Tensor and Emission Tomography Problems on the Plane

Tensor and Emission Tomography Problems on the Plane A. L. Bukhgeim S. G. Kazantsev A. A. Bukhgeim Sobolev Institute of Mathematics Novosibirsk, RUSSIA

Overview 1) Transmission tomography • inversion formula on the basis of SVD of the Radon transform • scalar, vector and tensor cases 2) Emission tomography • the first explicit inversion formula (A. L. Bukhgeim, S. G. Kazantsev, 1997) • recent results that follow from it • scalar and vector cases • 2 D problems, in a unit disc on the plane • Isotropic case • Fan-beam scanning geometry

Transmission Tomography (Scalar Case)

Helmholtz Decomposition = 2 -D Vector Field + = Solenoidal Part + Potential Part

Vectorial Radon Transform Normal Flow Radon Transform

Vectorial Radon Transform Solenoidal Part of the Vector Field Normal Flow Radon Transform

Vectorial Radon Transform Solenoidal Part of the Vector Field Normal Flow Radon Transform Potential Part of the Vector Field

Tensorial Radon Transform Consider a unit disk on the plane: Covariant symmetric tensor field of rank m: Due to symmetry it has m+1 independent components. By analogy with the vector case: • similar decomposition into the solenoidal and potential parts, • define tensorial Radon transform. Refer to: V. A. Sharafutdinov “Integral Geometry of Tensor Fields” Utrecht: VSP, 1994.

SVD of the Radon Transform SVD is one of the methods for solving ill-posed problems: Consider two Hilbert spaces: H with O. N. B K with O. N. S. Singular value decomposition of an operator Then its generalized inverse operator will look like: - can be unbounded. - truncated SVD. and

SVD of the Radon Transform The presence of a singular value decomposition allows to: • describe the image of the operator, • invert the operator, • estimate its level of incorrectness. The first SVD of the Radon transform for the parallel-beam geometry was derived by Herlitz in 1963 and Cormack in 1964 (scalar case only). Bukhgeim A. A. , Kazantsev S. G. “Singular-value decomposition of the fan-beam Radon transform of tensor fields in a disc” // Preprint of Russian Academy of Sciences, Siberian Branch. No. 86. Novosibirsk: Institute of Mathematics Press, October 2001. 34 pages.

SVD of the Radon Transform (scalar case)

Singular Values Radon Transform Inverse Radon Transform Integration Operator Differentiation Operator

Transmission Tomography: Numerical Examples (Scalar Case) Compare with the talk of Emmanuel Candes ! original image reconstruction from 512 300 noisy fan-projections; N=298 N=446 N=382 N=318 N=254 N=510 (noise level: 20%)

Transmission Tomography: Numerical Examples (Scalar Case) original image reconstruction from 2048 reconstruction from noisy fan-projections; N=1022 N=2046 N=510 1024 128 256 512 16 32 64 8 fan-projections; N=126 N=254 N=510 N=14 N=30 N=62 N=6 (noise level: 5% in L 2 -norm)

Transmission Tomography: Numerical Examples (Vector Case) original (solenoidal) vector field reconstruction from noisy (3%) projections reconstruction from non-uniform projections

Emission Tomography Attenuated Radon Transform • Assume, that the attenuation map of the object is known • Inject a radioactive solution into the patient, it is then spread all over the body with the blood • Place detectors around and measure how many radioactive particles go through it in the given directions • Reconstruct the Emission Map

Formulation of the emission tomography problem Consider a unit disc on the plane: Let and represent an attenuation map represent an emission map, both given in The fan-beam Radon transform The fan-beam attenuated Radon transform Emission tomography problem: reconstruct from its known attenuated Radon transform provided that the attenuation map is known.

Attenuated Vectorial Radon Transform Attenuated Normal Flow Radon Transform

Servey of the Results in Emission Tomography SCALAR CASE: • 1980, O. J. Tretiak, C. Metz. The first inversion formula for emission tomography with constant attenuation. • 1997, A. L. Bukhgeim, S. G. Kazantsev. The first explicit inversion formula for emission tomography (in the fan-beam formulation) with arbitrary nonconstant attenuation (based on theory of A-analytic functions). • 2000, R. G. Novikov (and then F. Natterer in 2001). Inversion formula for emission tomography in the parallel-beam formulation which then was numerically implemented by L. A. Kunyansky in 2001. VECTOR CASE: • 1997, K. Stråhlén. Inversion formula for full reconstruction of a vector field from both Exponential Vectorial Radon Transform and Exponential Normal Flow Transform, attenuation coefficient is constant. • 2002, A. A. Bukhgeim, S. G. Kazantsev. Full reconstruction of a vector field only from its Attenuated Vectorial Radon Transform, arbitrary nonconstant attenuation function is allowed.

Inversion formula (sketch)

Inversion formula (scalar case)

Equivalence of the Inversion Formulae Angular Hilbert Transform

Inversion formula (vector case) - components of the vector field being reconstructed, - a known attenuation function: • For the full reconstruction of a vector field it’s sufficient to know only one transform: either Vectorial Attenuated Radon Transform or the Normal Flow Attenuated Radon Transform; • Arbitrary non-constant attenuation is allowed.

Emission Tomography: Numerical Examples (Scalar Case) 360 degree Medium Attenuation No Noise

Emission Tomography: Numerical Examples (Scalar Case) 360 degree Medium Attenuation Large Noise

Emission Tomography: Numerical Examples (Scalar Case) 360 degree XXL Attenuation [6, 14] No Noise

Emission Tomography: Numerical Examples (Scalar Case) 270 degree Medium Attenuation No Noise

Emission Tomography: Numerical Examples (Scalar Case) 180 degree Medium Attenuation No Noise

Emission Tomography: Numerical Examples (Scalar Case) 90 degree ! Medium Attenuation No Noise

Emission Tomography: Numerical Examples (Scalar Case) 180 degree Large Attenuation [4, 7] No Noise

Emission Tomography: Numerical Examples (Scalar Case) 180 degree Large Attenuation [4, 7] With Noise

Emission Tomography: Numerical Examples (Vector Case) Sinogram Original Vector Field Reconstruction from 128 fan-projections

Emission Tomography: Numerical Examples (Vector Case) Sinogram Original Vector Field Reconstruction from 256 fan-projections

Emission Tomography: Numerical Examples (Vector Case) Sinogram Original Vector Field Reconstruction from 512 fan-projections

Emission Tomography: Numerical Examples (Vector Case) Sinogram Original Vector Field Reconstruction from 256 fan-projections

Conclusion 1) SVD of the Radon transform of tensor fields • description of the image of the operator, • inversion formula, • estimation of incorrectness of the inverse problem, • unified formula (for reconstruction of scalar, vector and tensor fieds), • numerical implementation; 2) The very first inversion formula (by A. L. Bukhgeim, S. G. Kazantsev) was re-derived • shows equivalence of the first inversion formula to the formulae obtained later by Novikov and Natterer, • yields a pathbreaking inversion formula for the vectorial attenuated Radon transfom, • numerical implementation.