Xavier Pennec

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Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are(More)
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(More)
We propose an efficient non-parametric diffeomorphic image registration algorithm based on Thirion's demons algorithm. In the first part of this paper, we show that Thirion's demons algorithm can be seen as an optimization procedure on the entire space of displacement fields. We provide strong theoretical roots to the different variants of Thirion's demons(More)
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;(More)
In medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of(More)
We investigate in this article the rigid registration of large sets of points, generally sampled from surfaces. We formulate this problem as a general Maximum-Likelihood (ML) estimation of the transformation and the matches. We show that, in the specific case of a Gaussian noise, it corresponds to the Iterative Closest Point algorithm (ICP) with the(More)
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(More)
Modern morphometric studies use non-linear image registration to compare anatomies and perform group analysis. Recently, log-Euclidean approaches have contributed to promote the use of such computational anatomy tools by permitting simple computations of statistics on a rather large class of invertible spatial transformations. In this work, we propose a(More)
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(More)