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A discrete complexified quaternion Fourier transform is introduced. This is a generalization of the discrete quaternion Fourier transform to the case where either or both of the signal/image and the transform kernel are complex quaternion-valued. It is shown how to compute the transform using four standard complex Fourier transforms and the properties of… (More)

PGA, or Principal Geodesic Analysis, is an extension of the classical PCA (Principal Component Analysis) to the case of data taking values on a Riemannian manifold. In this paper a new and original algorithm, for the exact computation of the PGA of data on the rotation group SO(3), is presented. Some properties of this algorithm are illustrated, with tests… (More)

—Noncommutative harmonic analysis is used to solve a nonparametric estimation problem stated in terms of compound Poisson processes on compact Lie groups. This problem of decom-pounding is a generalization of a similar classical problem. The proposed solution is based on a characteristic function method. The treated problem is important to recent models of… (More)

—This paper studies Brownian distributions on compact Lie groups. These are defined as the marginal distributions of Brownian processes and are intended as a natural extension of the well-known normal distributions to compact Lie groups. It is shown that this definition preserves key properties of normal distributions. In particular, Brownian distributions… (More)

The current paper deals with filtering problems where the observation process, conditioned on the unknown signal, is an elliptic diffusion in a differentiable manifold. Precisely, the observation model is given by a stochastic differential equation in the underlying manifold. The main new idea is to use a Le Jan-Watanabe connection, instead of a usual… (More)

The Riemannian geometry of the space P m , of m × m symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision and radar signal processing. Still, an open challenge remains, which consists of extending these tools to correctly handle the presence of outliers (or abnormal data), arising from excessive… (More)

This paper considers the problem of optimal filtering for partially observed signals taking values on the rotation group. More precisely, one or more components are considered not to be available in the measurement of the attitude of a 3D rigid body. In such cases, the observed signal takes its values on a Stiefel manifold. It is demonstrated how to filter… (More)