Inverse anisotropic diffusion from power density measurements in two dimensions

This paper concerns the reconstruction of an anisotropic diffusion tensor $\gamma=(\gamma_{ij})_{1\leq i,j\leq 2}$ from knowledge of internal functionals of the form $\gamma\nabla u_i\cdot\nabla u_j$ with $u_i$ for $1\leq i\leq I$ solutions of the elliptic equation $\nabla \cdot \gamma \nabla u_i=0$ on a two dimensional bounded domain with appropriate boundary conditions. We show that for I=4 and appropriately chosen boundary conditions, $\gamma$ may uniquely and stably be reconstructed from such internal functionals, which appear in coupled-physics inverse problems involving the ultrasound modulation of electrical or optical coefficients. Explicit reconstruction procedures for the diffusion tensor are presented and implemented numerically.Comment: 27 pages, 6 figure

Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra

We present explicit filtration/backprojection-type formulae for the inversion of the spherical (circular) mean transform with the centers lying on the boundary of some polyhedra (or polygons, in 2D). The formulae are derived using the double layer potentials for the wave equation, for the domains with certain symmetries. The formulae are valid for a rectangle and certain triangles in 2D, and for a cuboid, certain right prisms and a certain pyramid in 3D. All the present inversion formulae yield exact reconstruction within the domain surrounded by the acquisition surface even in the presence of exterior sources.Comment: 9 figure