MASKS © 2004 Invitation to 3D vision

Lecture 6Lecture 6

Introduction to Algebra & Rigid-Body Introduction to Algebra & Rigid-Body MotionMotion

Allen Y. YangAllen Y. YangSeptember 18September 18thth, 2006, 2006

MASKS © 2004 Invitation to 3D vision

Outline

• Euclidean space1. Points and Vectors2. Cross products3. Singular value decomposition (SVD)

• Rigid-body motion1. Euclidean transformation2. Representation3. Canonical exponential coordinates4. 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

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:

Linear space

MASKS © 2004 Invitation to 3D vision

Change of basis

Summary:

MASKS © 2004 Invitation to 3D vision

Cross product

• Properties:

• Pop quiz:• Homework:

0ˆ =xyxT

Cross product between two vectors:

MASKS © 2004 Invitation to 3D vision

Rank

Pop Quiz: R is a rotation matrix, T is nontrivial. rank( )=?RT̂

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 motions preserve distances, angles, and orientations.

Goal: finding representation of SE(3). Translation T Rotation R

Rigid-Body Motion

MASKS © 2004 Invitation to 3D vision

Orthogonal change of coordinates

Collect coordinates of one reference framerelative to the other into a matrix R

Rotation

MASKS © 2004 Invitation to 3D vision

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

Degree of Freedom (DOF)

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

Velocity transformations Given

Twist coordinates

MASKS © 2004 Invitation to 3D vision

Summary

MASKS © 2004 Invitation to 3D vision

We will prove this if we have time