Lower Bounds for Accuracy of Estimation in Diffusion Tensor Imaging

A vector field is observed at random locations with additive noise. The corresponding integral curve is to be estimated based on the data. The focus of the current paper is to obtain lower bounds for the functions of deviations between true and estimated integral curves. In particular, we show that the estimation procedure in [Koltchinskii, Sakhanenko, and Cai, Ann. Statist., 35 (2007), pp. 1576–1607] yields estimates, which have the optimal rate of convergence in a minimax sense. Overall, this work is motivated by diffusion tensor imaging, which is a modern brain imaging technique. The integral curves are used to model axonal fibers in the brain. In medical research, it is important to estimate and map these fibers. The paper addresses statistical aspects pertinent to such an estimation problem.