A Riemannian Framework for Tensor Computing by Xavier Pennec, Pierre Fillard, and Nicholas Ayache introduces a Riemannian metric for symmetric positive definite matrices (tensors) to enable geometric data processing. The framework provides an affine-invariant Riemannian metric, leading to strong theoretical properties, such as a regular manifold of constant curvature without boundaries. This metric allows for the definition of geodesics and means of tensors, and enables generalization of statistical and geometric algorithms to tensor fields. The paper demonstrates how to perform interpolation, filtering, diffusion, and restoration of missing data on tensor fields using this framework. It also introduces intrinsic numerical schemes for computing gradient and Laplacian operators, and proposes least-squares criteria for data fidelity. The framework is applied to various tasks, including tensor interpolation, filtering, and regularization, with practical implementations and examples. The paper emphasizes the importance of the Riemannian metric in ensuring invariance and consistency in tensor computations.A Riemannian Framework for Tensor Computing by Xavier Pennec, Pierre Fillard, and Nicholas Ayache introduces a Riemannian metric for symmetric positive definite matrices (tensors) to enable geometric data processing. The framework provides an affine-invariant Riemannian metric, leading to strong theoretical properties, such as a regular manifold of constant curvature without boundaries. This metric allows for the definition of geodesics and means of tensors, and enables generalization of statistical and geometric algorithms to tensor fields. The paper demonstrates how to perform interpolation, filtering, diffusion, and restoration of missing data on tensor fields using this framework. It also introduces intrinsic numerical schemes for computing gradient and Laplacian operators, and proposes least-squares criteria for data fidelity. The framework is applied to various tasks, including tensor interpolation, filtering, and regularization, with practical implementations and examples. The paper emphasizes the importance of the Riemannian metric in ensuring invariance and consistency in tensor computations.