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Point Source
For a point source,
2
cosi i
d r dI
dw dA
2
cosi idAdw
r
2
cosi idAd I
r
2
cos i
i
dIrradiance I
dA r
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A Point Source at Infinity
The radiosity due to a point source at infinity is
( )xN( )xS
x
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Local Shading Models for Point Sources
The radiosity due to light generated by a set of point sources is
Radiosity due to source s
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Local Shading Models for Point Sources
If all the sources are point sources at infinity, then
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Ambient Illumination
For some environments, the total irradiance a patch obtains from other patches is roughly constant and roughly uniformly distributed across the input hemisphere.
In such an environment, it is possible to model the effect of other patches by adding an ambient illumination term to each patch’s radiosity.
+ B0
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Photometric Stereo
If we are given a set of images of the same scene taken under different given lighting sources, can we recover the 3D shape of the scene?
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Photometric Stereo
For a point source and a Lambertian surface, we can write the image intensity as
Suppose we are given the intensities under three lighting conditions:
Camera and object are fixed, so a particular pixel intensity is only a function of lighting direction si.
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Photometric Stereo
Stack the pixel intensities to get a vector
The surface normal can be found as
Since n is a unit vector
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Photometric Stereo
If we have more than three sources, we can find the least squares estimate using the pseudo inverse:
As a result, we can find the surface normal of each point, hence the 3D shape
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Photometric Stereo
When the source directions are not given, they can be estimated from three known surface normals.
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Photometric Stereo
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Photometric Stereo
Surface normals 3D shape
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Photometric Stereo
(by Xiaochun Cao)
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Photometric Stereo (by Xiaochun Cao)