EXPONENTIAL TENSORS: A FRAMEWORK FOR EFFICIENT HIGHER-ORDER DT-MRI COMPUTATIONS

In diffusion tensor magnetic resonance image (DT-MRI) processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the 3D volume image. These tensors are symmetric positive definite matrices and the appropriate constraints required in algorithms for processing them makes these algorithms complex and significantly increases their computational complexity. In this paper we present a novel parameterization of the diffusivity function using which the positive definite property of the function is guaranteed without any increase in computation. This parameterization can be used for any order tensor approximations; we present Cartesian tensor approximations of order 2, 4, 6 and 8 respectively, of the diffusivity function all of which retain the positivity property in this parameterization without the need for any explicit enforcement. Furthermore, we present an efficient framework for computing distances and geodesies in the space of the coefficients of our proposed diffusivity function. Distances & geodesies are useful for performing interpolation, computation of statistics etc. on high rank positive definite tensors. We validate our model using simulated and real diffusion weighted MR data from excised, perfusion-fixed rat optic chiasm.