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 Norton’s setup for vector field tomography 1 Ray and Radon transforms on a Riemannian manifold 2 An inversion method relying on Beylkin’s theory 3 Numerical tests taking refraction into account 4 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 Norton’s setup for vector field tomography 1 Ray and Radon transforms on a Riemannian manifold 2 An inversion method relying on Beylkin’s theory 3 Numerical tests taking refraction into account 4 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 ⊂ R 2 convex bounded domain Determine a 2D flow field u ( x ) = u ( x 1 , x 2 ) ∈ R 2 x ∈ Ω with the help of ultrasound time-of-flight (TOF) measurements c ( x ) = c ( x 1 , x 2 ) sound velocity in x c ( x ) = c 0 const in R 2 \ 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 (W ELLS ET AL ., 1977) Industry: Reconstruction of gaz flows in a furnace (S IELSCHOTT , 1995) Oceanography: Estimation of flow velocities (R OUSEFF , W INTERS , E WERT , 1991) Plasma physics: Determination of the curl of velocity fields in a plasma from the measurements of Doppler shifts (E FREMOV , P AMENTOV , K HARACHENKO , 1995) Photoelasticity: Computation of the strain tensor of a transparent probe from the polarization of light (S HARAFUTDINOV , 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 N ORTON , Geophysics Journal, 1988 B RAUN , H AUCK , IEEE Trans. Med. Imag., 1991 J UHLIN , Tech. Rep., Lund Inst. Math, Lund, 1992 S HARAFUTDINOV , VSP Utrecht, 1994 S PARR ET AL ., Inverse Problems, 1995 S., Inverse Problems, 2000, 2001 N ATTERER , W ÜBBELING , SIAM, 2001 B UKGHEIM , K AZANTSEV , Tech. Rep. Sob. Inst. Math., Novosibirsk, 2002 A NDERSSON , Inverse problems, 2005 D EREVTSOV , JIIP , 2005 S., Inverse Problems, 2005, JIIP , 2006 S TEFANOV , U HLMANN 2007 S HARAFUTDINOV , JIIP , 2007 S., T HEIS , L OUIS , 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 N ORTON , Geophysics Journal, 1988 TOF : ✻ ◗◗◗◗◗◗◗◗◗◗ d ℓ ( x ) � t ( a , b ) = a c ( x ) + u ( x ) · τ L L ( a , b ) q s ◗ ◗ τ L b ✲ q ◗ u ( x 1 , x 2 ) L ( a , b ) 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 N ORTON , Geophysics Journal, 1988 TOF : ✻ ◗◗◗◗◗◗◗◗◗◗ d ℓ ( x ) � t ( a , b ) = a c ( x ) + u ( x ) · τ L L ( a , b ) q ◗ s ◗ Taylor - expansion about u / c : τ L b ✲ q ◗ 1 � c ( x ) − u ( x ) · τ L u ( x 1 , x 2 ) � � L ( a , b ) t ( a , b ) ≈ d ℓ ( x ) c 2 ( x ) L ( a , b ) 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 c ( x ) = 2 � d ℓ ( x ) � t ( a , b ) + t ( b , a ) = 2 (1) n ( x ) d ℓ ( x ) c 0 L ( a , b ) L ( a , b ) � u ( x ) · τ L � t ( a , b ) − t ( b , a ) = − 2 (2) c 2 ( x ) d ℓ ( x ) = f ( x ) · τ L d ℓ ( x ) L ( a , b ) L ( a , b ) n ( x ) = c 0 / 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 c ( x ) = 2 � d ℓ ( x ) � t ( a , b ) + t ( b , a ) = 2 (1) n ( x ) d ℓ ( x ) c 0 L ( a , b ) L ( a , b ) � u ( x ) · τ L � t ( a , b ) − t ( b , a ) = − 2 (2) c 2 ( x ) d ℓ ( x ) = f ( x ) · τ L d ℓ ( x ) L ( a , b ) L ( a , b ) n ( x ) = c 0 / 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 d s 2 = n ( x ) 2 d x µ d x µ 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 c 0 be the speed of sound in a reference medium and n ( x ) = c 0 / c ( x ) the corresponding index of refraction. Let ( M , g ) be a Riemannian manifold with metric tensor g ij = n 2 ( x ) δ ij and the associated element of length d s 2 = g µν d x µ d x ν = n 2 ( x ) d x µ d x µ . Then an ultrasound signal propagates along geodesics of d s. 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 Norton’s setup for vector field tomography 1 Ray and Radon transforms on a Riemannian manifold 2 An inversion method relying on Beylkin’s theory 3 Numerical tests taking refraction into account 4 T. Schuster Inversion of the vectorial ray transform on Riemannian manifolds
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