This paper presents processing and visualization techniques for Diffusion Tensor Magnetic Resonance Imaging (DT-MRI). In DT-MRI, each voxel is assigned a tensor that describes local water diffusion. The geometric nature of diffusion tensors enables us to quantitatively characterize the local structure in tissues such as bone, muscle, and white matter of the brain. This makes DT-MRI an interesting modality for image analysis. In this paper we present a novel analytical solution to the Stejskal-Tanner diffusion equation system whereby a dual tensor basis, derived from the diffusion sensitizing gradient configuration, eliminates the need to solve this equation for each voxel. We further describe decomposition of the diffusion tensor based on its symmetrical properties, which in turn describe the geometry of the diffusion ellipsoid. A simple anisotropy measure follows naturally from this analysis. We describe how the geometry or shape of the tensor can be visualized using a coloring scheme based on the derived shape measures. In addition, we demonstrate that human brain tensor data when filtered can effectively describe macrostructural diffusion, which is important in the assessment of fiber-tract organization. We also describe how white matter pathways can be monitored with the methods introduced in this paper. DT-MRI tractography is useful for demonstrating neural connectivity (in vivo) in healthy and diseased brain tissue.