Lie Groups and Algebras for optimisation and …cmei/talks/reading_group_lie.pdfDefinitions...
Transcript of 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
DefinitionsRepresenting motion and geometric objects
InterpolationMinimisationUncertainty
Why use Lie Groups ?
Some uses...Interpolation
Motion representation
General theory for the minimal representation of geometricobjects
Representation of PDFs over groups
AVL/MRG Reading Group Lie Groups 2008 - 2/27
DefinitionsRepresenting motion and geometric objects
InterpolationMinimisationUncertainty
Outline
1 Definitions
2 Representing motion and geometric objects
3 Interpolation
4 Minimisation
5 Uncertainty
AVL/MRG Reading Group Lie Groups 2008 - 3/27
DefinitionsRepresenting motion and geometric objects
InterpolationMinimisationUncertainty
Outline
1 Definitions
2 Representing motion and geometric objects
3 Interpolation
4 Minimisation
5 Uncertainty
AVL/MRG Reading Group Lie Groups 2008 - 4/27
DefinitionsRepresenting motion and geometric objects
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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
AVL/MRG Reading Group Lie Groups 2008 - 5/27
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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.
AVL/MRG Reading Group Lie Groups 2008 - 6/27
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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
AVL/MRG Reading Group Lie Groups 2008 - 7/27
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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)
AVL/MRG Reading Group Lie Groups 2008 - 8/27
DefinitionsRepresenting motion and geometric objects
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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)
AVL/MRG Reading Group Lie Groups 2008 - 9/27
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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.
AVL/MRG Reading Group Lie Groups 2008 - 10/27
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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).
AVL/MRG Reading Group Lie Groups 2008 - 11/27
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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
AVL/MRG Reading Group Lie Groups 2008 - 12/27
DefinitionsRepresenting motion and geometric objects
InterpolationMinimisationUncertainty
Outline
1 Definitions
2 Representing motion and geometric objects
3 Interpolation
4 Minimisation
5 Uncertainty
AVL/MRG Reading Group Lie Groups 2008 - 13/27
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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}
AVL/MRG Reading Group Lie Groups 2008 - 14/27
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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]×)
AVL/MRG Reading Group Lie Groups 2008 - 15/27
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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
AVL/MRG Reading Group Lie Groups 2008 - 16/27
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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]
AVL/MRG Reading Group Lie Groups 2008 - 17/27
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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
AVL/MRG Reading Group Lie Groups 2008 - 18/27
DefinitionsRepresenting motion and geometric objects
InterpolationMinimisationUncertainty
Outline
1 Definitions
2 Representing motion and geometric objects
3 Interpolation
4 Minimisation
5 Uncertainty
AVL/MRG Reading Group Lie Groups 2008 - 19/27
DefinitionsRepresenting motion and geometric objects
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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.
AVL/MRG Reading Group Lie Groups 2008 - 20/27
DefinitionsRepresenting motion and geometric objects
InterpolationMinimisationUncertainty
Outline
1 Definitions
2 Representing motion and geometric objects
3 Interpolation
4 Minimisation
5 Uncertainty
AVL/MRG Reading Group Lie Groups 2008 - 21/27
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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)
AVL/MRG Reading Group Lie Groups 2008 - 22/27
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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...).
AVL/MRG Reading Group Lie Groups 2008 - 23/27
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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
AVL/MRG Reading Group Lie Groups 2008 - 24/27
DefinitionsRepresenting motion and geometric objects
InterpolationMinimisationUncertainty
Outline
1 Definitions
2 Representing motion and geometric objects
3 Interpolation
4 Minimisation
5 Uncertainty
AVL/MRG Reading Group Lie Groups 2008 - 25/27
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BCF
Baker-Campbell-Hausdorff formula
Solution to Z = log (eXeY) :
Z = X + Y +12
[X, Y] +112
[X, [X, Y]]−112
[Y, [X, Y]] + . . .
AVL/MRG Reading Group Lie Groups 2008 - 26/27
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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.
AVL/MRG Reading Group Lie Groups 2008 - 27/27