Nonparametric model for a tensor field based on high angular resolution diffusion imaging (HARDI)

We develop a nonparametric technique for the estimation of curve trajectories using HARDI data. For various regions of the brain, we consider the imaging signal process and apply multivariate kernel smoothing techniques to a general function f describing the signal process obtained from the MRI image. At each location in the brain we search for the direction of maximum diffusion on the unit sphere, and then trace the integral curve driven by the vector field to obtain the estimates of curve trajectories. We establish the convergence of the properly normalized curve estimators to a Gaussian process. This method is computationally efficient as with each step of the curve tracing we construct a pointwise confidence ellipsoid region as opposed to exhaustive iterative sampling methods. These curve trajectories are models of axonal fibers whose location and geometry are important in neuroscience.