Backprojection filtering algorithms for reconstruction of vector and second order tensor fields

Tensor tomography is being investigated as a technique for reconstruction of in vivo tensor fields which will enable development of more accurate models of the properties of biological tissue. This paper presents backprojection filtering algorithms for reconstructing three-dimensional vector and second order tensor fields with the property of being independent of the coordinate z. The backprojection of a complete set of directional projection measurements gives straightforward results that are fairly simple extensions of the usual scalar reconstruction methods. Both vector fields and tensor fields are represented as a sum of solenoidal and irrotational components. The irrotational component of the vector field can be reconstructed from a single set of directional measurements. The reconstruction of the solenoidal and irrotational components for a tensor field each require six directional projection measurements. Results of computer simulations that demonstrate the validity of the mathematical formulations are presented. A simulated cardiac diffusion tensor field shows that the principal component of the tensor field could be specified utilizing half of the number of MRI measurements that are presently being utilized