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An SVD in Spherical Surface Wave Tomography Chemnitz University of Technology, Faculty of Mathematics An SVD in Spherical Surface Wave Tomography Michael Quellmalz (joint work with Ralf Hielscher and Daniel Potts) Chemnitz University of
An SVD in Spherical Surface Wave Tomography Chemnitz University of Technology, Faculty of Mathematics
An SVD in Spherical Surface Wave Tomography
Michael Quellmalz (joint work with Ralf Hielscher and Daniel Potts)
Chemnitz University of Technology Faculty of Mathematics
New Trends in Parameter Identification for Mathematical Models Chemnitz Symposium on Inverse Problems on Tour Rio de Janeiro, 2 November 2017
2 November 2017 · Michael Quellmalz 1 / 24 tu-chemnitz.de/∼qmi
An SVD in Spherical Surface Wave Tomography
Content
Motivation
Definition Singular value decomposition
Arcs starting in a fixed point Recovery of local functions Arcs with fixed length
2 November 2017 · Michael Quellmalz 2 / 24 tu-chemnitz.de/∼qmi
Introduction
Content
Motivation
Definition Singular value decomposition
Arcs starting in a fixed point Recovery of local functions Arcs with fixed length
2 November 2017 · Michael Quellmalz 3 / 24 tu-chemnitz.de/∼qmi
Introduction
Funk–Radon transform
◮ Sphere S2 = {ξ ∈ R3 : ξ = 1} ◮ Function f : S2 → C ◮ Funk–Radon transform (a.k.a. Funk
transform or spherical Radon transform) F : C(S2) → C(S2), Ff(ξ) =
f(η) dλ(η)
Theorem
[Funk 1911]
Any even function f can be reconstructed from Ff.
2 November 2017 · Michael Quellmalz 4 / 24 tu-chemnitz.de/∼qmi
Introduction
Funk–Radon transform
◮ Sphere S2 = {ξ ∈ R3 : ξ = 1} ◮ Function f : S2 → C ◮ Funk–Radon transform (a.k.a. Funk
transform or spherical Radon transform) F : C(S2) → C(S2), Ff(ξ) =
f(η) dλ(η)
Theorem
[Funk 1911]
Any even function f can be reconstructed from Ff.
2 November 2017 · Michael Quellmalz 4 / 24 tu-chemnitz.de/∼qmi
Introduction
Funk–Radon transform
◮ Sphere S2 = {ξ ∈ R3 : ξ = 1} ◮ Function f : S2 → C ◮ Funk–Radon transform (a.k.a. Funk
transform or spherical Radon transform) F : C(S2) → C(S2), Ff(ξ) =
f(η) dλ(η)
Theorem
[Funk 1911]
Any even function f can be reconstructed from Ff.
2 November 2017 · Michael Quellmalz 4 / 24 tu-chemnitz.de/∼qmi
Introduction Motivation
Spherical surface wave tomography
◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S2
Method
◮ Measure the traveltimes of surface waves between many pairs of
epicenter and detector
◮ Reconstruct the local speed of propagation
Assumption
A wave propagates along the arc of a great circle.
2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/∼qmi
Introduction Motivation
Spherical surface wave tomography
◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S2
Method
◮ Measure the traveltimes of surface waves between many pairs of
epicenter and detector
◮ Reconstruct the local speed of propagation
Assumption
A wave propagates along the arc of a great circle.
2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/∼qmi
Introduction Motivation
Spherical surface wave tomography
◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S2
Method
◮ Measure the traveltimes of surface waves between many pairs of
epicenter and detector
◮ Reconstruct the local speed of propagation
Assumption
A wave propagates along the arc of a great circle.
2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/∼qmi
Introduction Motivation
Spherical surface wave tomography
◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S2
Method
◮ Measure the traveltimes of surface waves between many pairs of
epicenter and detector
◮ Reconstruct the local speed of propagation
Assumption
A wave propagates along the arc of a great circle.
2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/∼qmi
Introduction Motivation
Spherical surface wave tomography
◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S2
Method
◮ Measure the traveltimes of surface waves between many pairs of
epicenter and detector
◮ Reconstruct the local speed of propagation
Assumption
A wave propagates along the arc of a great circle.
2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/∼qmi
Introduction Motivation
Spherical surface wave tomography
◮ Seismic waves propagate along the surface of the earth ◮ Speed of propagation depends on the position on S2
Method
◮ Measure the traveltimes of surface waves between many pairs of
epicenter and detector
◮ Reconstruct the local speed of propagation
Assumption
A wave propagates along the arc of a great circle.
2 November 2017 · Michael Quellmalz 5 / 24 tu-chemnitz.de/∼qmi
Introduction Selected references
¨ Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien.
Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms.
Parametrizing surface wave tomographic models with harmonic spherical splines.
An SVD in spherical surface wave tomography In B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in Parameter Identification for Mathematical Models. Birkh¨ auser, Basel, 2018. https://arxiv.org/abs/1706.05284
2 November 2017 · Michael Quellmalz 6 / 24 tu-chemnitz.de/∼qmi
Introduction Selected references
¨ Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien.
Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms.
Parametrizing surface wave tomographic models with harmonic spherical splines.
An SVD in spherical surface wave tomography In B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in Parameter Identification for Mathematical Models. Birkh¨ auser, Basel, 2018. https://arxiv.org/abs/1706.05284
2 November 2017 · Michael Quellmalz 6 / 24 tu-chemnitz.de/∼qmi
Introduction Selected references
¨ Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien.
Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms.
Parametrizing surface wave tomographic models with harmonic spherical splines.
An SVD in spherical surface wave tomography In B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in Parameter Identification for Mathematical Models. Birkh¨ auser, Basel, 2018. https://arxiv.org/abs/1706.05284
2 November 2017 · Michael Quellmalz 6 / 24 tu-chemnitz.de/∼qmi
Introduction Selected references
¨ Uber Fl¨ achen mit lauter geschlossenen geod¨ atischen Linien.
Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms.
Parametrizing surface wave tomographic models with harmonic spherical splines.
An SVD in spherical surface wave tomography In B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in Parameter Identification for Mathematical Models. Birkh¨ auser, Basel, 2018. https://arxiv.org/abs/1706.05284
2 November 2017 · Michael Quellmalz 6 / 24 tu-chemnitz.de/∼qmi
Arc transform
Content
Motivation
Definition Singular value decomposition
Arcs starting in a fixed point Recovery of local functions Arcs with fixed length
2 November 2017 · Michael Quellmalz 7 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform
◮ Function f : S2 → R
◮ Surface waves: f = 1
c
(c ... speed of sound)
◮ ξ, ζ ∈ S2 not antipodal ◮ γ(ξ, ζ) great circle arc
Definition
t(ξ, ζ) =
f dγ
2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform
◮ Function f : S2 → R
◮ Surface waves: f = 1
c
(c ... speed of sound)
◮ ξ, ζ ∈ S2 not antipodal ◮ γ(ξ, ζ) great circle arc
Definition
t(ξ, ζ) =
f dγ
2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform
◮ Function f : S2 → R
◮ Surface waves: f = 1
c
(c ... speed of sound)
◮ ξ, ζ ∈ S2 not antipodal ◮ γ(ξ, ζ) great circle arc
Definition
t(ξ, ζ) =
f dγ
2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform
◮ Function f : S2 → R
◮ Surface waves: f = 1
c
(c ... speed of sound)
◮ ξ, ζ ∈ S2 not antipodal ◮ γ(ξ, ζ) great circle arc
Definition
t(ξ, ζ) =
f dγ
2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform
◮ Function f : S2 → R
◮ Surface waves: f = 1
c
(c ... speed of sound)
◮ ξ, ζ ∈ S2 not antipodal ◮ γ(ξ, ζ) great circle arc
Definition
t(ξ, ζ) =
f dγ
2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform
◮ Function f : S2 → R
◮ Surface waves: f = 1
c
(c ... speed of sound)
◮ ξ, ζ ∈ S2 not antipodal ◮ γ(ξ, ζ) great circle arc
Definition
t(ξ, ζ) =
f dγ t is not continuous on S2 × S2
2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform
◮ Function f : S2 → R
◮ Surface waves: f = 1
c
(c ... speed of sound)
◮ ξ, ζ ∈ S2 not antipodal ◮ γ(ξ, ζ) great circle arc
Definition
t(ξ, ζ) =
f dγ t is not continuous on S2 × S2 We choose a different parameterization
2 November 2017 · Michael Quellmalz 8 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform: alternative parameterization
◮ ψ = arccos(ξ⊤ζ) ... length of γ ◮ Q ∈ SO(3) such that
◮ Qξ = e−ψ/2 and ◮ Qζ = eψ/2,
where eψ = (sin ψ, cos ψ, 0)
Definition
A: C(S2) → C(SO(3) × [0, 2π]), Af(Q, ψ) = ψ/2
−ψ/2
f(Q−1eϕ) dϕ
2 November 2017 · Michael Quellmalz 9 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform: alternative parameterization
◮ ψ = arccos(ξ⊤ζ) ... length of γ ◮ Q ∈ SO(3) such that
◮ Qξ = e−ψ/2 and ◮ Qζ = eψ/2,
where eψ = (sin ψ, cos ψ, 0)
Definition
A: C(S2) → C(SO(3) × [0, 2π]), Af(Q, ψ) = ψ/2
−ψ/2
f(Q−1eϕ) dϕ
2 November 2017 · Michael Quellmalz 9 / 24 tu-chemnitz.de/∼qmi
Arc transform Definition
The arc transform: alternative parameterization
◮ ψ = arccos(ξ⊤ζ) ... length of γ ◮ Q ∈ SO(3) such that
◮ Qξ = e−ψ/2 and ◮ Qζ = eψ/2,
where eψ = (sin ψ, cos ψ, 0)
Definition
A: C(S2) → C(SO(3) × [0, 2π]), Af(Q, ψ) = ψ/2
−ψ/2
f(Q−1eϕ) dϕ
2 November 2017 · Michael Quellmalz 9 / 24 tu-chemnitz.de/∼qmi
Arc transform Singular value decomposition
Notation: On the sphere S2
◮ Spherical coordinates
ξ(ϕ, ϑ) = sin(ϑ)eϕ + cos ϑ
1
Y k
n (ϕ, ϑ) =
4π (n − k)! (n + k)! P k
n(cos ϑ) eikϕ,
k = −n, . . . , n
◮ P k n ... associated Legendre function
2 November 2017 · Michael Quellmalz 10 / 24 tu-chemnitz.de/∼qmi
Arc transform Singular value decomposition
Notation: On the sphere S2
◮ Spherical coordinates
ξ(ϕ, ϑ) = sin(ϑ)eϕ + cos ϑ
1
Y k
n (ϕ, ϑ) =
4π (n − k)! (n + k)! P k
n(cos ϑ) eikϕ,
k = −n, . . . , n
◮ P k n ... associated Legendre function
2 November 2017 · Michael Quellmalz 10 / 24 tu-chemnitz.de/∼qmi
Arc transform Singular value decomposition
Notation: On the rotation group SO(3)
◮ Rotation group
SO(3) = {Q ∈ R3×3 : Q−1 = Q⊤, det(Q) = 1}
◮ Orthogonal basis on L2(SO(3)): rotational harmonics (Wigner
D-functions) Dj,k
n (Q) =
n (Q−1ξ) Y j n (ξ) dξ
2 November 2017 · Michael Quellmalz 11 / 24 tu-chemnitz.de/∼qmi
Arc transform Singular value decomposition
Notation: On the rotation group SO(3)
◮ Rotation group
SO(3) = {Q ∈ R3×3 : Q−1 = Q⊤, det(Q) = 1}
◮ Orthogonal basis on L2(SO(3)): rotational harmonics (Wigner
D-functions) Dj,k
n (Q) =
n (Q−1ξ) Y j n (ξ) dξ
2 November 2017 · Michael Quellmalz 11 / 24 tu-chemnitz.de/∼qmi
Arc transform Singular value decomposition
Theorem
[Dahlen & Tromp 1998]
Let n ∈ N and k ∈ {−n, . . . , n}. Then AY k
n (Q, ψ) = n
n(0) Dj,k n (Q) sj(ψ),
where sj(ψ) =
j = 0
2 sin(jψ/2) j
, j = 0 and
n(0) =
n+j 2
4π (n−j−1)!!(n+j−1)!! (n−j)!!(n+j)!!
, n + j even 0, n + j odd.
2 November 2017 · Michael Quellmalz 12 / 24 tu-chemnitz.de/∼qmi
Arc transform Singular value decomposition
Singular value decomposition
[Hielscher, Potts, Q. 2017]
The operator A: L2(S2) → L2(SO(3) × [0, 2π]) is compact with the singular value decomposition AY k
n = σn Ek n,
n ∈ N, k ∈ {−n, . . . , n}, with singular values σn =
2n + 1
3
n(0)
+
n
1 j2
n(0)
∈ O 1 √n
En
k = σ−1 n n
n(0) Dj,k n (Q) sj(ψ).
2 November 2017 · Michael Quellmalz 13 / 24 tu-chemnitz.de/∼qmi
Special cases
Content
Motivation
Definition Singular value decomposition
Arcs starting in a fixed point Recovery of local functions Arcs with fixed length
2 November 2017 · Michael Quellmalz 14 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs starting in a fixed point
Arcs from the north pole
◮ Fix one endpoint of the arcs as the north
pole e3: Bf(ξ(ϕ, ϑ)) =
f dγ
◮ If f is differentiable, it can be recovered
from Bf by f(ξ(ϕ, ϑ)) = d dϑBf(ξ(ϕ, ϑ)).
2 November 2017 · Michael Quellmalz 15 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs starting in a fixed point
Arcs from the north pole
◮ Fix one endpoint of the arcs as the north
pole e3: Bf(ξ(ϕ, ϑ)) =
f dγ
◮ If f is differentiable, it can be recovered
from Bf by f(ξ(ϕ, ϑ)) = d dϑBf(ξ(ϕ, ϑ)).
2 November 2017 · Michael Quellmalz 15 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs starting in a fixed point
Arcs between two sets
More general Theorem
[Amirbekyan 2007]
Let S be an open subset of S2 and A, B ⊂ S nonempty sets with A ∪ B = S. If f ∈ C(S2) and
f dγ = 0 ∀ξ ∈ A, ζ ∈ B, then f ≡ 0 on S.
2 November 2017 · Michael Quellmalz 16 / 24 tu-chemnitz.de/∼qmi
Special cases Recovery of local functions
Theorem
[Hielscher, Potts, Q. 2017]
Let f ∈ C(S2) and Ω be a convex subset of S2 whose closure Ω is strictly contained in a hemisphere, i.e., there exists a ζ ∈ S2 such that ξ, ζ > 0 for all ξ ∈ Ω. If
f dγ = 0 for all ξ, η ∈ ∂Ω, (1) then f = 0 on Ω.
Proof
◮ Extend f to zero outside Ω ◮ (1) implies that the Funk–Radon transform of f must vanish ◮ f must be odd ◮ Because supp f ⊂ ¯
Ω is contained in a hemisphere, f must vanish
2 November 2017 · Michael Quellmalz 17 / 24 tu-chemnitz.de/∼qmi
Special cases Recovery of local functions
Theorem
[Hielscher, Potts, Q. 2017]
Let f ∈ C(S2) and Ω be a convex subset of S2 whose closure Ω is strictly contained in a hemisphere, i.e., there exists a ζ ∈ S2 such that ξ, ζ > 0 for all ξ ∈ Ω. If
f dγ = 0 for all ξ, η ∈ ∂Ω, (1) then f = 0 on Ω.
Proof
◮ Extend f to zero outside Ω ◮ (1) implies that the Funk–Radon transform of f must vanish ◮ f must be odd ◮ Because supp f ⊂ ¯
Ω is contained in a hemisphere, f must vanish
2 November 2017 · Michael Quellmalz 17 / 24 tu-chemnitz.de/∼qmi
Special cases Recovery of local functions
Theorem
[Hielscher, Potts, Q. 2017]
Let f ∈ C(S2) and Ω be a convex subset of S2 whose closure Ω is strictly contained in a hemisphere, i.e., there exists a ζ ∈ S2 such that ξ, ζ > 0 for all ξ ∈ Ω. If
f dγ = 0 for all ξ, η ∈ ∂Ω, (1) then f = 0 on Ω.
Proof
◮ Extend f to zero outside Ω ◮ (1) implies that the Funk–Radon transform of f must vanish ◮ f must be odd ◮ Because supp f ⊂ ¯
Ω is contained in a hemisphere, f must vanish
2 November 2017 · Michael Quellmalz 17 / 24 tu-chemnitz.de/∼qmi
Special cases Recovery of local functions
Theorem
[Hielscher, Potts, Q. 2017]
Let f ∈ C(S2) and Ω be a convex subset of S2 whose closure Ω is strictly contained in a hemisphere, i.e., there exists a ζ ∈ S2 such that ξ, ζ > 0 for all ξ ∈ Ω. If
f dγ = 0 for all ξ, η ∈ ∂Ω, (1) then f = 0 on Ω.
Proof
◮ Extend f to zero outside Ω ◮ (1) implies that the Funk–Radon transform of f must vanish ◮ f must be odd ◮ Because supp f ⊂ ¯
Ω is contained in a hemisphere, f must vanish
2 November 2017 · Michael Quellmalz 17 / 24 tu-chemnitz.de/∼qmi
Special cases Recovery of local functions
Theorem
[Hielscher, Potts, Q. 2017]
Let f ∈ C(S2) and Ω be a convex subset of S2 whose closure Ω is strictly contained in a hemisphere, i.e., there exists a ζ ∈ S2 such that ξ, ζ > 0 for all ξ ∈ Ω. If
f dγ = 0 for all ξ, η ∈ ∂Ω, (1) then f = 0 on Ω.
Proof
◮ Extend f to zero outside Ω ◮ (1) implies that the Funk–Radon transform of f must vanish ◮ f must be odd ◮ Because supp f ⊂ ¯
Ω is contained in a hemisphere, f must vanish
2 November 2017 · Michael Quellmalz 17 / 24 tu-chemnitz.de/∼qmi
Special cases Recovery of local functions
Theorem
[Hielscher, Potts, Q. 2017]
Let f ∈ C(S2) and Ω be a convex subset of S2 whose closure Ω is strictly contained in a hemisphere, i.e., there exists a ζ ∈ S2 such that ξ, ζ > 0 for all ξ ∈ Ω. If
f dγ = 0 for all ξ, η ∈ ∂Ω, (1) then f = 0 on Ω.
Proof
◮ Extend f to zero outside Ω ◮ (1) implies that the Funk–Radon transform of f must vanish ◮ f must be odd ◮ Because supp f ⊂ ¯
Ω is contained in a hemisphere, f must vanish
2 November 2017 · Michael Quellmalz 17 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Arcs with fixed length
We fix the arclength ψ ∈ [0, 2π] and define Aψ = A(·, ψ): L2(S2) → L2(SO(3)).
2 November 2017 · Michael Quellmalz 18 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Arcs with fixed length
We fix the arclength ψ ∈ [0, 2π] and define Aψ = A(·, ψ): L2(S2) → L2(SO(3)).
2 November 2017 · Michael Quellmalz 18 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular Value Decomposition
[Hielscher, Potts, Q. 2017]
Let ψ ∈ (0, 2π) be fixed. The operator Aψ : L2(S2) → L2(SO(3)) has the SVD AψY k
n = µn(ψ) Zk n,ψ,
n ∈ N, k ∈ {−n, . . . , n}, with singular values µn(ψ) =
8π2 2n + 1
n(0)
sj(ψ)2 and singular functions Zk
n,ψ =
1 µn(ψ)
n
n(0) sj(ψ) Dj,k n
∈ L2(SO(3)). Hence Aψ is injective.
2 November 2017 · Michael Quellmalz 19 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on n
ψ = 0.02 π 5 10 15 20 25 30 35 40 45 50 0.2 0.4 0.6 0.8 polynomial degree n (n + 1
2) µn(ψ)2
2 November 2017 · Michael Quellmalz 20 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on n
ψ = 0.10 π 5 10 15 20 25 30 35 40 45 50 2 4 polynomial degree n (n + 1
2) µn(ψ)2
2 November 2017 · Michael Quellmalz 20 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on n
ψ = 0.20 π 5 10 15 20 25 30 35 40 45 50 2 4 6 8 polynomial degree n (n + 1
2) µn(ψ)2
2 November 2017 · Michael Quellmalz 20 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on n
ψ = 0.40 π 5 10 15 20 25 30 35 40 45 50 5 10 15 polynomial degree n (n + 1
2) µn(ψ)2
2 November 2017 · Michael Quellmalz 20 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on n
ψ = 1.00 π (half circle) 5 10 15 20 25 30 35 40 45 50 32 34 36 38 40 polynomial degree n (n + 1
2) µn(ψ)2
2 November 2017 · Michael Quellmalz 20 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on n
ψ = 1.04 π 5 10 15 20 25 30 35 40 45 50 35 40 45 polynomial degree n (n + 1
2) µn(ψ)2
2 November 2017 · Michael Quellmalz 20 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on n
ψ = 1.90 π 5 10 15 20 25 30 35 40 45 50 50 100 150 polynomial degree n (n + 1
2) µn(ψ)2
2 November 2017 · Michael Quellmalz 20 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on n
ψ = 2.00 π (Funk–Radon transform) 5 10 15 20 25 30 35 40 45 50 50 100 150 polynomial degree n (n + 1
2) µn(ψ)2
2 November 2017 · Michael Quellmalz 20 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on arc-length ψ
π 2π 50 100 150 ψ (n + 1
2) µn(ψ)2
n even limit n = 0 π 2π 10 20 30 40 ψ n odd limit n = 1
2 November 2017 · Michael Quellmalz 21 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on arc-length ψ
π 2π 50 100 150 ψ (n + 1
2) µn(ψ)2
n even limit n = 2 π 2π 10 20 30 40 ψ n odd limit n = 3
2 November 2017 · Michael Quellmalz 21 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on arc-length ψ
π 2π 50 100 150 ψ (n + 1
2) µn(ψ)2
n even limit n = 4 π 2π 10 20 30 40 ψ n odd limit n = 5
2 November 2017 · Michael Quellmalz 21 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on arc-length ψ
π 2π 50 100 150 ψ (n + 1
2) µn(ψ)2
n even limit n = 6 π 2π 10 20 30 40 ψ n odd limit n = 7
2 November 2017 · Michael Quellmalz 21 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on arc-length ψ
π 2π 50 100 150 ψ (n + 1
2) µn(ψ)2
n even limit n = 8 π 2π 10 20 30 40 ψ n odd limit n = 9
2 November 2017 · Michael Quellmalz 21 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values µn(ψ): dependency on arc-length ψ
π 2π 50 100 150 ψ (n + 1
2) µn(ψ)2
n even limit n = 100 π 2π 10 20 30 40 ψ n odd limit n = 101
2 November 2017 · Michael Quellmalz 21 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
Singular values: asymptotic behavior
Theorem
[Hielscher, Potts, Q. 2017]
The singular values µn(ψ) of Aψ satisfy for odd n = 2m − 1 lim
m→∞
4m − 1 4 µ2m−1(ψ)2 =
ψ ∈ [0, π] 4π2 − 2πψ, ψ ∈ [π, 2π], and for even n = 2m lim
m→∞
4m + 1 4 µ2m(ψ)2 =
ψ ∈ [0, π] 12πψ − 2π2, ψ ∈ [π, 2π].
2 November 2017 · Michael Quellmalz 22 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
ψ = 2π: Great circles
[Funk 1911]
◮ Funk–Radon transform
F : C(S2) → C(S2), Ff(ξ) =
f(η) dγ(η)
◮ Injective only for even functions
ψ = π: Half-circle transform
◮ Injective for all functions [Groemer 1998], [Goodey & Weil 2006] ◮ [Rubin 2017] half circles in one hemisphere
2 November 2017 · Michael Quellmalz 23 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
ψ = 2π: Great circles
[Funk 1911]
◮ Funk–Radon transform
F : C(S2) → C(S2), Ff(ξ) =
f(η) dγ(η)
◮ Injective only for even functions
ψ = π: Half-circle transform
◮ Injective for all functions [Groemer 1998], [Goodey & Weil 2006] ◮ [Rubin 2017] half circles in one hemisphere
2 November 2017 · Michael Quellmalz 23 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
ψ = 2π: Great circles
[Funk 1911]
◮ Funk–Radon transform
F : C(S2) → C(S2), Ff(ξ) =
f(η) dγ(η)
◮ Injective only for even functions
ψ = π: Half-circle transform
◮ Injective for all functions [Groemer 1998], [Goodey & Weil 2006] ◮ [Rubin 2017] half circles in one hemisphere
2 November 2017 · Michael Quellmalz 23 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
ψ = 2π: Great circles
[Funk 1911]
◮ Funk–Radon transform
F : C(S2) → C(S2), Ff(ξ) =
f(η) dγ(η)
◮ Injective only for even functions
ψ = π: Half-circle transform
◮ Injective for all functions [Groemer 1998], [Goodey & Weil 2006] ◮ [Rubin 2017] half circles in one hemisphere
2 November 2017 · Michael Quellmalz 23 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
ψ = 2π: Great circles
[Funk 1911]
◮ Funk–Radon transform
F : C(S2) → C(S2), Ff(ξ) =
f(η) dγ(η)
◮ Injective only for even functions
ψ = π: Half-circle transform
◮ Injective for all functions [Groemer 1998], [Goodey & Weil 2006] ◮ [Rubin 2017] half circles in one hemisphere
2 November 2017 · Michael Quellmalz 23 / 24 tu-chemnitz.de/∼qmi
Special cases Arcs with fixed length
ψ = 2π: Great circles
[Funk 1911]
◮ Funk–Radon transform
F : C(S2) → C(S2), Ff(ξ) =
f(η) dγ(η)
◮ Injective only for even functions
ψ = π: Half-circle transform
◮ Injective for all functions [Groemer 1998], [Goodey & Weil 2006] ◮ [Rubin 2017] half circles in one hemisphere
2 November 2017 · Michael Quellmalz 23 / 24 tu-chemnitz.de/∼qmi
The end
2 November 2017 · Michael Quellmalz 24 / 24 tu-chemnitz.de/∼qmi