Log‐Euclidean metrics for fast and simple calculus on diffusion tensors

Diffusion tensor imaging (DT-MRI or DTI) is an emerging imaging modality whose importance has been growing considerably. However, the processing of this type of data (i.e., symmetric positive-definite matrices), called "tensors" here, has proved difficult in recent years. Usual Euclidean operations on matrices suffer from many defects on tensors, which have led to the use of many ad hoc methods. Recently, affine-invariant Riemannian metrics have been proposed as a rigorous and general framework in which these defects are corrected. These metrics have excellent theoretical properties and provide powerful processing tools, but also lead in practice to complex and slow algorithms. To remedy this limitation, a new family of Riemannian metrics called Log-Euclidean is proposed in this article. They also have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations. This new approach is based on a novel vector space structure for tensors. In this framework, Riemannian computations can be converted into Euclidean ones once tensors have been transformed into their matrix logarithms. Theoretical aspects are presented and the Euclidean, affine-invariant, and Log-Euclidean frameworks are compared experimentally. The comparison is carried out on interpolation and regularization tasks on synthetic and clinical 3D DTI data.