MASKS © 2004 Invitation to 3D vision Lecture 6 Introduction to Algebra & Rigid-Body Motion Allen Y. Yang September 18 th, 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 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: Cross product between two vectors:

MASKS © 2004 Invitation to 3D vision Rank Pop Quiz: R is a rotation matrix, T is nontrivial. rank( )=?

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 frame relative 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