Computer Vision
Radiometry
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.
Bahadir K. Gunturk
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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
d
r
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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.
dA  rd r sin  d  r 2 sin  d d
dA
2
TotalArea 

 
r 2 sin  d d  4 r 2
0 0
dA
dw  2  sin  d d
r
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Solid Angle

Similarly, solid angle due to a line segment is

dl
d
r
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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/m2
W 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
Radiance = Power / (solid angle x foreshortened area)
Radiance at x1 leaving to x2
Illuminated
surface
d
r 2 d
L(x1 , x1  x 2 ) 

dw cos1dA1 dA2 cos 2 cos1dA1
dw 
Source
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dA2 cos  2
r2
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Radiance
Radiance = Power / (solid angle x foreshortened area)
Power at x1 leaving to x2
Illuminated
surface
d  L(x1 , x1  x2 )dw cos1dA1

Source
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L(x1 , x1  x 2 )dA2 cos  2 cos 1dA1
r2
dw 
dA2 cos  2
r2
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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
L(x1 , x1  x 2 )dA2 cos  2 cos 1dA1
d 
2
r
Illuminated
surface
Source
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Let the radiance arriving at x2
from the direction of x1 is
d
r 2 d
L(x 2 , x1  x 2 ) 

dw cos 2 dA2 dA1 cos1 cos 2 dA2
dw 
dA1 cos1
r2
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Radiance

Radiance is constant along a straight line.
Illuminated
surface
L(x1 , x1  x2 )  L(x2 , x1  x2 )
Source
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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.
Radiance intensity = Power / (solid angle)
Illuminated
surface
d
r 2 d
I

dw dA2 cos 2
dw 
dA2 cos  2
r2
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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?
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Irradiance

What is the irradiance due to source from angle
?
dA
Irradiance 
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d Li ( x, i , i )dw cos i dA

 Li ( x, i , i )dw cos i
dA
dA
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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,
d
r 2 d
I

dw dAi cos i
dw 
dAi cos  i
d  I
r2
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dAi cos  i
r2
cos i
d
Irradiance 
I 2
dAi
r
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The Relationship Between Image Intensity
and Object Radiance
Diameter of
lens
We assume that there is no power
loss in the lens.
The power emitted to the lens is
d  Lobject dA0 cos  dw0
Radiance of object
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The Relationship Between Image Intensity
and Object Radiance
Diameter of
lens
The solid angle for the entire lens is
dw0
d


2
/ 4  cos
r2
The power emitted to the lens is
d  Lobject dA0 cos  dw0
 d 2 cos
 Lobject dA0 cos 
4r 2
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The Relationship Between Image Intensity
and Object Radiance
Diameter of
lens
The solid angle at O can be written in
two ways.
dA0 cos  dAp cos

2
2
r
OA '
Note that
OA '  f / cos
Therefore
3
dA0 cos  dAp cos 

2
r
f2
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The Relationship Between Image Intensity
and Object Radiance
Diameter of
lens
Combine
3
dA0 cos  dAp cos 

2
r
f2
d  Lobject dA0 cos 
 d 2 cos
4r 2
to get
2
d
 
d    Lobject   cos 4  dAp
4
f
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The Relationship Between Image Intensity
and Object Radiance
Diameter of
lens
Therefore the irradiance on the
image plane is
2
d
d   
Irradiance 
   Lobject   cos 4 
dAp  4 
 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.
constant

If the light source is a point source, a pixel intensity
will only be a function of
Remember, for a point source
Bahadir K. Gunturk
Irradiance 
cos
d
I 2 i
dAi
r
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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
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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.
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Part II
Shading
Point Source

For a point source,
d
r 2 d
I

dw dAi cos i
dw 
dAi cos  i
d  I
r2
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dAi cos  i
r2
cos i
d
Irradiance 
I 2
dAi
r
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A Point Source at Infinity

The radiosity due to a point source at infinity is
S( x)
N( x)
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
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3D shape
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Photometric Stereo
(by Xiaochun Cao)
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Photometric Stereo
Bahadir K. Gunturk
(by Xiaochun Cao)
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