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In this paper, we consider the extension of the Fourier transform to biquaternion-valued signals. We introduce a transform that we call the biquaternion Fourier transform (BiQFT). After giving some general properties of this transform, we show how it can be used to generalize the notion of analytic signal to complex-valued signals. We introduce the notion(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)
Data, which lie in the space <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}_{m\,}$ </tex-math></inline-formula>, of <inline-formula> <tex-math notation="LaTeX">$m \times m$ </tex-math></inline-formula> symmetric positive definite matrices, (sometimes called <italic>tensor data</italic>), play a fundamental role in applications, including medical(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 decompounding 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)
The polymorphism at codon 72 of the TP53 gene has been extensively studied for its involvement in cancerogenesis and loss of heterozygosity (LOH) detection. Usually, the exon 4 of the TP53 gene is amplified by polymerase chain reaction (PCR) on DNA extracted from blood and tumor tissues, then digested by AccII. In the case of heterozygosity, the comparison(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)
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 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)