Vladimir Sharafutdinov

Slice-by-slice reconstruction algorithm for vector tomography with incomplete data

For a finite collection of planes, we consider the problem of recovering the solenoidal part of a vector (symmetric second rank tensor) field on from ray integrals known over all lines parallel to one of the planes. Two different planes are sufficient for the uniqueness in the case of vector fields, but three planes in the general position are needed for the stable reconstruction. In the case of symmetric second rank tensor fields, three (six) planes in the general position are needed for the unique (stable) reconstruction. The reconstruction algorithm is presented for each of these cases. The main ingredients of the algorithm are the 3D Fourier transform and multiple application of the 2D back-projection operator.

Finiteness theorem for the ray transform on a Riemannian manifold

The ray transform I on a compact Riemannian manifold M with boundary is the operator sending a symmetric tensor field f to the set of integrals of f over all geodesics joining boundary points. A field f is called potential if it can be represented as the symmetric part of the covariant derivative of another tensor field vanishing on the boundary: The main result asserts that the space of potential tensor fields is a subspace of a finite codimension in Ker I if M is simple. A Riemannian manifold is called simple if every two points are joined by a unique geodesic.

On the problem of polarization tomography: I

The polarization tomography problem consists of recovering a matrix function f from the fundamental matrix of the equation

Tomography of small residual stresses

D\eta/dt=\pi_{\dot\gamma}f\eta

The Reshetnyak formula and Natterer stability estimates in tensor tomography

known for every geodesic

Zeta-invariants of the Steklov spectrum of a planar domain

\gamma

The problem of polarization tomography. II

of a given Riemannian metric. Here

THE MODIFIED HORIZONTAL DERIVATIVE AND SOME OF ITS APPLICATIONS

\pi_{\dot\gamma}

Killing tensor fields on the 2-torus

is the orthogonal projection onto the hyperplan

The John equation for tensor tomography in three-dimensions*

\dot\gamma^{\perp}