Inversion of the vectorial ray transform on Riemannian manifolds T. - - PowerPoint PPT Presentation

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Inversion of the vectorial ray transform on Riemannian manifolds

• T. Pfitzenreiter, T. Schuster

Helmut Schmidt University 22043 Hamburg, Germany

AIP Conference Vienna, 23.07.2009

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Overview

1

Norton’s setup for vector field tomography

2

Ray and Radon transforms on a Riemannian manifold

3

An inversion method relying on Beylkin’s theory

4

Numerical tests taking refraction into account

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Übersicht

1

Norton’s setup for vector field tomography

2

Ray and Radon transforms on a Riemannian manifold

3

An inversion method relying on Beylkin’s theory

4

Numerical tests taking refraction into account

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Aim of 2D vector field tomography

Ω ⊂ R2 convex bounded domain Determine a 2D flow field u(x) = u(x1, x2) ∈ R2 x ∈ Ω with the help of ultrasound time-of-flight (TOF) measurements c(x) = c(x1, x2) sound velocity in x c(x) = c0 const in R2\Ω

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Applications

Medical diagnosis: Early detection of malign tumuours (WELLS ET AL., 1977) Industry: Reconstruction of gaz flows in a furnace (SIELSCHOTT, 1995) Oceanography: Estimation of flow velocities (ROUSEFF, WINTERS, EWERT, 1991) Plasma physics: Determination of the curl of velocity fields in a plasma from the measurements of Doppler shifts (EFREMOV, PAMENTOV, KHARACHENKO, 1995) Photoelasticity: Computation of the strain tensor of a transparent probe from the polarization of light (SHARAFUTDINOV, 1994)

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

References

NORTON, Geophysics Journal, 1988 BRAUN, HAUCK, IEEE Trans. Med. Imag., 1991 JUHLIN, Tech. Rep., Lund Inst. Math, Lund, 1992 SHARAFUTDINOV, VSP Utrecht, 1994 SPARR ET AL., Inverse Problems, 1995 S., Inverse Problems, 2000, 2001 NATTERER, WÜBBELING, SIAM, 2001 BUKGHEIM, KAZANTSEV, Tech. Rep. Sob. Inst. Math., Novosibirsk, 2002 ANDERSSON, Inverse problems, 2005 DEREVTSOV, JIIP , 2005 S., Inverse Problems, 2005, JIIP , 2006 STEFANOV, UHLMANN 2007 SHARAFUTDINOV, JIIP , 2007 S., THEIS, LOUIS, J. Biomed. Imag., 2008

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Data acquisition via TOF 1

NORTON, Geophysics Journal, 1988

✲ ✻ ◗◗◗◗◗◗◗◗◗◗ ◗ ◗ ◗ s q q

a b u(x1, x2) L(a, b) τL

TOF : t(a, b) =

• L(a,b)

dℓ(x) c(x) + u(x) · τL

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Data acquisition via TOF 1

NORTON, Geophysics Journal, 1988

✲ ✻ ◗◗◗◗◗◗◗◗◗◗ ◗ ◗ ◗ s q q

a b u(x1, x2) L(a, b) τL

TOF : t(a, b) =

• L(a,b)

dℓ(x) c(x) + u(x) · τL Taylor - expansion about u/c: t(a, b) ≈

• L(a,b)
• 1

c(x) − u(x) · τL c2(x)

• dℓ(x)

if |u(x)|/c(x) ≪ 1

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Data acquisition via TOF 2

t(a, b) + t(b, a) = 2

• L(a,b)

dℓ(x) c(x) = 2 c0

• L(a,b)

n(x)dℓ(x) (1) t(a, b) − t(b, a) = −2

• L(a,b)

u(x) · τL c2(x) dℓ(x) =

• L(a,b)

f(x) · τL dℓ(x) (2) n(x) = c0/c(x) refractive index

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Data acquisition via TOF 2

t(a, b) + t(b, a) = 2

• L(a,b)

dℓ(x) c(x) = 2 c0

• L(a,b)

n(x)dℓ(x) (1) t(a, b) − t(b, a) = −2

• L(a,b)

u(x) · τL c2(x) dℓ(x) =

• L(a,b)

f(x) · τL dℓ(x) (2) n(x) = c0/c(x) refractive index Idea: Compute n(x) from (1) by conventional CT Compute u(x) from (2) by an inversion method of VFT

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Can the influence of refraction be neglected?

Norton: ’We show that VFT reduces to the above reconstruction problem under the assumption of straight line propagation of the acoustic signal (i.e. neglecting refraction)’

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Can the influence of refraction be neglected?

Norton: ’We show that VFT reduces to the above reconstruction problem under the assumption of straight line propagation of the acoustic signal (i.e. neglecting refraction)’ But: The refractive index n(x) affects the propagation of the ultrasound signal!

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Can the influence of refraction be neglected?

Norton: ’We show that VFT reduces to the above reconstruction problem under the assumption of straight line propagation of the acoustic signal (i.e. neglecting refraction)’ But: The refractive index n(x) affects the propagation of the ultrasound signal! Remedy: Use the Riemannian metric ds2 = n(x)2 dxµ dxµ instead the Euclidean one!

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Fermat’s principle

Theorem (Fermat’s principle) Let c0 be the speed of sound in a reference medium and n(x) = c0/c(x) the corresponding index of refraction. Let (M, g) be a Riemannian manifold with metric tensor gij = n2(x)δij and the associated element of length ds2 = gµνdxµdxν = n2(x)dxµ dxµ . Then an ultrasound signal propagates along geodesics of ds.

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Ray propagation in homogeneous and inhomogeneous media

homogeneous medium inhomogeneous medium

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Übersicht

1

Norton’s setup for vector field tomography

2

Ray and Radon transforms on a Riemannian manifold

3

An inversion method relying on Beylkin’s theory

4

Numerical tests taking refraction into account

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Radon and ray transforms on a RM 1

Let ω = ω(ϕ) = (cos ϕ, sin ϕ) ∈ S1 be a unit vector, (M, g) be a 2D Riemannian manifold and γs,ϕ : [τ−, 0] → M s ∈ R a geodesic with γs,ϕ(0) = b, ˙ γs,ϕ(0) = ω(ϕ)⊥ and τ− such that γs,ϕ(τ−) = a The Radon transform on a RM is defined by Rf(s, ϕ) =

• τ−

f(γs,ϕ(t)) dt and R : L2(M) → L2(R × [0, 2π]) is continuous (see SHARAFUTDINOV, 1994)

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Radon and ray transforms on a RM 2

The longitudinal and transversal ray transform for a vector field f = fi on M are defined by Df(s, ϕ) =

• τ−

fi(γs,ϕ(t))˙ γi

s,ϕ(t) dt

D⊥f(s, ϕ) =

• τ−

fi(γs,ϕ(t))ni

s,ϕ(t) dt

where ns,ϕ is the unit normal vector at the geodesic D, D⊥ : L2(S1, M) → L2(R × [0, 2π]) are continuous (see SHARAFUTDINOV, 1994)

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Connection between R and D, D⊥

Lemma (Pfitzenreiter, S.) Let f ∈ C1(S1, M). Then ∂ ∂sDf(s, ϕ) = R[curl f](s, ϕ) ∂ ∂sD⊥f(s, ϕ) = R[div f](s, ϕ)

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Übersicht

1

Norton’s setup for vector field tomography

2

Ray and Radon transforms on a Riemannian manifold

3

An inversion method relying on Beylkin’s theory

4

Numerical tests taking refraction into account

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Implicit representation of geodesics 1

Lemma

Let {γs,ϕ}s∈R : [τ−, 0] → M be a family of geodesics that do not intersect. Then there exists a phase function φ : Ω ⊂ R2 × S1 → R , such that the geodesic γs,ϕ is implicitly defined by φ(x, ω) = s in the sense that φ

• ψ(γs,ϕ(t)), ω(ϕ)
• = s ,

where ψ : U → V ⊂ Ω is a chart and U ⊂ M is a neighborhood of γs,ϕ(t).

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Implicit representation of geodesics 2

Example: If gij = δij (i.e. n(x) = 1) is the Euclidean metric, then the geodesics are straight lines γs,ϕ(t) = sω(ϕ) +

• t + t(b)
• ω(ϕ)⊥

The phase function is then given as φ(x, ω) = x, ω , where ω(ϕ)⊥ = ω(ϕ − π

2)

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Implicit representation of geodesics 3

φ(x, ω) = s Ru(s, ϕ) =

• φ(x,ω(ϕ))=s

u(x) dσ(x) dσ(x) can be expressed by φ

ω s ϕ ω⊥

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Implicit representation of geodesics 3

φ(x, ω) = s Ru(s, ϕ) =

• φ(x,ω(ϕ))=s

u(x) dσ(x) dσ(x) can be expressed by φ Weighted dual Radon transform: b ≥ 0 on Ω × S1 R∗

bw(y) = 2π

• w
• φ(y, ω), ϕ
• b(y, ω(ϕ)) dϕ

ω s ϕ ω⊥

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Beylkin’s theorem

Theorem (Beylkin 1984) Let U(s) ∈ C∞(R) be even, |∂k

s U(s)| ≤ C(k)(1 + s2)(m−k)/2, and define

K(s) = 1 2(2π)2

∞

• −∞

|r|U(r)eırs dr . Then R∗

b

• K ∗s R
• = F ,

where F is the Fourier integral operator Fu(y) = 1 (2π)2

• R2
• Ω

eı(φ(x,θ)−φ(y,θ))b(y, θ)U(θ)u(x) d2x dθ If U = 1 and gij = δij, then F = I.

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Characterization of F

Theorem (Beylkin 1984) The operator F is a PDO. Letting b(y, θ) = det ∂2φ(y, θ) ∂yj∂θk

• > 0

and U = 1, F can be extended to an operator L2(Ω) → L2

loc(Ω)

F = I + T , T =

∞

• l=1

l

• m=0

T m

l

where T is compact. We have that R∗

b(K ∗s R)u = u + Tu ≈ u

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Computing curl f and div f via FBP

From R∗

b(K ∗s R)u ≈ u we approximate

curl f ≈ R∗ K ∗s ∂ ∂sDf

• div f

≈ R∗ K ∗s ∂ ∂sD⊥f

• These computations can be done by standard filtered

backprojection algorithms

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Übersicht

1

Norton’s setup for vector field tomography

2

Ray and Radon transforms on a Riemannian manifold

3

An inversion method relying on Beylkin’s theory

4

Numerical tests taking refraction into account

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Numerical calculation of geodesics 1

Choose gij(x) = n2(x)δij leading to the Riemannian metric ds2 = n2(x) dxµdxµ , µ = 1, 2 The geodesics solve the 2nd order initial value problem ¨ γν

s,ϕ(t) + Γν ρk(x)˙

γk

s,ϕ(t)˙

γρ

s,ϕ(t) = 0

γs,ϕ(0) = x0(s, ϕ), ˙ γs,ϕ(0) = ω(ϕ)⊥ Γν

ρk are the Christoffel symbols corresponding to gij

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Numerical calculation of geodesics 2

n1(x) =

• κ2

1 sin2(x1) + κ2 1 cos2(x2) + 1

κ1 = 0.2 corresponds to ray propagation in a fluid consisting of water and oil

−1 −0.5 0.5 1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 −1 −0.5 0.5 1

Geodesics corresponding to the refractive index n1(x) for scanning angles ϕ = 0 (left) and ϕ = π/3 (right) Black: n1(x) = 1.15, white: n1(x) = 1.6

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Numerical calculation of geodesics 2

n2(x) =

• (κ2x1)2 + (κ2x2)2 + 1

κ2 = 0.2 corresponds to ray propagation in saltwater

−1 −0.5 0.5 1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 −1 −0.5 0.5 1

Geodesics corresponding to the refractive index n2(x) for scanning angles ϕ = 0 (left) and ϕ = π/3 (right) Black: n1(x) = 1.15, white: n1(x) = 1.6

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Reconstructions of synthetic data 1

f(x1, x2) =

• − cos(6x1)

sin(3x2) cos(4x2)

• e−κ(x2

1 +x2 2 )

Compute: curl f ≈ R∗

b

• K ∗s ∂

∂s Df

• div f ≈ R∗

b

• K ∗s ∂

∂s D⊥f

• using FBP with Shepp - Logan filter

−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Reconstructions of synthetic data 2

−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

−1 −0.5 0.5 1 −1 −0.5 0.5 1 −1 −0.5 0.5 1

−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

Exact curl and divergence of f (top), reconstruction of curl f, div f using n1 as refractive index (bottom)

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Reconstructions of synthetic data 3

−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

−1 −0.5 0.5 1 −1 −0.5 0.5 1 −1 −0.5 0.5 −1 −0.5 0.5 1 −1 −0.5 0.5 1 −1 −0.5 0.5 1

Reconstruction of curl f, div f using n2 as refractive index (top), comparison between reconstruction of curl f on lines and with refractive index n1 (bottom)

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Summary

We proposed the following method to reconstruct curl f from TOF measurements taking refractions into account: Compute n(x) from TOF measurements using CT Compute w = Df from TOF measurements Approximate curl f ≈ (I + T)curl f = R∗

b

• K ∗s

∂ ∂sw

• using a FBP algorithm with the Shepp - Logan filter

This method is suited to reconstruct the curl of a velocity field of a multi-phase flow in a very efficient manner

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds

Norton’s setup for vector field tomography Ray and Radon transforms on a Riemannian manifold An inversion method relying on Beylkin’s theory Numerical tests taking refraction into account

Future work

Exact reconstruction by solving the Fredholm integral equation (I + T)curl f = R∗

b

• K ∗s

∂ ∂sw

• ,

w = Df Computation of reconstruction kernels to apply the method

• f approximate inverse (LOUIS, 1996/1999)

Reconstruction of f Inversion formula for D on Riemannian manifolds

• T. Schuster

Inversion of the vectorial ray transform on Riemannian manifolds