Rigid Transformations and Manifolds: Understanding SO(2) and SO(3), Study Guides, Projects, Research of Algebra

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Institute of Visual Computing

Rigid Transformations

--- the geometry of SO(3) & SE(3) ---

Luca Ballan

Mathematical Foundations of

Computer Graphics and Vision

Motivation

it is very thin!!

( unconstrained minimization problem )

( unconstrained minimization problem with functions as domain)

( constrained minimization problem )

 Rigid Registration

 Camera pose estimation

 Input: two images (with known intrinsics)  Compute correspondences between these images  Estimate the essential matrix  Factorize E in (R,t)  Compute the 3D structure  Bundle-Adjustment

Motivation

Motivation

 The trajectory of a rigid object

is a (smooth) curve in

 3D Rigid Object or Camera Tracking

Content

 Rigid transformations

 Linear Matrix Groups

 Manifolds

 Lie Groups/Lie Algebras

 Charts on SO(2) and SO(3)

is a transformation

Rigid Transformations

Taxonomy

Affine maps (^) Conformal maps

Isometry

Rigid

Affine + Conformal

Linear

Isometries which does not preserve the orientation

Representation

if A is a finite dimensional space (e.g. )

a rigid transformation

can be written as

 R orthogonal (isometry)  (preserve orientation)

Projective space

Note: in this space, F is also linear

Rotation matrix

Content

 Rigid transformations

 Matrix Groups

 Manifolds

 Lie Groups/Lie Algebras

 Charts on SO(2) and SO(3)

Matrix Groups

 The set of all the nxn invertible matrices is a group w.r.t. the matrix multiplication

 GL(n) is isomorphic to the group of linear and invertible transformations in with the composition as operation

General linear group

 It exists an isomorphism , such that

Matrix Groups

 The set of all the nxn orthogonal matrices with determinant equal to 1 is a group w.r.t. the matrix multiplication

 SO(n) is isomorphic to the group of linear rigid transformations in with the composition as operation

Special orthogonal group

 It exists an isomorphism , such that

Affine maps (^) Conformal maps

Isometry Rigid

Affine + Conformal

Linear

Groups of Matrices: Summary

General linear group of order n

GL(n) O(n)

SO(n)

O(n)/SO(n)

= vector space of all the nxn matrices

Orthogonal group of order n

Special orthogonal group of order n Set of orthogonal matrices which do not preserve orientation (not a group)

Special Euclidean group

 The Cartesian product is a group w.r.t. a “weird” operation

 The “weird” operation is define in such a way that the group SE(n) is isomorphic to the group of rigid transformations in with the composition as operation

 It exists an isomorphism , such that

Special Euclidean group

Commutative??

The Geometry of these Groups

 GL(n), O(n), SO(n) and SE(n) are all subset of a vector space

 GL(n), O(n), SO(n) and SE(n) are all smooth manifolds (surfaces, curves, solids, etc... immerse in some big vector space)

GL(n) O(n) SO(n) O(n)/SO(n)

SE(n)