Transcript Lecture 6 Introduction to Algebra & Rigid-Body Motion Allen Y. Yang September 18th, 2006 MASKS © 2004 Invitation to 3D vision.

Lecture 6
Introduction to Algebra & Rigid-Body Motion
Allen Y. Yang
September 18th, 2006
MASKS © 2004
Invitation to 3D vision
Outline
• Euclidean space
1. Points and Vectors
2. Cross products
3. Singular value decomposition (SVD)
• Rigid-body motion
1. Euclidean transformation
2. Representation
3. Canonical exponential coordinates
4. Velocity transformations
MASKS © 2004
Invitation to 3D vision
Euclidean space

Points and vectors are different!

Bound vector & free vector:
MASKS © 2004
Invitation to 3D vision
Linear space

The set of all free vectors, V, forms a linear space over
the field R. (points don’t)

Closed under “+” and “*”

V is completely determined by a basis, B:

Change of basis:
MASKS © 2004
Invitation to 3D vision
Change of basis
Summary:
MASKS © 2004
Invitation to 3D vision
Cross product
Cross product between two vectors:
• Properties:
• Pop quiz:
x yˆ x  0
T
• Homework:
MASKS © 2004
Invitation to 3D vision
Rank
Pop Quiz: R is a rotation matrix, T is nontrivial.
rank( TˆR )=?
MASKS © 2004
Invitation to 3D vision
Singular Value Decomposition (SVD)
MASKS © 2004
Invitation to 3D vision
Fixed-Rank Approximation
MASKS © 2004
Invitation to 3D vision
Geometric Interpretation
A
MASKS © 2004
Invitation to 3D vision
Rigid-Body Motion


To describe an object movement, one should specify the
trajectory of all points on the object.
For rigid-body objects, it is sufficient to specify the motion of
one point, and the local coordinate axes attached at it.
MASKS © 2004
Invitation to 3D vision
Rigid-Body Motion

Rigid-body motions preserve distances, angles, and orientations.

Goal: finding representation of SE(3).


Translation T
Rotation R
MASKS © 2004
Invitation to 3D vision
Rotation
Orthogonal change of coordinates
Collect coordinates of one reference frame
relative to the other into a matrix R
MASKS © 2004
Invitation to 3D vision
Degree of Freedom (DOF)




Translation T has 3 DOF
.
Rotation R has 3 DOF. Can be specified by three space angles.
Summary: R in SO(3) has 3 DOF. g in SE(3) has 6 DOF.
Homogeneous representation
MASKS © 2004
Invitation to 3D vision
Homogeneous representation (summary)

Points

Vectors

Transformation

Representation
MASKS © 2004
Invitation to 3D vision
Canonical Exponential Coordinates
MASKS © 2004
Invitation to 3D vision
Canonical Exponential Coordinates



One such solution:
Yet the solution is NOT unique!
when w is a unit vector.
Multiplication:
MASKS © 2004
Invitation to 3D vision
Canonical Exponential Coordinates
Canonical exponential coordinates for rigid-body motions.
Similar to rotation:

(twist)

Hence,
MASKS © 2004
Invitation to 3D vision
Canonical Exponential Coordinates
Twist coordinates
Velocity transformations Given
MASKS © 2004
Invitation to 3D vision
Summary
MASKS © 2004
Invitation to 3D vision
We will prove this if we have time
MASKS © 2004
Invitation to 3D vision