ECE 549CS 543: COMPUTER VISON LECTURE 4

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ECE 549/CS 543 COMPUTER VISON LECTURE
4 CAMERAS AND CAMERA MODELS III

• Analytical Euclidean Geometry
• The Intrinsic Parameters of a Camera
• The Extrinsic Parameters of a Camera
• The Calibration Problem

Reading Chapters 2 and 3 Slides http//www-cvr.a
i.uiuc.edu/ponce/fall04/lect4.ppt

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Coordinate Changes Pure Translations
OBP OBOA OAP , BP AP BOA
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Coordinate Changes Pure Rotations
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Coordinate Changes Rotations about the z Axis
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A rotation matrix is characterized by the
following properties

• Its inverse is equal to its transpose, and
• its determinant is equal to 1.

Or equivalently

• Its rows (or columns) form a right-handed
• orthonormal coordinate system.

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Coordinate Changes Pure Rotations
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Coordinate Changes Rigid Transformations
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Block Matrix Multiplication
What is AB ?
Homogeneous Representation of Rigid
Transformations
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Rigid Transformations as Mappings
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Rigid Transformations as Mappings Rotation about
the k Axis
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The Intrinsic Parameters of a Camera
Units
k,l pixel/m
f m
a,b
pixel
Physical Image Coordinates
Normalized Image Coordinates
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The Intrinsic Parameters of a Camera
Calibration Matrix
The Perspective Projection Equation
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The Extrinsic Parameters of a Camera
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Explicit Form of the Projection Matrix
Note
M is only defined up to scale in this setting!!
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Theorem (Faugeras, 1993)
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Calibration Problem
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Linear Systems
Square system
A
x
b

• unique solution
• Gaussian elimination

Rectangular system ??

• underconstrained
• infinity of solutions

A
x
b

• overconstrained
• no solution

Minimize Ax-b
2
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How do you solve overconstrained linear equations
??
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Homogeneous Linear Systems
Square system
A
x
0

• unique solution 0
• unless Det(A)0

Rectangular system ??

• 0 is always a solution

A
x
0

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Minimize Ax under the constraint x 1
2
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How do you solve overconstrained homogeneous
linear equations ??
The solution is e .
1
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Example Line Fitting
Problem minimize with respect to (a,b,d).

• Minimize E with respect to d

n

• Minimize E with respect to a,b

where

• Done !!