Groupwise registration and atlas construction of 4th-order tensor fields using the R+ Riemannian metric

Registration of Diffusion-Weighted MR Images (DW-MRI) can be achieved by registering the corresponding 2nd-order Diffusion Tensor Images (DTI). However, it has been shown that higher-order diffusion tensors (e.g. order-4) outperform the traditional DTI in approximating complex fiber structures such as fiber crossings. In this paper we present a novel method for unbiased group-wise non-rigid registration and atlas construction of 4th-order diffusion tensor fields. To the best of our knowledge there is no other existing method to achieve this task. First we define a metric on the space of positive-valued functions based on the Riemannian metric of real positive numbers (denoted by R+). Then, we use this metric in a novel functional minimization method for non-rigid 4th-order tensor field registration. We define a cost function that accounts for the 4th-order tensor re-orientation during the registration process and has analytic derivatives with respect to the transformation parameters. Finally, the tensor field atlas is computed as the minimizer of the variance defined using the Riemannian metric. We quantitatively compare the proposed method with other techniques that register scalar-valued or diffusion tensor (rank-2) representations of the DWMRI.