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This paper concerns the reconstruction of an anisotropic diffusion tensor$\gamma=(\gamma_{ij})_{1\leq i,j\leq 2}$ from knowledge of internal functionalsof 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 twodimensional bounded domain with appropriate boundary conditions. We show thatfor I=4 and appropriately chosen boundary conditions, $\gamma$ may uniquely andstably be reconstructed from such internal functionals, which appear incoupled-physics inverse problems involving the ultrasound modulation ofelectrical or optical coefficients. Explicit reconstruction procedures for thediffusion tensor are presented and implemented numerically.

Inverse anisotropic diffusion from power density measurements in two dimensions