This paper deals with the problem of regularizing noisy fields of diffusion tensors, considered as symmetric and semi-positive definite n × n matrices (as for instance 2D structure tensors or DT-MRI medical images). We first propose a simple anisotropic PDE-based scheme that acts directly on the matrix coefficients and preserve the semi-positive constraint thanks to a specific reprojection step. The limitations of this algorithm lead us to introduce a more effective approach based on constrained spectral regularizations acting on the tensor orientations (eigenvectors) and diffusivities (eigenvalues), while explicitely taking the tensor constraints into account. The regularization of the orientation part uses orthogonal matrices diffusion PDE?s and local vector alignment procedures and will be particularly developed. For the interesting 3D case, a special implementation scheme designed to numerically fit the tensor con-straints is also proposed. Experimental results on synthetic and real DT-MRI data sets finally illustrates the proposed tensor regularization framework.