Robust interpolation of DT-MRI data using Tensor Splines

In this paper, we present a novel and robust spline interpolation algorithm given a noisy symmetric positive definite (SPD) tensor field. We construct a B-spline surface using the Riemannian metric of the manifold of SPD tensors. Each point of this surface corresponds to a diffusion tensor. We develop an algorithm for fitting such a Tensor Spline to a given tensor field and also an algorithm for evaluation of the spline at any intermediate points. The algorithm was tested on DTI data acquired from the isolated rabbit heart slice preparation. We also present validation results, which conclusively demonstrate superior interpolation performance of our algorithm over other existing methods. Motivation The presence of outliers is common in DT-MRI data due to noise. Hence, robust smoothing and interpolation is necessary. Interpolation of DTIs has many other applications e.g., registration and atlas construction, both of which require an interpolation module. In order to interpolate in the space of DTs, which is negatively curved, an appropriate metric needs to be defined. We define a metric in this space and use it to perform smooth and robust interpolation between points (tensors) in this space. Existing Methods Tensor interpolation can be done by evaluating the Riemannian geodesic between two neighbor tensors in a tensor field [1]. An approximation to this method is the Log-Euclidean geodesic interpolation which is quite efficient [2]. However, both these methods are sensitive to outliers and are not sufficiently smooth. Tensor Splines We develop a robust algorithm for constructing a tensor product of B-splines for interpolating DT-MRI data, using the Riemannian metric. Our method involves a two step procedure wherein the first step uses Riemannian distances in the well known DeBoor's spline evaluation algorithm [3] and the second step involves minimization of a robust distance measure between the evaluated spline curve and the given data. These two steps are alternated to achieve a smooth tensor spline approximation to the given tensor field. Experimental Results In fig. 1 we present comparison of our algorithm with two existing methods applied to synthetically generated noisy DTI data with outliers. Our results show significant improvement in the comparisons, specifically in the presence of noise and outliers. Figure 2 and 3 demonstrate the performance of our algorithm applied on DTI data acquired from the isolated rabbit heart slice preparation. The first image in both figures is a T2-weighted MR image obtained using a zero gradient field. The bottom row of figure 3 depicts an ellipsoidal visualization (in color) of the presented tensor fields. Figures are explained with more details in their captions. Acknowledgement This research was supported in part by the grant NIH R01 NS42075. References [1] Fletcher, P., Joshi, S.: Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. Proc. of the workshop on Computer Vision Approaches to Medical Image Analysis (CVAMIA) (2004) 87–98 [2] Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Fast and simple calculus on tensors in the log-euclidean framework. In: Proceedings of MICCAI. LNCS (2005) 259–267 [3] de Boor, C.: On calculating with b-splines. J. Approx. Theory 6 (1972) 50–62 E rr or 10 20 30 0 0.5 1 Riemannian geodesic