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 manyad hocmethods. 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. Magn Reson Med 56, 2006.

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 bi‐invariant 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 g...

The primary objective of this study is to perform a blinded evaluation of a group of retrospective image registration techniques using as a gold standard a prospective, marker-based registration method. To ensure blindedness, all retrospective registrations were performed by participants who had no knowledge of the gold standard results until after their results had been submitted. A secondary goal of the project is to evaluate the importance of correcting geometrical distortion in MR images by comparing the retrospective registration error in the rectified images, i.e., those that have had the distortion correction applied, with that of the same images before rectification. Image volumes of three modalities (CT, MR, and PET) were obtained from patients undergoing neurosurgery at Vanderbilt University Medical Center on whom bone-implanted fiducial markers were mounted. These volumes had all traces of the markers removed and were provided via the Internet to project collaborators outside Vanderbilt, who then performed retrospective registrations on the volumes, calculating transformations from CT to MR and/ or from PET to MR. These investigators communicated their transformations again via the Internet to Vanderbilt, where the accuracy of each registration was evaluated. In this evaluation, the accuracy is measured at multiple volumes of interest (VOIs), i.e., areas in the brain that would commonly be areas of neurological interest. A VOI is defined in the MR image and its centroid c is determined. Then, the prospective registration is used to obtain the corresponding point c' in CT or PET. To this point, the retrospective registration is then applied, producing c" in MR. Statistics are gathered on the target registration error (TRE), which is the distance between the original point c and its corresponding point c". This article presents statistics on the TRE calculated for each registration technique in this study and provides a brief description of each technique and an estimate of both preparation and execution time needed to perform the registration. Our results indicate that retrospective techniques have the potential to produce satisfactory results much of the time, but that visual inspection is necessary to guard against large errors.