Diffusion Tensor Imaging (DTI) calculates a tensor for each voxel, representing the mean diffusive characteristics in volume-averaged tissue. Gradients that phase-encode spins according to the amount of their diffusion are usually applied uniformly over a sphere during a DTI procedure for minimal bias of tensor information. If prior knowledge of diffusion direction exists, the angular precision for determining the principle eigenvector of cylindrically-symmetric ("prolate") tensors can be improved by specifying gradients non-uniformly. Improvements in precision of 30-40% can be achieved using a restricted band of zenith angle values for gradient directions. Sensitivity to the a priori angular range of the principle eigenvector can be adjusted with the width of the band. Simulations and phantom data are in agreement; a preliminary validation is presented.