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In this paper, we propose the DT-REFinD algorithm for the diffeomorphic nonlinear registration of diffusion tensor images. Unlike scalar images, deforming tensor images requires choosing both a reorientation strategy and an interpolation scheme. Current diffusion tensor registration algorithms that use full tensor information face difficulties in computing the differential of the tensor reorientation strategy and consequently, these methods often approximate the gradient of the objective function. In the case of the finite-strain (FS) reorientation strategy, we borrow results from the pose estimation literature in computer vision to derive an analytical gradient of the registration objective function. By utilizing the closed-form gradient and the velocity field representation of one parameter subgroups of diffeomorphisms, the resulting registration algorithm is diffeomorphic and fast. We contrast the algorithm with a traditional FS alternative that ignores the reorientation in the gradient computation. We show that the exact gradient leads to significantly better registration at the cost of computation time. Independently of the choice of Euclidean or Log-Euclidean interpolation and sum of squared differences dissimilarity measure, the exact gradient achieves better alignment over an entire spectrum of deformation penalties. Alignment quality is assessed with a battery of metrics including tensor overlap, fractional anisotropy, inverse consistency and closeness to synthetic warps. The improvements persist even when a different reorientation scheme, preservation of principal directions, is used to apply the final deformations.