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In this paper, we present novel algorithms for statistically robust interpolation and approximation of diffusion tensors&#x2014;which are symmetric positive definite (SPD) matrices&#x2014;and use them in developing a significant extension to an existing probabilistic algorithm for scalar field segmentation, in order to segment diffusion tensor magnetic resonance imaging (DT-MRI) datasets. Using the Riemannian metric on the space of SPD matrices, we present a novel and robust higher order (cubic) continuous tensor product of   $B$   -splines algorithm to approximate the SPD diffusion tensor fields. The resulting approximations are appropriately dubbed  tensor splines . Next, we segment the diffusion tensor field by jointly estimating the label (assigned to each voxel) field, which is modeled by a Gauss Markov measure field (GMMF) and the parameters of each smooth tensor spline model representing the labeled regions. Results of interpolation, approximation, and segmentation are presented for synthetic data and real diffusion tensor fields from an isolated rat hippocampus, along with validation. We also present comparisons of our algorithms with existing methods and show significantly improved results in the presence of noise as well as outliers.

Tensor Splines for Interpolation and Approximation of DT-MRI With Applications to Segmentation of Isolated Rat Hippocampi