Diffusion tensor regularization with constraints preservation

The paper deals with the problem of regularizing noisy fields of diffusion tensors, considered as symmetric and semi-positive definite n × n matrices (such 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 preserves 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 explicitly taking the tensor constraints into account. The regularization of the orientation part uses orthogonal matrix diffusion PDE´s and local vector alignment procedures. For the interesting 3D case, a special implementation scheme designed to numerically fit the tensor constraints is also proposed. Experimental results on synthetic and real DT-MRI data sets finally illustrates the proposed tensor regularization framework.