shape analysis on lie groups and homogeneous manifolds ... · shape analysis on lie groups and...
TRANSCRIPT
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Shape analysis on Lie groups and HomogeneousManifolds with applications in computer
animation
Elena Celledoni
Department of Mathematical Sciences, NTNU, Trondheim, Norwayjoint work with Markus Eslitzbichler and Alexander Schmeding
Geometric Numerical IntegrationOberwolfach, March 21st 2016
Elena Celledoni Geometric animation of character motion
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Outline
• Analysis of shapes in vector spaces.
• Character animation and skeletal animation.
• Analysis of shapes on Lie groups.
• Applications in:• curve blending,• projection from open curves to closed curves, and• distances between curves.
• Examples in computer animation.
Elena Celledoni Geometric animation of character motion
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Analysis of shapes
Shapes are unparametrized curves in a vector space or on amanifold.
Example: object recognition, objects can be represented by theircontours, i.e. closed planar curves.
Image recognition: a tennis player a tennis racket and a ball.
Elena Celledoni Geometric animation of character motion
![Page 4: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/4.jpg)
Analysis of shapes
Shapes are unparametrized curves in a vector space or on amanifold.
Example: object recognition, objects can be represented by theircontours, i.e. closed planar curves.
Image recognition: a tennis player a tennis racket and a ball.
Elena Celledoni Geometric animation of character motion
![Page 5: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/5.jpg)
Analysis of shapes
Shapes are unparametrized curves in a vector space or on amanifold.
Example: object recognition, objects can be represented by theircontours, i.e. closed planar curves.
Image recognition: a tennis player a tennis racket and a ball.
Elena Celledoni Geometric animation of character motion
![Page 6: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/6.jpg)
Shapes
Shapes are unparametrized curves in a vector space or on amanifold.
Definition of shapes via an equivalence relation: let I ⊂ R aninterval, consider
P ∶= Imm(I,M) = {c ∈ C∞(I,M) ∣ c(t) ≠ 0},P is called pre-shape space (an infinite dimensional manifold).Let c0, c1 ∈ P then
c0 ∼ c1 ⇐⇒ ∃ ϕ ∶ c0 = c1 ○ ϕwith ϕ ∈ Diff+(I) a orientation preserving diffeomorphism on I
shape:[c0]
Shape space:S ∶= Imm(I,M)/ ∼
Elena Celledoni Geometric animation of character motion
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Shapes
Shapes are unparametrized curves in a vector space or on amanifold.
Definition of shapes via an equivalence relation: let I ⊂ R aninterval, consider
P ∶= Imm(I,M) = {c ∈ C∞(I,M) ∣ c(t) ≠ 0},P is called pre-shape space (an infinite dimensional manifold).Let c0, c1 ∈ P then
c0 ∼ c1 ⇐⇒ ∃ ϕ ∶ c0 = c1 ○ ϕwith ϕ ∈ Diff+(I) a orientation preserving diffeomorphism on I
shape:[c0]
Shape space:S ∶= Imm(I,M)/ ∼
Elena Celledoni Geometric animation of character motion
![Page 8: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/8.jpg)
Shapes
Shapes are unparametrized curves in a vector space or on amanifold.
Definition of shapes via an equivalence relation: let I ⊂ R aninterval, consider
P ∶= Imm(I,M) = {c ∈ C∞(I,M) ∣ c(t) ≠ 0},P is called pre-shape space (an infinite dimensional manifold).Let c0, c1 ∈ P then
c0 ∼ c1 ⇐⇒ ∃ ϕ ∶ c0 = c1 ○ ϕwith ϕ ∈ Diff+(I) a orientation preserving diffeomorphism on I
shape:[c0]
Shape space:S ∶= Imm(I,M)/ ∼
Elena Celledoni Geometric animation of character motion
![Page 9: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/9.jpg)
Reparametrization invariance of shapes
LetDiff+(I) = {ϕ ∈ C∞(I, I) ∣ϕ′(t) > 0}
be the set of smooth, orientation preserving, invertible maps.Open curves. I = [0,1]
cos(t)0.6 0.8 1
sin
(t)
0
0.5
1
t0 0.5 1
phi(t)
0
0.5
1
cos(phi(t))0.6 0.8 1
sin
(phi(t)
)
0
0.5
1
(cos(t), sin(t)) , ϕ ∶ t ↦ 34t2 + t
4, (cos(ϕ(t)), sin(ϕ(t)))
Closed curves. Diff+(S1) = {ϕ ∈ C∞(S1,S1) ∣ϕ′(t) > 0}
0 2π0
2π
Elena Celledoni Geometric animation of character motion
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Reparametrization invariance of shapes
LetDiff+(I) = {ϕ ∈ C∞(I, I) ∣ϕ′(t) > 0}
be the set of smooth, orientation preserving, invertible maps.Open curves. I = [0,1]
cos(t)0.6 0.8 1
sin
(t)
0
0.5
1
t0 0.5 1
phi(t)
0
0.5
1
cos(phi(t))0.6 0.8 1
sin
(phi(t)
)
0
0.5
1
(cos(t), sin(t)) , ϕ ∶ t ↦ 34t2 + t
4, (cos(ϕ(t)), sin(ϕ(t)))
Closed curves. Diff+(S1) = {ϕ ∈ C∞(S1,S1) ∣ϕ′(t) > 0}
0 2π0
2π
Elena Celledoni Geometric animation of character motion
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Distance function on S
Applications often require a distance function to measuresimilarities between shapes. Let dP be a distance function on P(on parametrized curves) then
Distance on S:
dS([c0], [c1]) ∶= infϕ∈Diff+(I)
dP(c0, c1 ○ ϕ).
But we need also to check that dS is well defined, i.e. isindependent on the choice of representatives c0 for [c0] and c1 for[c1].
Computational methods for dS :
• gradient flows
• dynamic programming
Elena Celledoni Geometric animation of character motion
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Distance function on S
Applications often require a distance function to measuresimilarities between shapes. Let dP be a distance function on P(on parametrized curves) then
Distance on S:
dS([c0], [c1]) ∶= infϕ∈Diff+(I)
dP(c0, c1 ○ ϕ).
But we need also to check that dS is well defined, i.e. isindependent on the choice of representatives c0 for [c0] and c1 for[c1].
Computational methods for dS :
• gradient flows
• dynamic programming
Elena Celledoni Geometric animation of character motion
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Distance on P
dP(c0, c1) it is usually obtained by defining a Riemannian metricon the infinite dimensional manifold P of parametrized curves:
Gc ∶ TcP ×TcP → R
and taking the length of the geodesic α between c0 ∈ P and c1 ∈ Pwrt Gc , i.e.
dP(c0, c1) ∶= length(α(c0, c1))
α(c0, c1) is the shortest path between c0 and c1 wrt Gc
We will proceed differently and use the SRV transform
R ∶ P → C,
with C a vector space where we can (and will) use the L2 metric.
Elena Celledoni Geometric animation of character motion
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Virtual characters and skeletal animation
• Virtual character: a closed surface in R3 (triangle mesh).
• Motions of characters via skeletal animation approach.
• Skeleton consisting of bones connected by joints.
The coordinates of the vertices of the triangle mesh are specifiedin a coordinate system aligned with the bone. In the animationthe movement of the vertices is determined by the bones.
Elena Celledoni Geometric animation of character motion
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Virtual characters and skeletal animation
• Virtual character: a closed surface in R3 (triangle mesh).
• Motions of characters via skeletal animation approach.
• Skeleton consisting of bones connected by joints.
The coordinates of the vertices of the triangle mesh are specifiedin a coordinate system aligned with the bone. In the animationthe movement of the vertices is determined by the bones.
Elena Celledoni Geometric animation of character motion
![Page 16: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/16.jpg)
Virtual characters and skeletal animation
• Virtual character: a closed surface in R3 (triangle mesh).
• Motions of characters via skeletal animation approach.
• Skeleton consisting of bones connected by joints.
The coordinates of the vertices of the triangle mesh are specifiedin a coordinate system aligned with the bone. In the animationthe movement of the vertices is determined by the bones.
Elena Celledoni Geometric animation of character motion
![Page 17: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/17.jpg)
Character animation and skeletal animation
• Skeleton: rooted tree madeof bones and joints.
• Configuration spaceJ = SE(3)n or J = SO(3)n
(for human caracters).
• Character’s pose specifiedby assigning values to thedegrees of freedom.
• Animation: α ∶ [a,b]→ J , acurve on J , [a,b] intervalof time.
• Motion capturing (recordingcurves on J ).
• Motion manipulationconsider entire animationsas shapes belonging to S.
root
left hipjoint
left femur
left tibia
left foot
right hipjoint
right femur
right tibia
right foot
lowerback
upperback
thorax
lowerneck
upperneck
left clavicle
left humerus
left radius
left wristleft hand
rright clavicle
right humerus
right radius
right wrist
right hand
right toesleft toes
head
left fingers
right fingers
left thumbright thumb
Figure: Skeleton: each node correspondsto a Q(t) ∈ SO(3)
Elena Celledoni Geometric animation of character motion
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Motion capturing
Generating the data: α ∶ [a,b]→ J , a curve on J = SO(3)n
motion capturing with and without markers
Elena Celledoni Geometric animation of character motion
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Shape analysis on Lie groups: spaces and metrics
P ∶= Imm(I,G) smooth functions with first derivative≠ 0
S ∶= Imm(I,G)/Diff+(I) shape space
Plan: we aim to obtain a distance function on S by
dS([c0], [c1]) ∶= infϕ∈Diff+(I)
dP(c0, c1 ○ ϕ) (1)
with dP a distance on P.
Definition. We say dP is a reparametrization invariant distancefunction on P iff
dP(c0, c1) = dP(c0 ○ ϕ, c1 ○ ϕ) ∀ϕ ∈ Diff+(I). (2)
Proposition
If dP is a reparametrization invariant distance function on P thendS([c0], [c1]) as defined in (1) is independent of the choice ofrepresentatives of [c0] and [c1].
Elena Celledoni Geometric animation of character motion
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Shape analysis on Lie groups: spaces and metrics
P ∶= Imm(I,G) smooth functions with first derivative≠ 0
S ∶= Imm(I,G)/Diff+(I) shape space
Plan: we aim to obtain a distance function on S by
dS([c0], [c1]) ∶= infϕ∈Diff+(I)
dP(c0, c1 ○ ϕ) (1)
with dP a distance on P.
Definition. We say dP is a reparametrization invariant distancefunction on P iff
dP(c0, c1) = dP(c0 ○ ϕ, c1 ○ ϕ) ∀ϕ ∈ Diff+(I). (2)
Proposition
If dP is a reparametrization invariant distance function on P thendS([c0], [c1]) as defined in (1) is independent of the choice ofrepresentatives of [c0] and [c1].
Elena Celledoni Geometric animation of character motion
![Page 21: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/21.jpg)
Shape analysis on Lie groups: spaces and metrics
P ∶= Imm(I,G) smooth functions with first derivative≠ 0
S ∶= Imm(I,G)/Diff+(I) shape space
Plan: we aim to obtain a distance function on S by
dS([c0], [c1]) ∶= infϕ∈Diff+(I)
dP(c0, c1 ○ ϕ) (1)
with dP a distance on P.
Definition. We say dP is a reparametrization invariant distancefunction on P iff
dP(c0, c1) = dP(c0 ○ ϕ, c1 ○ ϕ) ∀ϕ ∈ Diff+(I). (2)
Proposition
If dP is a reparametrization invariant distance function on P thendS([c0], [c1]) as defined in (1) is independent of the choice ofrepresentatives of [c0] and [c1].
Elena Celledoni Geometric animation of character motion
![Page 22: Shape analysis on Lie groups and Homogeneous Manifolds ... · Shape analysis on Lie groups and Homogeneous Manifolds with applications in computer animation Elena Celledoni Department](https://reader030.vdocuments.net/reader030/viewer/2022041102/5edd070bad6a402d6667f609/html5/thumbnails/22.jpg)
Tools to work with curves on G
C∞
∗(I ,G) is an infinite dimensional Lie group, C∞(I ,g) infinite
dimensional Lie algebra
The evolution operator of a regular (e.g. finite dimensional) Lie groupG with Lie algebra g:
Evol ∶ C∞(I,g)→ C∞
∗(I ,G) ∶= {c ∈ C∞(I,G) ∶ c(0) = e}
Evol(q)(t) ∶= c(t), where∂c
∂t= Rc(t)∗(q(t)), c(0) = e,
Evol ∶ C∞(I ,g)→ C∞
∗(I ,G)
is a diffeomorphism.The inverse of the evolution operator is the so called right logarithmicderivative
δr ∶C∞
∗(I ,G)→ C∞(I ,g),
δrg ∶= R−1g∗(g).
H. Glockner, arXiv:1502.05795v3, March 2015.
A. Kriegl and P. W. Michor, The convenient setting of global analysis.
Elena Celledoni Geometric animation of character motion
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Tools to work with curves on G
C∞
∗(I ,G) is an infinite dimensional Lie group, C∞(I ,g) infinite
dimensional Lie algebraThe evolution operator of a regular (e.g. finite dimensional) Lie groupG with Lie algebra g:
Evol ∶ C∞(I,g)→ C∞
∗(I ,G) ∶= {c ∈ C∞(I,G) ∶ c(0) = e}
Evol(q)(t) ∶= c(t), where∂c
∂t= Rc(t)∗(q(t)), c(0) = e,
Evol ∶ C∞(I ,g)→ C∞
∗(I ,G)
is a diffeomorphism.
The inverse of the evolution operator is the so called right logarithmicderivative
δr ∶C∞
∗(I ,G)→ C∞(I ,g),
δrg ∶= R−1g∗(g).
H. Glockner, arXiv:1502.05795v3, March 2015.
A. Kriegl and P. W. Michor, The convenient setting of global analysis.
Elena Celledoni Geometric animation of character motion
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Tools to work with curves on G
C∞
∗(I ,G) is an infinite dimensional Lie group, C∞(I ,g) infinite
dimensional Lie algebraThe evolution operator of a regular (e.g. finite dimensional) Lie groupG with Lie algebra g:
Evol ∶ C∞(I,g)→ C∞
∗(I ,G) ∶= {c ∈ C∞(I,G) ∶ c(0) = e}
Evol(q)(t) ∶= c(t), where∂c
∂t= Rc(t)∗(q(t)), c(0) = e,
Evol ∶ C∞(I ,g)→ C∞
∗(I ,G)
is a diffeomorphism.The inverse of the evolution operator is the so called right logarithmicderivative
δr ∶C∞
∗(I ,G)→ C∞(I ,g),
δrg ∶= R−1g∗(g).
H. Glockner, arXiv:1502.05795v3, March 2015.
A. Kriegl and P. W. Michor, The convenient setting of global analysis.
Elena Celledoni Geometric animation of character motion
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SRVT for curves on G and distance on P
SRVT: square root velocity transform. Let ⟨⋅, ⋅⟩ be a right-invariantmetric on G and ∥ ⋅ ∥ the induced norm on tangent spaces,
R ∶ {c ∈ Imm(I,G) ∣ c(0) = e}→ {v ∈ C∞(I,g) ∣ ∥v(t)∥ ≠ 0}
q(t) =R(c)(t) ∶= δrc√∥δrc∥
=R−1
c(t)∗(c(t))√∥c(t)∥
with inverseR−1 ∶ {v ∈ C∞(I,g) ∣ ∥v(t)∥ ≠ 0}→ {c ∈ Imm(I,G) ∣ c(0) = e},
R−1(q)(t) = Evol(q∥q∥) = c(t).
Reparametrization invariant distance on P ∶
dP(c0, c1) ∶= dL2(R(c0),R(c1)) = (∫I∥q0(t) − q1(t)∥2dt)
12
Proposition
dP is reparametrization invariant.
Distance on S
dS([c0], [c1]) ∶= infϕ∈Diff+
(I)(∫
I∥q0(t) − q1(t)∥2dt)
12
Elena Celledoni Geometric animation of character motion
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SRVT for curves on G and distance on P
SRVT: square root velocity transform. Let ⟨⋅, ⋅⟩ be a right-invariantmetric on G and ∥ ⋅ ∥ the induced norm on tangent spaces,
R ∶ {c ∈ Imm(I,G) ∣ c(0) = e}→ {v ∈ C∞(I,g) ∣ ∥v(t)∥ ≠ 0}
q(t) =R(c)(t) ∶= δrc√∥δrc∥
=R−1
c(t)∗(c(t))√∥c(t)∥
with inverseR−1 ∶ {v ∈ C∞(I,g) ∣ ∥v(t)∥ ≠ 0}→ {c ∈ Imm(I,G) ∣ c(0) = e},
R−1(q)(t) = Evol(q∥q∥) = c(t).Reparametrization invariant distance on P ∶
dP(c0, c1) ∶= dL2(R(c0),R(c1)) = (∫I∥q0(t) − q1(t)∥2dt)
12
Proposition
dP is reparametrization invariant.
Distance on S
dS([c0], [c1]) ∶= infϕ∈Diff+
(I)(∫
I∥q0(t) − q1(t)∥2dt)
12
Elena Celledoni Geometric animation of character motion
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Connections to the elastic metric
Elastic metric on P ∶= Imm(I,G). Using the right invariantmetric ⟨⋅, ⋅⟩ on G we can define
G ∶ TP ×TP ↦ R
where
Gc(h, k) ∶ = ∫I[a2⟨Dsh, v⟩⟨Dsk , v⟩] ds (tangential)
(normal) + ∫I[b2 (∣Dsh − ⟨Dsh, v⟩v ∣2 ∣Dsk − ⟨Dsk, v⟩v ∣2)] ds,
here the integration is with respect to arc-length, ds = ∣c ′(t)∣dt,and
v ∶= c ′(t)∣c ′(t)∣ , Dsh ∶=
∂h(t(s))∂s
= 1
∣c ′(t)∣∂h
∂t.
A family of Sobolev type metrics of order one.
Elena Celledoni Geometric animation of character motion
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Connections to the elastic metric
Assume I = [0,1], R ∶ Imm(I,G)→ C∞(I,g)
Theorem
The pullback of the L2 inner product on C∞(I,g) to P = Imm(I,G), bythe SRV transform R is the elastic metric
Gc(h,h) = ∫I∣Dsh − ⟨Dsh, v⟩ v ∣2 +
1
4⟨Dsh, v⟩2 ds,
and
Dsh =h
∥c∥ , v = c
∥c∥ .
Ds denotes differentiation with respect to arc length, v is the unittangent vector of c and ds denotes integration with respect to arc length.
For computational purposes, we can transform the curves with R and
then use the L2 metric.
Elena Celledoni Geometric animation of character motion
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Interpolation of curves (motion blending)
Motion blending on G
Geodesic paths α ∶ [0,1]→ P between two parametrized curves onthe Lie group G :
c = α(0, t), d = α(1, t)
we get (visually convincing) deformations from c to d
α(x , t) =R−1((1 − x)R(c) + xR(d))(t).
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Original curves Reparametrized
No reparam.
Reparametrized
Figure: Interpolation between two curves in SO(3) with and withoutreparametrization.
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Obstacle example (motion blending)
Elastic reparam. Feature reparam.
SO(3) no reparam. SO(3) reparam.
t
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Closed curves
Using SRVT R, we can identify curves c ∈ P = Imm(I ,G) withR(c) ∈ C∞(I,g)Open curves
Co ∶=R(Imm(I ,G))
= {q ∈ C∞(I ,g) ∣ ∥q∥ ≠ 0}
Closed curves
Cc ∶=R({c ∈ Imm(I ,G) ∣ c(0) = c(1) = e})
Cc = r−1(e), r ∶= ev1 ○Evol ○ sc, r ∶ Co → G
ev1 is the evaluation operator evaluating a curve at t = 1, while scis the map sc(q) ∶= q ∥q∥, and we notice that r(q) =R−1(q)(1).
Theorem
Cc is a submanifold of finite codimension of C∞(I ,g)
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Closed curves
Using SRVT R, we can identify curves c ∈ P = Imm(I ,G) withR(c) ∈ C∞(I,g)Open curves
Co ∶=R(Imm(I ,G)) = {q ∈ C∞(I ,g) ∣ ∥q∥ ≠ 0}
Closed curves
Cc ∶=R({c ∈ Imm(I ,G) ∣ c(0) = c(1) = e})
Cc = r−1(e), r ∶= ev1 ○Evol ○ sc, r ∶ Co → G
ev1 is the evaluation operator evaluating a curve at t = 1, while scis the map sc(q) ∶= q ∥q∥, and we notice that r(q) =R−1(q)(1).
Theorem
Cc is a submanifold of finite codimension of C∞(I ,g)
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Closed curves
Using SRVT R, we can identify curves c ∈ P = Imm(I ,G) withR(c) ∈ C∞(I,g)Open curves
Co ∶=R(Imm(I ,G)) = {q ∈ C∞(I ,g) ∣ ∥q∥ ≠ 0}
Closed curves
Cc ∶=R({c ∈ Imm(I ,G) ∣ c(0) = c(1) = e})
Cc = r−1(e), r ∶= ev1 ○Evol ○ sc, r ∶ Co → G
ev1 is the evaluation operator evaluating a curve at t = 1, while scis the map sc(q) ∶= q ∥q∥, and we notice that r(q) =R−1(q)(1).
Theorem
Cc is a submanifold of finite codimension of C∞(I ,g)
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Curve closing
Projection from Co onto Cc by means of a constrainedminimization problem
minq∈Cc
1
2∥q − q0∥, q0 ∈ Co
Instead of minimizing the distance from closed curves to q0 weminimize the closure constraint:
Measuring closedness.Consider φ ∶ Co → R
φ(q) ∶= 1
2∥ log(r(q))∥2, r ∶= ev1 ○Evol ○ sc
andφ(q) = 0⇐⇒ q ∈ Cc
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Curve closing
Projection from Co onto Cc by means of a constrainedminimization problem
minq∈Cc
1
2∥q − q0∥, q0 ∈ Co
Instead of minimizing the distance from closed curves to q0 weminimize the closure constraint:
Measuring closedness.Consider φ ∶ Co → R
φ(q) ∶= 1
2∥ log(r(q))∥2, r ∶= ev1 ○Evol ○ sc
andφ(q) = 0⇐⇒ q ∈ Cc
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Projection on the space of closed curves
Tqφ(f ) = ⟨grad(φ)(q), f ⟩L2 = ∫I⟨grad(φ)(q), f ⟩dx ,
Theorem
The gradient vector field wrt the L2 inner product is
grad(Φ)(q) = ∥q∥ α(q) + ⟨α(q), q
∥q∥⟩ q, (3)
where
α(q) ∶= AdTc(q)−1 AdT
r(q) (log(r(q))) ∈ C∞(I ,g)c(q) ∶=R−1(q) ∈ C∞(I ,G)r(q) ∶=R−1(q)(1) ∈ G .
Projection form Co to Cc :Gradient flow
∂u
∂τ= −grad(φ)(u), u(t,0) = q(t).
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Discrete curves: piecewise continuous in G
c based on discrete points {ci ∶= c(θi)}ni=0
c(t) ∶=n−1∑k=0
χ[θk ,θk+1)(t) exp( t − θkθk+1 − θk
log(ck+1c−1k )) ck , (4)
where χ is the characteristic function.
SRV transform: q =R(c) piecewise constant function in g:q = {qi}n−1i=0
qi ∶=ηi√∥ηi∥
, ηi ∶=log(ci+1c−1i ),θi+1 − θi
.
The inverse SRV transform: c =R−1(q):
ci+1 = exp (qi∥qi∥ (θi+1 − θi)) ⋅ ci
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Numerical algorithms
Curve reparametrizationApplying a reparametrization ϕ ∈ Diff(I) to the discrete curve c gives cwith {ci}ni=0:
ci ∶= cj exp(s log(cj+1c−1j )), s ∶= ϕ(θi) − θjθj+1 − θj
, i = 0, . . . ,n,
where j is an index such that θj ≤ ϕ(θi) < θj+1. Note that c0 = c0 andcn = cnCurve interpolation
[0,1] ×P ×P → P(s, c0, c1)↦R−1((1 − s)R(c0) + sR(c1)),
(5)
with interpolation parameter s.Curve closing in SO(3)
grad(φ)(q) = ∥q∥c log(c(1))cT + ⟨c log(c(1))cT , q
∥q∥⟩q, c =R−1(q)
uk+1 = uk − αk grad(φ)(uk),where every uk is a discrete curve as defined above, i.e., uk = {uki }ni=0.
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Jumping example (closure of curves)
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Handspring example (closure of curves)
Original
Discontinuities
t
Closed
t
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Curve closure: experiments
Original Closed
Figure: Discontinuities in the handspring animation.
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Curve closing
Original
t
Closed
t
Figure: Application of closing algorithm to a cartwheel animation.
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References
• E. Celledoni, M. Eslitzbichler ans A. Scmeding, Shape analysis on Lie groupswith applications in computer animation, arXiv:1506.00783.
• M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds,The Visual Computer, 2014.
• M. Bauer, M. Eslitzbichler, M. Grasmair, Landmark-Guided Elastic ShapeAnalysis of Human Character Motions, to appear, 2015.
• M. Bauer, M. Bruveris, S. Marsland, and P.W. Michor, Constructingreparametrization invariant metrics on spaces of plane curves, arxiv, 2012.
• Carnegie-Mellon. Carnegie-Mellon Mocap Database, 2003.
• H. Glockner. Regularity properties of infinite-dimensional Lie groups, andsemiregularity. arXiv:1208.0715 [math], August 2012.
• A. Kriegl and P. W. Michor. Regular infinite dimensional Lie groups. Journal ofLie Theory, 7:61–99, 1997.
• A. Srivastava, E. Klassen, S.H. Joshi, and I.H. Jermyn. Shape Analysis ofElastic Curves in Euclidean Spaces. IEEE Transactions on Pattern Analysis andMachine Intelligence, 33(7): 1415 –1428, July 2011.
Thank you for listening.
Elena Celledoni Geometric animation of character motion