Introduction
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Proof for singleton
Proof for open set
Broken ray tomography in the disk
Inverse Days 2012
Joonas Ilmavirta
University of Jyväskylä
18.12.2012
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Introduction
Results
Proof for singleton
Proof for open set
Broken rays
In a (bounded smooth) Euclidean domain Ω a broken ray is a
geodesic which starts and ends at ∂Ω and may have
reflections at ∂Ω.
For a fixed set E ⊂ ∂Ω, consider the following problem: If we
know the integral of an unknown function f : Ω → R over
every broken ray from E to E , what can we say about f ?
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Proof for singleton
Proof for open set
Broken ray transform
Given Ω and E ⊂ ∂Ω, the set of broken rays from E to E is
denoted by ΓE .
For f ∈ C (Ω̄, R) and γ ∈ ΓE we define
Gf (γ) =
f ds.
γ
The map G : C (Ω̄, R) → B(ΓE , R) is the broken ray transform.
Full reconstruction is possible ⇔ G is injective.
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Motivation
Some partial data problems for the Calderón problem can be
reduced to injectivity of the broken ray transform (Salo&Kenig
2012).
From partial Dircihlet to Neumann data for the magnetic
Schrödinger equation one can recover the integrals of the
electromagnetic potential over broken rays (Eskin 2004).
The broken ray problem may also arise more directly in
X-ray-like imaging with reflections.
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Proof for open set
Tomography from a singleton
Theorem
Suppose that Ω ⊂ Rn , n ≥ 2, is the unit ball, E ⊂ ∂Ω is a
singleton and f ∈ C (Ω̄, R). Then we can recover f (0) from the
broken ray transform Gf .
Remark: When n = 2, we can recover the integral average of f
over any circle ∂B(0, r ), r ∈ (0, 1].
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Proof for open set
Tomography from an open set
Theorem
Suppose that Ω ⊂ Rn , n ≥ 2, is the unit ball, E ⊂ ∂Ω is open and
f : Ω̄ → R is uniformly quasianalytic in the angular variable. Then
we can recover f everywhere from the broken ray transform Gf .
For example, functions of the form (in polar coordinates in the
plane)
K
X
f (r , θ) = a0 (r ) +
[ak (r ) cos(kθ) + bk (r ) sin(kθ)]
k=1
are uniformly quasianalytic in the angular variable θ provided that
the functions ak and bk are Hölder continuous.
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Proof for open set
Reduction to dimension two
Observation
It is enough to show the theorems when n = 2.
If n > 2, fix any plane P going through E and the origin. The
theorems in the case n = 2 provide reconstruction in B ∩ P.
Repeating for all P gives the theorems.
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Proof for singleton
Proof for open set
Statement
We wish to prove:
Theorem
Suppose that Ω ⊂ R2 is the unit disk, E ⊂ ∂Ω is a singleton and
f ∈ C (Ω̄, R). Then we can recover f (0) from the broken ray
transform Gf .
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Proof for open set
Long trajectories are spherically symmetric
If there are very many reflections, the trajectory is almost
spherically symmetric.
When the number of reflections goes to infinity, the broken
ray transform of f becomes an integral of f with a spherically
symmetric weight.
In this way we recover the integral
ˆ 1
a (r )
p 0
A0 a0 (z) = 2
dr ,
1 − (z/r )2
z
where
f dH1
a0 (r ) =
∂B(0,r )
is the angular average of f .
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Proof for open set
The Abel transform
The abel transform A0 ,
ˆ
A0 g (z) = 2
z
1
g (r )
p
dr ,
1 − (z/r )2
is injective.
From A0 a0 we can recover a0 , the angular average of f .
Finally f (0) = limr →0 a0 (r ).
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Introduction
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Proof for singleton
Proof for open set
Statement
We wish to prove:
Theorem
Suppose that Ω ⊂ R2 is the unit disk, E ⊂ ∂Ω is open and
f : Ω̄ → R is uniformly quasianalytic in the angular variable. Then
we can recover f everywhere from the broken ray transform Gf .
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Proof for singleton
Proof for open set
Fourier series
Write the unknown function as a Fourier series in the angular
variable:
f (r , θ) = a0 (r ) +
∞
X
[ak (r ) cos(kθ) + bk (r ) sin(kθ)].
k=1
Calculate the broken ray transform term by term. This
involves the generalized Abel transform
ˆ 1
g (r )
Ak g (z) = 2
Tk (z/r ) p
dr ,
1 − (z/r )2
z
where Tk are the Chebyshev polynomials.
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Generalized Abel transform
The generalized Abel transform is injective:
ˆ 1
g (r )
Ak g (z) = 2
dr ,
Tk (z/r ) p
1 − (z/r )2
z
ˆ 1
Ak g (z)
1 d
Tk (z/r ) p
dz.
g (r ) = −
π dr r
z (z/r )2 − 1
The Radon transform in the plane can be conveniently written
in terms of generalized Abel transforms of the Fourier
components (Cormack 1963).
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Proof for open set
Rotation symmetry
The integral of a function over a rotated trajectory equals the
integral of a rotated function over the original trajectory.
If Gf = 0, then also a slightly rotated versions of f have
vanishing broken ray transform.
n f = 0 for all n.
Thus Gf = 0 ⇒ G∂ang
Using this and the regularity assumptions we get Ak ak = 0
and Ak bk = 0 for all k.
This shows that f = 0 and concludes the proof.
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Proof for open set
Radon transform via Abel transforms
If
f (r , θ) =
X
ak (r )e ikθ ,
k∈Z
then
Rf (r , θ) =
X
Ak ak (r )e ikθ .
k∈Z
The Radon transform R is parametrized so that
ˆ
Rg (r , ϑ) =
g dH1 ,
Lr ,ϑ
where Lr ,ϑ = {x ∈ R2 : x1 cos ϑ + x2 sin ϑ = r }.
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Proof for open set
Broken ray transform via Abel transforms
We define the coefficient

sin(k(κγ − ιγ )/2)


sin(kαγ /2)
Sk (γ) =

kαγ

nγ (−1)(nγ +1) 2π +kmγ
when kαγ ∈
/ 2πZ
when kαγ ∈ 2πZ
for all k = 0, 1, . . . and γ ∈ ΓE .
If γ is symmetric w.r.t. to the angle zero, then
Gf (γ) = length(γ)−1
∞
X
k=0
Sk (γ)Ak ak (zγ ).
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Uniformly quasianalytic functions
M = (Mn )∞
n=0 is an increasing logarithmically convex sequence
of strictly positive real numbers.
#
A sequence (ak )∞
k=1 is in S (M) if there is a constant R > 0
such that for each n
∞
X
k 2n |ak |2 ≤ Mn R n .
k=0
The class S # (M) is a quasianalytic class of sequences if M
satisfies
s
∞
X
Mn
= ∞.
Mn+1
n=0
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Uniformly quasianalytic functions
A function f : R → R with a period 2π belongs to the class
C # (M) if it can be written as a uniformly convergent Fourier
series as
∞
X
f (x) =
(ak cos(kx) + bk sin(kx))
k=0
∞
such that the sequences (ak )∞
k=1 and (bk )k=1 are in the class
#
n
S (M) and each ∂ang f is Dini-Lipschitz continuous.
If the class S # (M) is quasianalytic, then C # (M) is a
quasianalytic class of functions.
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Proof for singleton
Proof for open set
Uniformly quasianalytic functions
A continuous function f : D̄ → R is uniformly quasianalytic in
the angular variable if for each R ∈ (0, 1] the collection of
functions {f (r , ·) : r ≥ R} belong to the same quasianalytic
class.
In dimensions three or higher, a function f : B̄ n → R, n ≥ 3,
is uniformly quasianalytic in the angular variables if its
restriction to any two dimensional plane intersecting the origin
is uniformly quasianalytic in the sense defined in two
dimensions.
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