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Chapter 7
Projective Geometry and
Camera Models
James Hays, Brown University
Department of Mechatronics
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Contents
Mapping between image and world coordinates
• Projective geometry
Vanishing points and lines
• Pinhole camera model
• Cameras & lenses
• Projection matrix
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Camera and World Geometry
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Projection can be tricky…
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Projection can be tricky…
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Projective Geometry
• What is lost?
Length
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Length is not preserved
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Projective Geometry
• What is lost?
Length
Angles
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Projective Geometry
• What is preserved?
Straight lines are still straight
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Vanishing points and lines
• Parallel lines in the world intersect in the image at a
“vanishing point”.
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Vanishing points and lines
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Vanishing points and lines
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Vanishing points and lines
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Note on estimating vanishing points
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How do we see the world
Graphical representation of the eye looking at a palm tree. Point
C is the optical center of the lens.
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Camera obscura: the pre-camera
• Known during classical period in China and Greece
(e.g. Mo-Ti, China, 470BC to 390BC)
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Lens Based Camera Obscura, 1568
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Parallel light rays which pass through a small aperture
begin to diverge and interfere with one another. This
becomes more significant as the size of the aperture
decreases relative to the wavelength of light passing
through, but occurs to some extent for any size of
aperture or concentrated light source.
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Cameras and Lenses
Adding Lenses!
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Cameras and Lenses
• A lens focuses light onto the film
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Cameras and Lenses
• A lens focuses light onto the film
Rays passing through the center are not deviated.
All parallel rays converge to one point on a plane
located at the focal length f.
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Cameras and Lenses
• A lens focuses light onto the film
There is a specific distance at which objects are “in
focus” [other points project to a “circle of confusion”
in the image].
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In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in
that medium
n = speed of light in a vacuum / speed of light in medium
http://en.wikipedia.org/wiki/Refractive_index#Typical_values
Cameras and Lenses
• Laws of geometric optics:
Light travels in straight lines in homogeneous medium.
Reflection upon a surface: incoming ray, surface normal,
and reflection are co-planar.
Refraction: when a ray passes from one medium to
another.
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Large (top) and small (bottom) apertures
The aperture is not independent, it
must be closely matched to the
focal length to get the best lighting
effect.
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Field of View (Zoom, focal length)
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This simulation shows how adjusting the angle of
view of a camera, while varying the camera distance,
keeping the object in frame, results in vastly differing
images. At narrow angles, large distances, light rays
are nearly parallel, resulting in a "flattened" image. At
wide angles, short distances, the object appears
distorted.
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Mathematically, for a linear system, F, defined by F(x) = y, where x is some sort of stimulus (input) and
y is some sort of response (output), the superposition (i.e., sum) of stimuli yields a superposition of
the respective responses:
F ( x1 + x2 + ...)
= F ( x1 ) + F ( x2 ) + ...
In the field of electrical engineering, where the x and y signals are allowed to be complex-valued (as
is common in signal processing), a linear system must satisfy the superposition property, which
requires the system to be additive and homogeneous
F(x1 + x2) = F(x1) + F(x2) F(ax) = aF(x)
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Projection
World coordinates Image coordinates
X
Optical . P = Y
Center Z
(u0, v0)
. f
. Z Y
v
Camera
.u
Center
(tx, ty,
u tz)
p=
v
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Homogeneous coordinates
Conversion
Converting to homogeneous coordinates
homogeneous image homogeneous scene
coordinates coordinates
Converting from homogeneous coordinates
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Finally, the camera coordinate system may be skewed due to manufacturing error, so that angle θ
between two image axes is not equal to 90º.
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