Lie Groups and Algebras for optimisation and …cmei/talks/reading_group_lie.pdfDeﬁnitions...

DefinitionsRepresenting motion and geometric objects

InterpolationMinimisationUncertainty

Lie Groups and Algebras for optimisation andmotion representation

AVL/MRG Reading Group

Tuesday 6th May 2008

Chapter 2, An invitation to 3D vision, Ma & al

Chapter 5, PhD Mei 2007

Computing MAP trajectories by representing, propagatingand combining PDFs over groups, Smith & al, ICCV 2003

AVL/MRG Reading Group Lie Groups 2008 - 1/27



Why use Lie Groups ?

Some uses...Interpolation

Motion representation

General theory for the minimal representation of geometricobjects

Representation of PDFs over groups




Outline

1 Definitions

2 Representing motion and geometric objects

3 Interpolation

4 Minimisation

5 Uncertainty




Outline

1 Definitions


3 Interpolation

4 Minimisation

5 Uncertainty




Matrix Lie group (1/2)

Properties of a group (G, ◦) :

closed : (g1, g2) ∈ G2 ⇒ g1 ◦ g2 ∈ G,

associative :∀(g1, g2, g3) ∈ G3, (g1 ◦ g2) ◦ g3 = g1 ◦ (g2 ◦ g3),

has a neutral (unit) element e : ∀g ∈ G3, e ◦ g = g ◦ e = g,

◦ is invertible : ∀g ∈ G,∃g−1 ∈ G|g ◦ g−1 = g−1 ◦ g = e




Matrix Lie group (2/2)

Lie group (G, ◦)

(G, ◦) is a group,

G is a smooth manifold, ie has the topology of Rn, (the

inverse function is differentiable everywhere)

All closed subgroups of the general linear group GL(n) (groupof all invertible matrices) are Lie groups.




Matrix exponential (1/2)

eA

eA = In +∑

p≥1

Ap

p!=

∑

p≥0

Ap

p!, beware : eXeY 6= eX+Y

This series is absolutely convergent and thus well-defined.

log A

Under the condition ‖A− I‖ < 1, the logarithm of A isdefined as :

log A =∑

p≥0

(−1)p+1 (A− I)p

p




Matrix exponential (2/2)

Calculating eA in practice

explicit formulas (eg. Rodrigues’ formula for SO(3)). Ageneral way of finding explicit formulas is to use theCayley–Hamilton theorem.

diagonalisation (not generally a good idea),

Nineteen dubious ways to compute the exponential of amatrix, Moler and Loan, 1978 (or 2003)

The scaling and squaring method for the matrixexponential revisited, N. Higham, 2005 (expm in Matlab)


expm



Lie algebra

Lie algebra g of the Lie group G

The set of all matrices X such that etX is in G for all realnumbers t.

g is an algebra (vector space+ ring)

Real vector space∀t , tX ∈ GX + Y ∈ G

[X, Y] = XY− YX ∈ G (Lie bracket)




Lie groups and algebras

Exponential map

If G is a matrix Lie group with Lie algebra g, then theexponential mapping for G is the map :

exp : g→ G

In general the mapping is neither one-to-one nor onto butprovides the link between the group and the Lie algebra.

There exists a neighborhood v about zero in g and aneighborhood V of I in G such that exp : v → V is smooth andone-to-one onto with smooth inverse.




Paths

Path-connectedness

G is path-connected if given any two matrices A and B in G,there exists a continuous path A(t), a ≤ t ≤ b, lying in G withA(a) = A and A(b) = B.

SO(n), SL(n) and SE(n) are connected (O(n) is not).




Generators

Generators

Let g(ti) = exp(tiAi) define a subgroup of G, then

Ai =∂g(ti)

∂ti

∣∣∣∣ti=0

is a generator of g.

The set of generators form a basis and any element x ∈ g

can be written :

A(x) =n∑

i=1

xiAi




Outline

1 Definitions


3 Interpolation

4 Minimisation

5 Uncertainty




Special Orthogonal Group

[det(eA) = etrace(A)

]

SO(3) = {R ∈ GL(3) | RR⊤ = I, det(R) = +1}

preserves orientation (not a reflexion)

Associated Lie algebra :

so(3) = {[ω]× ∈ R3×3|ω ∈ R

3}




Lie algebra representation and Euler angles

The Lie algebra representation :

(x1, x2, x3) 7−→ exp(x1 [e1]× + x2 [e2]× + x3 [e3]×)

Euler angles :

(x1, x2, x3) 7−→ exp(x1 [e1]×) exp(x2 [e2]×) exp(x3 [e3]×)




Special Euclidean Group

SE(3) =

{[R t0 1

]∈ GL(4) | R ∈ SO(3), t ∈ R

3}

preserves distances

preserves orientation (not a reflexion)

Associated Lie algebra twist :

se(3) = {

[[ω]× v

0 0

]|ω, v ∈ R

3}

v is the linear velocity

ω is the angular velocity




Expressing velocity

Velocity of a point in homogeneous coordinates (x(t) ∈ se(3)) :

X(t) = x(t)X(t)

If Y(t) = TX(t) with T ∈ SE(3) (change of coordinates) :

Y(t) = Tx(t)T−1Y(t)

Adjoint map on se(3) :

AdT : se(3) −→ se(3)

x 7−→ TxT−1

Adjoint representation of se(3) (eadX = AdeX) :

adX : se(3) −→ se(3)Y 7−→ [X, Y]




Another example : Special Linear Group

SL(3) = {H ∈ GL(3) | det(H) = +1}

ensures an invertible matrix with a minimal amount ofparameters,

subgroups include affine transforms or translations that aredirectly obtained by choosing the correct generators

This representation for a homography leads to “better” resultsthan the “standard” minimal representation :

H =

h1 h2 h3

h4 h5 h6

h7 h8 1




Outline

1 Definitions


3 Interpolation

4 Minimisation

5 Uncertainty




Interpolation

Interpolation

Let T1 = ex1 ∈ SE(3) and T2 = ex2 ∈ SE(3), a smooth trajectorycan be obtained as T(x) = eλx1+(1−λ)x2 with λ = 0..1.




Outline

1 Definitions


3 Interpolation

4 Minimisation

5 Uncertainty




A generic minimisation problem...

Let :f : G −→ R

g 7−→ f (g)

We want to solve, with f ∈ R :

g = ming

d(f (g), f)

Gradient descent update :

g← g + gk

g has no reason to still belong to G ! ! ! (eg. rotation)




Using Lie algebras...

h : Rn −→ g −→ R

x 7−→ G(x) 7−→ f (g ◦ eG(x))

The parameterisation only needs to be valid locally.New update :

g← g ◦ eG(xk )

g is guaranteed to still belong to the group.Important condition : the initial value and optimal value have tobe path-connected (in the case of O, there are twocomponents...).




Example...

Pose estimation (Lu et al.) :

minx,tx

n∑

i=1

‖(I−Qi) (R(x)Rpi + t + tx ) ‖2

Jacobians :

∇x fi = (I−Qi)[

[e1]× [e2]× [e3]×]

3×3×3 Rpi

∇tx fi = (I−Qi)

Rk+1 ← R(x)Rk

tk+1 ← tk + tx




Outline

1 Definitions


3 Interpolation

4 Minimisation

5 Uncertainty




BCF

Baker-Campbell-Hausdorff formula

Solution to Z = log (eXeY) :

Z = X + Y +12

[X, Y] +112

[X, [X, Y]]−112

[Y, [X, Y]] + . . .




Further reading

Geometric Means in a Novel Vector Space Structure onSymmetric Positive-Definite Matrices, V. Arsigny et al.,SIAM Journal on Matrix Analysis and Applications, 2006.

Processing Data in Lie Groups : An Algebraic Approach.Application to Non-Linear Registration and Difusion TensorMRI., V. Arsigny, PhD, 2006.

An Elementary Introduction to Groups andRepresentations, Brian C. Hall.


Lie Groups and Algebras for optimisation and …cmei/talks/reading_group_lie.pdfDeﬁnitions...

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