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. Key words: DT-MRI; Riemannian metrics; vector space; interpolation; regularizationDiffusion tensor imaging (DT-MRI or DTI or equivalently DT imaging) (1) is an emerging imaging modality whose importance has been growing considerably. In particular, most attempts to reconstruct noninvasively the connectivity of the brain are based on DTI (see (2-7) and references within for classical fiber tracking algorithms). Other applications of DT-MRI also include the study of diseases such as stroke, multiple sclerosis, dyslexia, and schizophrenia (8).The diffusion tensor is a simple and powerful model used to analyze the content of diffusion-weighted images (DWMRIs). It is based on the assumption that the motion of water molecules can be well approximated by a Brownian motion in each voxel of the image. This Brownian motion is entirely characterized by a symmetric and positive-definite matrix, called the "diffusion tensor" (1). In this article, we restrict the term tensor to mean a symmetric and positivedefinite matrix.With the increasing use of DT-MRI, there has been a growing need to generalize to the tensor case many usual vector processing tools. In particular, regularization techniques

In this work we present a new generalization of the geometric mean of positive numbers on symmetric positive-definite matrices, called Log-Euclidean. The approach is based on two novel algebraic structures on symmetric positive-definite matrices: first, a lie group structure which is compatible with the usual algebraic properties of this matrix space; second, a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure. From biinvariant metrics on the Lie group structure, we define the Log-Euclidean mean from a Riemannian point of view. This notion coincides with the usual Euclidean mean associated with the novel vector space structure. Furthermore, this means corresponds to an arithmetic mean in the domain of matrix logarithms. We detail the invariance properties of this novel geometric mean and compare it to the recently introduced affine-invariant mean. The two means have the same determinant and are equal in a number of cases, yet they are not identical in general. Indeed, the Log-Euclidean mean has a larger trace whenever they are not equal. Last but not least, the Log-Euclidean mean is much easier to compute.

Abstract. In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain.

Abstract. Computations on tensors have become common with the use of DT-MRI. But the classical Euclidean framework has many defects, and affine-invariant Riemannian metrics have been proposed to correct them. These metrics have excellent theoretical properties but lead to complex and slow algorithms. To remedy this limitation, we propose new metrics called Log-Euclidean. They also have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations. Indeed, Log-Euclidean computations are Euclidean computations in the domain of matrix logarithms. Theoretical aspects are presented and experimental results for multilinear interpolation and regularization of tensor fields are shown on synthetic and real DTI data.

Abstract-Diffusion tensor MRI (DT-MRI or DTI) is an imaging modality that is gaining importance in clinical applications.However, in a clinical environment, data have to be acquired rapidly, often at the expense of the image quality. This often results in DTI datasets that are not suitable for complex postprocessing like fiber tracking. We propose a new variational framework to improve the estimation of DT-MRI in this clinical context. Most of the existing estimation methods rely on a log-Gaussian noise (Gaussian noise on the image logarithms), or a Gaussian noise, that do not reflect the Rician nature of the noise in MR images with low SNR. With these methods, the Rician noise induces a shrinking effect: the tensor volume is underestimated when other noise models are used for the estimation. In this paper, we propose a maximum likelihood strategy that fully exploits the assumption of a Rician noise. To further reduce the influence of the noise, we optimally exploit the spatial correlation by coupling the estimation with an anisotropic prior previously proposed on the spatial regularity of the tensor field itself, which results in a maximum a posteriori estimation. Optimizing such a non-linear criterion requires adapted tools for tensor computing. We show that Riemannian metrics for tensors, and more specifically the Log-Euclidean metrics, are a good candidate and that this criterion can be efficiently optimized. Experiments on synthetic data show that our method correctly handles the shrinking effect even with very low SNR, and that the positive definiteness of tensors is always insured. Results on real clinical data demonstrate the truthfulness of the proposed approach and show promising improvements of fiber tracking in the brain and the spinal cord.