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Algorithmic Treatment of Nonlinear DataP. GrohsETH Zurich, Seminar for Applied Mathematics
Jun-15-2012, Liege University
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Outline
MotivationWeighted Averages in ManifoldsWaveletsGeodesic Finite Elements
P. Grohs Jun-15-2012, Liege University p. 2
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Nonlinear Data
Source: X. Pennec
P. Grohs Jun-15-2012, Liege University p. 3
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Nonlinear Data
Source: I. UrRahman
P. Grohs Jun-15-2012, Liege University p. 4
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Nonlinear Data
P. Grohs Jun-15-2012, Liege University p. 5
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Nonlinear Data
Source: A. Ivancevic
P. Grohs Jun-15-2012, Liege University p. 6
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Nonlinear Data
Source: G. Sundaramoorthi
P. Grohs Jun-15-2012, Liege University p. 7
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And many more!
Spherical data (SAR inferogram)Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)Quaternions (motion design). . .
P. Grohs Jun-15-2012, Liege University p. 8
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And many more!Spherical data (SAR inferogram)
Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)Quaternions (motion design). . .
P. Grohs Jun-15-2012, Liege University p. 8
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And many more!Spherical data (SAR inferogram)Grassmann manifold (array signal processing)
Stiefel manifold (low rank approximations)Quaternions (motion design). . .
P. Grohs Jun-15-2012, Liege University p. 8
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And many more!Spherical data (SAR inferogram)Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)
Quaternions (motion design). . .
P. Grohs Jun-15-2012, Liege University p. 8
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And many more!Spherical data (SAR inferogram)Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)Quaternions (motion design)
. . .
P. Grohs Jun-15-2012, Liege University p. 8
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And many more!Spherical data (SAR inferogram)Grassmann manifold (array signal processing)Stiefel manifold (low rank approximations)Quaternions (motion design). . .
P. Grohs Jun-15-2012, Liege University p. 8
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Goals/Challenges
Interpolation and Approximation (also for inverseproblems)CompressionNoise RemovalFeature Extraction
P. Grohs Jun-15-2012, Liege University p. 9
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Goals/ChallengesInterpolation and Approximation (also for inverseproblems)
CompressionNoise RemovalFeature Extraction
P. Grohs Jun-15-2012, Liege University p. 9
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Goals/ChallengesInterpolation and Approximation (also for inverseproblems)Compression
Noise RemovalFeature Extraction
P. Grohs Jun-15-2012, Liege University p. 9
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Goals/ChallengesInterpolation and Approximation (also for inverseproblems)CompressionNoise Removal
Feature Extraction
P. Grohs Jun-15-2012, Liege University p. 9
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Goals/ChallengesInterpolation and Approximation (also for inverseproblems)CompressionNoise RemovalFeature Extraction
P. Grohs Jun-15-2012, Liege University p. 9
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Desiderata
Objectivity (respecting natural invariances)Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)
In short, generalize linear tools to manifold-valued case.
P. Grohs Jun-15-2012, Liege University p. 10
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DesiderataObjectivity (respecting natural invariances)
Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)
In short, generalize linear tools to manifold-valued case.
P. Grohs Jun-15-2012, Liege University p. 10
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DesiderataObjectivity (respecting natural invariances)Universality (valid for general manifolds)
Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)
In short, generalize linear tools to manifold-valued case.
P. Grohs Jun-15-2012, Liege University p. 10
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DesiderataObjectivity (respecting natural invariances)Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)
Tractability (same overall complexity as linear case)In short, generalize linear tools to manifold-valued case.
P. Grohs Jun-15-2012, Liege University p. 10
![Page 23: Algorithmic Treatment of Nonlinear Datamishra/systmod_presentations/...P. Grohs Jun-15-2012, Liege University p. 10 Desiderata Objectivity (respecting natural invariances) Universality](https://reader031.vdocuments.us/reader031/viewer/2022011911/5f8923d8a37dd975e2229446/html5/thumbnails/23.jpg)
DesiderataObjectivity (respecting natural invariances)Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)
In short, generalize linear tools to manifold-valued case.
P. Grohs Jun-15-2012, Liege University p. 10
![Page 24: Algorithmic Treatment of Nonlinear Datamishra/systmod_presentations/...P. Grohs Jun-15-2012, Liege University p. 10 Desiderata Objectivity (respecting natural invariances) Universality](https://reader031.vdocuments.us/reader031/viewer/2022011911/5f8923d8a37dd975e2229446/html5/thumbnails/24.jpg)
DesiderataObjectivity (respecting natural invariances)Universality (valid for general manifolds)Theoretical Guarantees (same theoretical results asfor linear case)Tractability (same overall complexity as linear case)
In short, generalize linear tools to manifold-valued case.
P. Grohs Jun-15-2012, Liege University p. 10
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The Central Problem
f1 : Ω→ M f2 : Ω→ M
P. Grohs Jun-15-2012, Liege University p. 11
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The Central Problem
f1 : Ω→ M f2 : Ω→ M
f1 + f2 =??
P. Grohs Jun-15-2012, Liege University p. 12
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The Central Problem
f1 : Ω→ M f2 : Ω→ M
f1 + f2 =??linear functionalanalysis!
classicalapproximationtheory!anything based onvector spaces!
P. Grohs Jun-15-2012, Liege University p. 13
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The Central Problem
f1 : Ω→ M f2 : Ω→ M
f1 + f2 =??linear functionalanalysis!classicalapproximationtheory!
anything based onvector spaces!
P. Grohs Jun-15-2012, Liege University p. 13
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The Central Problem
f1 : Ω→ M f2 : Ω→ M
f1 + f2 =??linear functionalanalysis!classicalapproximationtheory!anything based onvector spaces!
P. Grohs Jun-15-2012, Liege University p. 13
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Weighted Two-Point Averages
f1 : Ω→ M f2 : Ω→ M
P. Grohs Jun-15-2012, Liege University p. 14
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Weighted Two-Point Averages
f1 : Ω→ M f2 : Ω→ M
tf1(x) + (1− t)f2(x) = γf2(x)f1(x) (t)
(γqp (·) is the (locally) unique geodesic connecting the points p ∈ M and q ∈ M)
P. Grohs Jun-15-2012, Liege University p. 15
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Weighted Two-Point Averages
f1 : Ω→ M f2 : Ω→ M
tf1(x) + (1− t)f2(x) = γf2(x)f1(x) (t)
(γqp (·) is the (locally) unique geodesic connecting the points p ∈ M and q ∈ M)
P. Grohs Jun-15-2012, Liege University p. 15
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Observation: Actually, many constructions can be formulatedentirely using weighted averages∑
i
wipi ,∑
i
wi = 1,wi ∈ R!
P. Grohs Jun-15-2012, Liege University p. 16
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Quadratic B-Splines
p−1
p0
p1
p2
P. Grohs Jun-15-2012, Liege University p. 17
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Quadratic B-Splines
q−2
q−1
q0
q1
q2
q2i = 34 pi + 1
4 pi+1
q2i+1 = 14 pi + 3
4 pi+1
P. Grohs Jun-15-2012, Liege University p. 18
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Quadratic B-Splines
q−2
q−1
q0
q1
q2
q2i = 34 pi + 1
4 pi+1
q2i+1 = 14 pi + 3
4 pi+1
P. Grohs Jun-15-2012, Liege University p. 18
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Quadratic B-Splines
Sp(i) =∑
j∈Z a(i − 2j)p(j) a = [. . . ,0, 14 ,
34 ,
34 ,
14 ,0, . . . ]
P. Grohs Jun-15-2012, Liege University p. 19
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Quadratic B-Splines∑
j∈Z a(i − 2j) = 1
P. Grohs Jun-15-2012, Liege University p. 20
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Quadratic B-Splines
limit function with control polygon
P. Grohs Jun-15-2012, Liege University p. 21
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Example:motion design/interpolation
limit functionwith control polygon??
P. Grohs Jun-15-2012, Liege University p. 22
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Example: finite elements
0 1
1
f (x) =∑
i ϕi (x)fiwith shape functions ϕi , satisfying∑
i ϕi (x) ≡ 1.
x-dependent weighted average of points fi !
P. Grohs Jun-15-2012, Liege University p. 23
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Example: finite elements
0 1
1
f (x) =∑
i ϕi (x)fiwith shape functions ϕi , satisfying∑
i ϕi (x) ≡ 1.
x-dependent weighted average of points fi !
P. Grohs Jun-15-2012, Liege University p. 23
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Weighted Averages in Manifolds
P. Grohs Jun-15-2012, Liege University p. 24
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Goal of this Section
Come up with geometric notion of weighted mean
p∗ =N⊕
i=1
wipi ∈ M
of points pi ∈ M with weights wi ∈ R, such that∑N
i=1 wi = 1.
Should also be efficiently computable!
P. Grohs Jun-15-2012, Liege University p. 25
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Goal of this Section
Come up with geometric notion of weighted mean
p∗ =N⊕
i=1
wipi ∈ M
of points pi ∈ M with weights wi ∈ R, such that∑N
i=1 wi = 1.
Should also be efficiently computable!
P. Grohs Jun-15-2012, Liege University p. 25
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Linear Averages Revisited
In Rd we have
p∗ =N∑
i=1
wipi ⇔N∑
i=1
wi−−→p∗pi = 0
Necessary: Computationof difference vectorbetween two points.
p∗
p1
p2
p3
p4
p∗ = 14 (p1 + p2 + p3 + p4)
P. Grohs Jun-15-2012, Liege University p. 26
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Linear Averages Revisited
In Rd we have
p∗ =N∑
i=1
wipi ⇔N∑
i=1
wi−−→p∗pi = 0
Necessary: Computationof difference vectorbetween two points.
p∗
p1
p2
p3
p4
p∗ = 14 (p1 + p2 + p3 + p4)
P. Grohs Jun-15-2012, Liege University p. 26
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Linear Averages Revisited
In Rd we have
p∗ =N∑
i=1
wipi ⇔N∑
i=1
wi−−→p∗pi = 0
Necessary: Computationof difference vectorbetween two points.
p∗
p1
p2
p3
p4
p∗ = 14 (p1 + p2 + p3 + p4)
P. Grohs Jun-15-2012, Liege University p. 26
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Computation of Difference Vectors
(a) Retraction pair basedon exponential map
(b) Retraction pair basedon closest point projection
(c) Retraction pair basedon vertical projection
Figure: Different retractions for the circle.
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Retraction Pairs
Definition (G (2012))
Two functions (f ,g),
f : TM → M, g : M ×M → TM,
such that
f (p,0) = p, f (p,g(p,q)) = q for all p, q ∈ M,
and∂
∂ξf (p,0) = Id
are called a retraction pair for the manifold M.
Standard example: f = expM and g = logM .
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Retraction Pairs
Definition (G (2012))
Two functions (f ,g),
f : TM → M, g : M ×M → TM,
such that
f (p,0) = p, f (p,g(p,q)) = q for all p, q ∈ M,
and∂
∂ξf (p,0) = Id
are called a retraction pair for the manifold M.
Standard example: f = expM and g = logM .
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Constructing operations from retraction pairs
Point-vector addition p ⊕ ξ for ξ ∈ TpMdefined by f (p, ξ).
Point-point difference q p ∈ TpM such that
p ⊕ (q p) = q
defined by g(p,q). Weighted average
p∗ =⊕i∈I
wipi ,∑i∈I
wi = 1, and pi ∈ M.
defined implicitly by ∑i∈I
wig(p∗,pi ) = 0.
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Constructing operations from retraction pairs Point-vector addition p ⊕ ξ for ξ ∈ TpM
defined by f (p, ξ). Point-point difference q p ∈ TpM such that
p ⊕ (q p) = q
defined by g(p,q). Weighted average
p∗ =⊕i∈I
wipi ,∑i∈I
wi = 1, and pi ∈ M.
defined implicitly by ∑i∈I
wig(p∗,pi ) = 0.
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Constructing operations from retraction pairs Point-vector addition p ⊕ ξ for ξ ∈ TpM
defined by f (p, ξ).
Point-point difference q p ∈ TpM such that
p ⊕ (q p) = q
defined by g(p,q). Weighted average
p∗ =⊕i∈I
wipi ,∑i∈I
wi = 1, and pi ∈ M.
defined implicitly by ∑i∈I
wig(p∗,pi ) = 0.
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Constructing operations from retraction pairs Point-vector addition p ⊕ ξ for ξ ∈ TpM
defined by f (p, ξ). Point-point difference q p ∈ TpM such that
p ⊕ (q p) = q
defined by g(p,q). Weighted average
p∗ =⊕i∈I
wipi ,∑i∈I
wi = 1, and pi ∈ M.
defined implicitly by ∑i∈I
wig(p∗,pi ) = 0.
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Constructing operations from retraction pairs Point-vector addition p ⊕ ξ for ξ ∈ TpM
defined by f (p, ξ). Point-point difference q p ∈ TpM such that
p ⊕ (q p) = q
defined by g(p,q).
Weighted average
p∗ =⊕i∈I
wipi ,∑i∈I
wi = 1, and pi ∈ M.
defined implicitly by ∑i∈I
wig(p∗,pi ) = 0.
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Constructing operations from retraction pairs Point-vector addition p ⊕ ξ for ξ ∈ TpM
defined by f (p, ξ). Point-point difference q p ∈ TpM such that
p ⊕ (q p) = q
defined by g(p,q). Weighted average
p∗ =⊕i∈I
wipi ,∑i∈I
wi = 1, and pi ∈ M.
defined implicitly by ∑i∈I
wig(p∗,pi ) = 0.
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Constructing operations from retraction pairs Point-vector addition p ⊕ ξ for ξ ∈ TpM
defined by f (p, ξ). Point-point difference q p ∈ TpM such that
p ⊕ (q p) = q
defined by g(p,q). Weighted average
p∗ =⊕i∈I
wipi ,∑i∈I
wi = 1, and pi ∈ M.
defined implicitly by ∑i∈I
wig(p∗,pi ) = 0.
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Well-Definedness
Theorem (G 2012)
The notion of average is locally well-defined for any retraction pair(f ,g).
M Riemannian manifold, f = expM , g = logM ; Karcher mean(Karcher ∼ 1970, Cartan ∼ 1930).
Computation is efficient as long as the compuation of f and g isefficient (simple algorithm: fixed-point iterationpn+1∗ ← f (pn
∗,∑
i wig(pn∗,pi )))!
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Well-Definedness
Theorem (G 2012)
The notion of average is locally well-defined for any retraction pair(f ,g).
M Riemannian manifold, f = expM , g = logM ; Karcher mean(Karcher ∼ 1970, Cartan ∼ 1930).
Computation is efficient as long as the compuation of f and g isefficient (simple algorithm: fixed-point iterationpn+1∗ ← f (pn
∗,∑
i wig(pn∗,pi )))!
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Well-Definedness
Theorem (G 2012)
The notion of average is locally well-defined for any retraction pair(f ,g).
M Riemannian manifold, f = expM , g = logM ; Karcher mean(Karcher ∼ 1970, Cartan ∼ 1930).
Computation is efficient as long as the compuation of f and g isefficient (simple algorithm: fixed-point iterationpn+1∗ ← f (pn
∗,∑
i wig(pn∗,pi )))!
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Examples of Retraction Pairs
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Sphere
M = x ∈ Rd : ‖x‖2 = 1TpM = ξ ∈ Rd : ξ>p = 0
expM(p, ξ) = cos(‖ξ‖2)p + sin(‖ξ‖2)ξ
‖ξ‖2
logM(p,q) = arccos(p>q)q − (p>q)p‖q − (p>q)p‖2
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The Lie Group of Euclidean Motions
SE(d) = A =
(R x0 1
): x ∈ Rd ,R ∈ SO(3)
TASE(d) = ξ =
(ξ1 ξ20 0
): ξ1 ∈ Rd , ξ2 ∈ sl3
expSE(d)(A, ξ) = A exp(ξ)
logSE(d)(A,B) = log(A−1B)
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The Symmetric Space of SPD-Matrices
SPD(d) = A ∈ Rd×d : A 0
TASPD(d) = ξ ∈ Rd×d : ξ> = ξ
expM(A, ξ) = A1/2 exp(A−1/2ξA−1/2)A1/2
logM(A,B) = A1/2 log(A−1/2BA−1/2)A1/2
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The Stiefel Manifold
St(n,p) = A ∈ Rn×p : A>A = Ip
TASt(n,p) = ξ ∈ Rn×p : ξ>A+A>ξ = 0p
expM(A, ξ) = some complicated expression
logM(A,B) = no explicit expression known!
Important: All operations computable inorder pn operations (not order n2).
Sources: TOP: [Lubich, Nonnenmacher
2010], Bottom: [Sundaramoorthi etal
2011]
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The Stiefel Manifold
St(n,p) = A ∈ Rn×p : A>A = Ip
TASt(n,p) = ξ ∈ Rn×p : ξ>A+A>ξ = 0p
expM(A, ξ) = some complicated expression
logM(A,B) = no explicit expression known!
Important: All operations computable inorder pn operations (not order n2).
Sources: TOP: [Lubich, Nonnenmacher
2010], Bottom: [Sundaramoorthi etal
2011]
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The Stiefel Manifold
St(n,p) = A ∈ Rn×p : A>A = Ip
TASt(n,p) = ξ ∈ Rn×p : ξ>A+A>ξ = 0p
expM(A, ξ) = some complicated expression
logM(A,B) = no explicit expression known!
Important: All operations computable inorder pn operations (not order n2).
Sources: TOP: [Lubich, Nonnenmacher
2010], Bottom: [Sundaramoorthi etal
2011]
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The Stiefel Manifold
St(n,p) = A ∈ Rn×p : A>A = Ip
TASt(n,p) = ξ ∈ Rn×p : ξ>A+A>ξ = 0p
expM(A, ξ) = some complicated expression
logM(A,B) = no explicit expression known!
Important: All operations computable inorder pn operations (not order n2).
Sources: TOP: [Lubich, Nonnenmacher
2010], Bottom: [Sundaramoorthi etal
2011]
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Closest Point Projection
Idea: Define f (A, ξ) by closest-point projection onto St(n,p) fromRn×p:
f (A, ξ) := π(A + ξ).
Algorithmic realization using polar decomposition of A + ξ:
π(A + ξ) = U,
whereA + ξ = UP
is the polar decomposition of A + ξ.
Efficient computation possible.
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Closest Point Projection
Idea: Define f (A, ξ) by closest-point projection onto St(n,p) fromRn×p:
f (A, ξ) := π(A + ξ).
Algorithmic realization using polar decomposition of A + ξ:
π(A + ξ) = U,
whereA + ξ = UP
is the polar decomposition of A + ξ.
Efficient computation possible.
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Computation of the Inverse
Theorem (G 2012)
Assume that −A>B is a Hadamard matrix (only eigenvalues in thenegative half-plane). Then
g(A,B) = 2B∫ ∞
0exp(−tA>B) exp(−tB>A)dt − A.
Fast computation is possible (for instance by alternating projections)!
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Computation of the Inverse
Theorem (G 2012)
Assume that −A>B is a Hadamard matrix (only eigenvalues in thenegative half-plane). Then
g(A,B) = 2B∫ ∞
0exp(−tA>B) exp(−tB>A)dt − A.
Fast computation is possible (for instance by alternating projections)!
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Summary
We are able to define computationally accessible and flexible notionsof geometric means in virtually all (structured) manifolds of practicalinterest.
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A first Application: Subdivision
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Subdivision in SE(3)
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Subdivision in SE(3)
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Subdivision in SE(3)
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Subdivision in SE(3)
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Interactive Motion Design
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Modeling in the Presence of Obstacles (Pottmann,Wallner 2006)
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Smoothness EquivalenceSmoothness equivalence property: nonlinear limit function is assmooth as corresponding linear one.
Theorem (G, SIAM J. Math. Anal. (2010))
The smoothness equivalence property is true for the geometricanalog.
The aim is to establish a (d , f )-proximity condition of the form
‖∆d (T − S)p‖∞ . Ωf (p),
Ωf (p) :=∑j∈Γf
f∏k=1
‖∆k p‖jk∞,
Γf :=
j ∈ Nf
0 :f∑
i=1
iji = f , ‖j‖1 > 1
,
∆p(i) := p(i + 1)− p(i).A perturbation result shows that Cr -smoothness requires(r − 1, r + 1).
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Smoothness EquivalenceSmoothness equivalence property: nonlinear limit function is assmooth as corresponding linear one.
Theorem (G, SIAM J. Math. Anal. (2010))
The smoothness equivalence property is true for the geometricanalog.
The aim is to establish a (d , f )-proximity condition of the form
‖∆d (T − S)p‖∞ . Ωf (p),
Ωf (p) :=∑j∈Γf
f∏k=1
‖∆k p‖jk∞,
Γf :=
j ∈ Nf
0 :f∑
i=1
iji = f , ‖j‖1 > 1
,
∆p(i) := p(i + 1)− p(i).A perturbation result shows that Cr -smoothness requires(r − 1, r + 1).
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Smoothness EquivalenceSmoothness equivalence property: nonlinear limit function is assmooth as corresponding linear one.
Theorem (G, SIAM J. Math. Anal. (2010))
The smoothness equivalence property is true for the geometricanalog.
The aim is to establish a (d , f )-proximity condition of the form
‖∆d (T − S)p‖∞ . Ωf (p),
Ωf (p) :=∑j∈Γf
f∏k=1
‖∆k p‖jk∞,
Γf :=
j ∈ Nf
0 :f∑
i=1
iji = f , ‖j‖1 > 1
,
∆p(i) := p(i + 1)− p(i).A perturbation result shows that Cr -smoothness requires(r − 1, r + 1).
P. Grohs Jun-15-2012, Liege University p. 46
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Smoothness Equivalence
DefinitionS with mask a has polynomial generation degree (PGD) (d , f ) if for allpolynomials p1, . . . ,pk with deg(
∏pj ) < f it holds that∏
j (Spj ) (i)− S(∏
j pj
)(i) = q(i) with some polynomial q of degree
< d .
TheoremAssume S has PGD (d , f ) and
‖∆j p‖∞ . Ω′j (p) j = 1, . . . ,d , (1)
then the geometric analog T satisfies a proximity condition of order(d , f ) with S. The geometric analog of a (stable) linear Cr scheme isCr provided that (1) holds.
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Proof Sketch:Step 1 Taylor expansion of ∆d (T − S)p in p.Step 2 Reduction to existence of ’rewriting rule’ of terms∑
i∈Zk
AiΨ (pi1 , . . . ,pik )
as as linear combinations of terms of the form
Ψ(∆j1pi1 , . . . ,∆
jk pik), ‖j‖1 = f ,
Ψ, Ψ k -multilinear forms.Step 3 Translation of rewriting rule to algebraic properties of the
generating function A(x) of (Ai)i∈Zk (vanishing derivatives at(1, . . . ,1)).
Step 4 Show that these algebraic properties are equivalent to PGD oforder (d , f ).
Step 5 Show that stable S ∈ Cr satisfies PGD of order (r − 1, r + 1).
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Main Result
TheoremThe smoothness equivalence true is true for the geometric analog.
Proof.Using the previous theorem we need to show that
‖∆j p‖∞ . Ω′j (p) j = 1, . . . , r − 1,
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Main Result
TheoremThe smoothness equivalence property is true for the Riemanniananalog.
Proof.Using the previous theorem we need to show that
‖∆jTp‖∞ . Ω′j (p) j = 1, . . . , r − 1,
This is shown using a bootstrapping argument.
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Main Result
TheoremThe smoothness equivalence property is true for the Riemanniananalog.
Proof.Using the previous theorem we need to show that
‖∆jTp‖∞ . Ω′j (p) j = 1, . . . , r − 1,
This is shown using a bootstrapping argument.
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Approximation Order
DefinitionS has approximation order r if for all f ∈ Cr∥∥∥f − S∞f
∣∣hZ
( ·h
)∥∥∥∞
. hr+1.
Theorem (G, J. Approx. Theory (2010))
Approximation order properties of the linear scheme S carry over tothe nonlinear scheme T .
These results imply that almost anything that can be done withB-splines can be done in general manifolds.
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Approximation Order
DefinitionS has approximation order r if for all f ∈ Cr∥∥∥f − S∞f
∣∣hZ
( ·h
)∥∥∥∞
. hr+1.
Theorem (G, J. Approx. Theory (2010))
Approximation order properties of the linear scheme S carry over tothe nonlinear scheme T .
These results imply that almost anything that can be done withB-splines can be done in general manifolds.
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Approximation Order
DefinitionS has approximation order r if for all f ∈ Cr∥∥∥f − S∞f
∣∣hZ
( ·h
)∥∥∥∞
. hr+1.
Theorem (G, J. Approx. Theory (2010))
Approximation order properties of the linear scheme S carry over tothe nonlinear scheme T .
These results imply that almost anything that can be done withB-splines can be done in general manifolds.
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LiteratureH. Karcher, “Mollifier Smoothing and Riemannian Center of Mass”,Comm. Pure and Appl. Anal., (1977).
J. Wallner and N. Dyn, “Convergence and C1-analysis of subdivisionschemes on manifolds by proximity.”, CAGD (2005).
P. Grohs, “A general proximity analysis of nonlinear subdivisionschemes”, SIAM J. Math. Anal. (2010).
O. Ebner, “Nonlinear Markov semigroups and refinement schemes onmetric spaces” arxiv:1112.6003 (2012).
P. Grohs, “Geometric multiscale decompositions of dynamic low-rankmatrices.”, SAM Report 2012-04, (2012).
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Wavelet Transforms
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Interpolatory Wavelets (Decomposition)given dense samples Pn(f ) := (f (i/2n))n∈Zof continuous function f
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Interpolatory Wavelets (Decomposition)
downsampling
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Interpolatory Wavelets (Decomposition)
prediction by interpolatory subdivision schemep(·) 7→ Sp(·) =
∑j∈Z a(· − 2j)p(j)
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Interpolatory Wavelets (Decomposition)
store error
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Interpolatory Wavelets (Decomposition)
next scale
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Geometric Multiscale Decomposition
Let Pnf (i) := f ( i2n ) and T a manifold-valued analog of the subdivision
scheme S (replace weighted averages by nonlinear averages).
DefinitionLet f : R→ M continuous. The geometric multiscale decompositionis defined as (
P0(f ),d1T (f ), . . .
),
wheredn
T (f ) := Pn(f ) TPn−1(f ).
We have
Pn(f ) = (. . .T (T (Pp0(f )⊕ d1T (f ))⊕ d2
T (f ))⊕ . . . )⊕ dnT (f ).
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Norm Equivalences
Theorem (G-Wallner, Appl. Comp. Harmon. Anal. (2009))
Given S interpolatory with polynomial reproduction ≥ d andsmoothness ≥ r . Denote T its geometric analog. Let f : Zk → Mcontinuous with decomposition (P0
T (f ),d1T (f ), . . . ). Then
(i) If f ∈ Bα∞,∞ with α < d, then
‖dnT (f )‖∞ . 2−αn.
(ii) If‖dn
T (f )‖∞ . 2−αn.
with α < r , then f ∈ Bα∞,∞.
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Stability
Is the nonlinear reconstruction procedure stable?
Yes, a typical result from (G, Constr. Approx. (2010)) for Riemannianmanifolds:Define for v, w ∈ TM the distance
dist(v, w) := ‖v − Ptπ(v)π(w)
(w)‖2.
Theorem (G, Constr. Approx. (2010))Given f ∈ Bα∞,∞ for some α > 0 with decomposition (P0(f ), d1
T (f ), . . . ). Let (Q0, e1, . . . ) be another sequence of coefficientswith ∥∥∥dist
(dn
T (f )(·), en(·))∥∥∥∞
. 2−αn.
Then there exists g ∈ Bα∞,∞ with
(P0(g), d1T (g), . . . ) = (Q0
, e1, . . . )
and‖f g(·)‖∞ . ‖P0(f ) Q0(·)‖∞ +
∑n≥1
∥∥∥dist(
dnT (f )(·), en(·)
)∥∥∥∞.
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Stability
Is the nonlinear reconstruction procedure stable?Yes, a typical result from (G, Constr. Approx. (2010)) for Riemannianmanifolds:Define for v, w ∈ TM the distance
dist(v, w) := ‖v − Ptπ(v)π(w)
(w)‖2.
Theorem (G, Constr. Approx. (2010))Given f ∈ Bα∞,∞ for some α > 0 with decomposition (P0(f ), d1
T (f ), . . . ). Let (Q0, e1, . . . ) be another sequence of coefficientswith ∥∥∥dist
(dn
T (f )(·), en(·))∥∥∥∞
. 2−αn.
Then there exists g ∈ Bα∞,∞ with
(P0(g), d1T (g), . . . ) = (Q0
, e1, . . . )
and‖f g(·)‖∞ . ‖P0(f ) Q0(·)‖∞ +
∑n≥1
∥∥∥dist(
dnT (f )(·), en(·)
)∥∥∥∞.
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Best N-term Approximation
Assume M a general manifold with operations ⊕, and weightedaverage induced by retraction pair.
Theorem (G 2012)
Let f : [−1,1]→ M, f∣∣[−1,0)
, f∣∣[0,1]∈ Cs. Then, using N wavelet
coefficients, one can approximate f up to an error of order N−s in theL2 norm.
This is as good as if there were no singularity!
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Best N-term Approximation
Assume M a general manifold with operations ⊕, and weightedaverage induced by retraction pair.
Theorem (G 2012)
Let f : [−1,1]→ M, f∣∣[−1,0)
, f∣∣[0,1]∈ Cs. Then, using N wavelet
coefficients, one can approximate f up to an error of order N−s in theL2 norm.
This is as good as if there were no singularity!
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Best N-term Approximation
Assume M a general manifold with operations ⊕, and weightedaverage induced by retraction pair.
Theorem (G 2012)
Let f : [−1,1]→ M, f∣∣[−1,0)
, f∣∣[0,1]∈ Cs. Then, using N wavelet
coefficients, one can approximate f up to an error of order N−s in theL2 norm.
This is as good as if there were no singularity!
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Proof Sketch
Step 1: Collect coefficients not contributing to the singularity up toscale log2(N) which needs O(N) coefficients. By a result of[G-Wallner ACHA (2009)] these coefficients decay of order 2js, wherej is the scale.
Step 2: For each scale j there are only a fixed number L ofcoefficients which feel the singularity. Collect also these up to scale2s log2(N). Coefficient count: 2Ls log2(N) = O(N).
Step 3: Away from an interval I of size ∼ 2−2s log2(N) we have onlythresholded coefficients which decay of order 2−sj . By a result of [GConstr. Approx. (2010)], the reconstruction precedure is stable, whichimplies that on [0,1] \ I there is a uniform (and thus L2-) error of orderN−s.
Step 4: On I we have a uniformly bounded error. Since I is of size2−2s log2(N) ∼ N−2s, the L2 error is of the same order.
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Proof SketchStep 1: Collect coefficients not contributing to the singularity up toscale log2(N) which needs O(N) coefficients. By a result of[G-Wallner ACHA (2009)] these coefficients decay of order 2js, wherej is the scale.
Step 2: For each scale j there are only a fixed number L ofcoefficients which feel the singularity. Collect also these up to scale2s log2(N). Coefficient count: 2Ls log2(N) = O(N).
Step 3: Away from an interval I of size ∼ 2−2s log2(N) we have onlythresholded coefficients which decay of order 2−sj . By a result of [GConstr. Approx. (2010)], the reconstruction precedure is stable, whichimplies that on [0,1] \ I there is a uniform (and thus L2-) error of orderN−s.
Step 4: On I we have a uniformly bounded error. Since I is of size2−2s log2(N) ∼ N−2s, the L2 error is of the same order.
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Proof SketchStep 1: Collect coefficients not contributing to the singularity up toscale log2(N) which needs O(N) coefficients. By a result of[G-Wallner ACHA (2009)] these coefficients decay of order 2js, wherej is the scale.
Step 2: For each scale j there are only a fixed number L ofcoefficients which feel the singularity. Collect also these up to scale2s log2(N). Coefficient count: 2Ls log2(N) = O(N).
Step 3: Away from an interval I of size ∼ 2−2s log2(N) we have onlythresholded coefficients which decay of order 2−sj . By a result of [GConstr. Approx. (2010)], the reconstruction precedure is stable, whichimplies that on [0,1] \ I there is a uniform (and thus L2-) error of orderN−s.
Step 4: On I we have a uniformly bounded error. Since I is of size2−2s log2(N) ∼ N−2s, the L2 error is of the same order.
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Proof SketchStep 1: Collect coefficients not contributing to the singularity up toscale log2(N) which needs O(N) coefficients. By a result of[G-Wallner ACHA (2009)] these coefficients decay of order 2js, wherej is the scale.
Step 2: For each scale j there are only a fixed number L ofcoefficients which feel the singularity. Collect also these up to scale2s log2(N). Coefficient count: 2Ls log2(N) = O(N).
Step 3: Away from an interval I of size ∼ 2−2s log2(N) we have onlythresholded coefficients which decay of order 2−sj . By a result of [GConstr. Approx. (2010)], the reconstruction precedure is stable, whichimplies that on [0,1] \ I there is a uniform (and thus L2-) error of orderN−s.
Step 4: On I we have a uniformly bounded error. Since I is of size2−2s log2(N) ∼ N−2s, the L2 error is of the same order.
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Proof SketchStep 1: Collect coefficients not contributing to the singularity up toscale log2(N) which needs O(N) coefficients. By a result of[G-Wallner ACHA (2009)] these coefficients decay of order 2js, wherej is the scale.
Step 2: For each scale j there are only a fixed number L ofcoefficients which feel the singularity. Collect also these up to scale2s log2(N). Coefficient count: 2Ls log2(N) = O(N).
Step 3: Away from an interval I of size ∼ 2−2s log2(N) we have onlythresholded coefficients which decay of order 2−sj . By a result of [GConstr. Approx. (2010)], the reconstruction precedure is stable, whichimplies that on [0,1] \ I there is a uniform (and thus L2-) error of orderN−s.
Step 4: On I we have a uniformly bounded error. Since I is of size2−2s log2(N) ∼ N−2s, the L2 error is of the same order.
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Remarks
This result is optimal and as good as the corresponding result forEuclidean data.
Wavelet trafo is nonadaptive, compression goes simply bythresholding small coefficients; the singularity is foundautomatically.
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Remarks
This result is optimal and as good as the corresponding result forEuclidean data.
Wavelet trafo is nonadaptive, compression goes simply bythresholding small coefficients; the singularity is foundautomatically.
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Remarks
This result is optimal and as good as the corresponding result forEuclidean data.
Wavelet trafo is nonadaptive, compression goes simply bythresholding small coefficients; the singularity is foundautomatically.
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Examples
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Flight Data US Air Compression
original data 10 : 1 compression
Data source: www-stat.stanford.edu/$\sim$symmlab
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Dynamical Low-Rank MatricesThe matrices of fixed rank form a smooth manifold:
Mn,p :=
A ∈ Rn×n : rank(A) = p.
Piecewise smooth curves γ : t 7→ A(t) ∈Mn,p can be decomposedinto smooth curves on the Stiefel manifold:
Theorem (Smooth Singular Value Decomposition)
Assume that γ as above is a (piecewise) Cs smooth curve. Thenthere exist (piecewise) Cs smooth curves
γU : t 7→ U(t) ∈ St(n,p), γV : t 7→ V(t) ∈ St(n,p), γS : t 7→ S(t) ∈ Rp×p
such thatA(t) = U(t)S(t)V(t)>.
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Occurences
Video compression Latent semantic indexing Low-rank (tensor) approximation of high-dimensional evolution
equations
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A Smooth Example
Smooth matrix curve
A(t) = U(t)Σ(t)V(t)> = sin (tE) + noise ∈ R10×10.
Use rank 2 approximation.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17
6
5
4
3
2
7
6
5
4
3
original data
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17
6
5
4
3
2
7
6
5
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thresholded coefficients
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0 20 40 60 80 100 120 140
0.16
0.18
0.2
0.22
0.24
0.26
coordinate plot
0 20 40 60 80 100 120 1402
4
6
8
10
12
14
16
18
20
22
two largest singular values
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A singular Example
Singular matrix curve
A(t) = U(t)Σ(t)V(t)> = sin (tE) +(1− t2)1/2
cos (tF) ∈ R10×10.
Use rank 3 approximation.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17
6
5
4
3
2
7
6
5
4
3
original data
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17
6
5
4
3
2
7
6
5
4
3
thresholded coefficients
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Real Data
A hyperspectral image is an image where intensities corresponding tomany different wavelengths are recorded separately. The result is aparametrized family A(t) of matrices, t being the wavelength. A(t) isa smooth curve!
Applications: Agriculture, Mineraology, Surveillance, ...
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Real Data
A hyperspectral image is an image where intensities corresponding tomany different wavelengths are recorded separately. The result is aparametrized family A(t) of matrices, t being the wavelength. A(t) isa smooth curve!Applications: Agriculture, Mineraology, Surveillance, ...
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Hyperspectral Image Compression
Given hyperspectal image A ∈ R144×144×128.1. Decompose A into small 12× 12× 128 blocks2. Approximate each block at each wavelength with a rank 2
approximation3. Perform wavelet thresholding on each low-rank approximation
Results in more than 90% saving with a relative error of order 0.01.
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Hyperspectral Image Compression
Given hyperspectal image A ∈ R144×144×128.1. Decompose A into small 12× 12× 128 blocks2. Approximate each block at each wavelength with a rank 2
approximation3. Perform wavelet thresholding on each low-rank approximation
Results in more than 90% saving with a relative error of order 0.01.
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Further Work
Feature detection, inpainting, ... Wavelet analysis in curve shape space
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LiteratureI. UrRahman etal., “Multiscale Representation of manifold-valueddata”, SIAM Mult. Mod. Sim., (2006).
P. Grohs and J. Wallner, “Interpolatory wavelets for manifold-valueddata”, ACHA (2009).
P. Grohs, “Stability of manifold-valued subdivision schemes andmultiscale transforms”, Constr. Approx. (2010).
O. Koch and Ch. Lubich, “Dynamical low-rank approximation” SIMAX(2007).
P. Grohs, “Geometric multiscale decompositions of dynamic low-rankmatrices.”, SAM Report 2012-04, (2012).
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Geodesic Finite Elements1
1ongoing joint work with Prof. Oliver Sander and Hanne Hardening (FU Berlin)
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Problem
Seek minimizer Φ : Ω→ M of manifold-valued variational problemargminJ(Φ), J : H1(Ω,M)→ R.
Applications:Liquid crystals: S2, PR2, SO(3)Cosserat shells: S2, SO(3)Image processing: S2
Problem: DiscretizationFinite elements cannot be used, because space M is nonlinearKey problem: enforce nonlinear constraints given by M
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Recall...
0 1
1
f (x) =∑
i ϕi (x)fiwith shape functions ϕi , satisfying∑
i ϕi (x) ≡ 1.
x-dependent weighted average of points fi !
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Geodesic Finite Elements
Let ϕpi be the p-th order scalar Lagrangian shape functions. ∆
reference simplex with coordinates ξ.
DefinitionGFE function
Υp(v , ξ) =⊕
i
ϕpi (ξ)vi .
GFE space associated to a simplicial decomposition: restriction toeach element must be GFE function. DOF’s: Control points vi ∈ M.
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Results (in preparation)
Yields H1-conforming discretization. Nonlinear Bramble-Hilbert lemma (optimal approximation order
of GFE spaces). Nonlinear Céa lemma (coerciveness⇒ minimization of J on
GFE space yields quasioptimal approximation)
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Harmonic Mappings
Harmonic energy
J(Φ) =
∫Ω
‖dΦ∣∣Φ(x)‖2
Φ(x)dx
with Dirichlet BC’s.
Used as simple model in liquid-crystal theory, coercivity results e.g.by [Haskins-Speight (2003)].
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Ω = [0,1]2, M = S2
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
0.01 0.1 1
err
or
mesh size
L2 errorh1 errorO(h^2)O(h^3)
Source: Oliver Sander
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Remarks
Discrete problems solved using Riemannian trust region schemedeveloped by Absil etal.
Method assumes existence of smooth solution which is notalways the case (but in many cases it is!)
Future work will also address time-dependent problems withpossible applications in image processing.
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LiteratureP.-A. Absil, R. Mahoney and R. Sepulchre “Optimization algorithmson matrix manifolds”, Princeton University Press, (2008).
O. Sander., “Geodesic finite elements for Cosserat rods”, Int. J. Num.Meth. Eng., (2011).
O. Sander, “Geodesic finite elements on simplicial grids”, Int. J. Num.Meth. Eng. (2012) to appear.
P. Grohs, “Finite elements of arbitrary order and quasiinterpolation forRiemannian data”, SAM report 2011-56 (2011).
P. Grohs, H. Hardening and O. Sander, “A convergence theory forGFEs” in preparation.
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The End
Questions?
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