Salem Said

Fast Complexified Quaternion Fourier Transform

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 of hyperanalytic signal. We also study the Hermitian symmetries of the BiQFT and their relation to the geometric nature of a biquaternion-valued signal. Finally, we present a fast algorithm for the computation of the BiQFT. This algorithm is based on a (complex) change of basis and four standard complex FFTs.

On Filtering with Observation in a Manifold: Reduction to a Classical Filtering Problem

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 Levi-Civita connection, in performing the operation of antidevelopment (both connections can be constructed from a stochastic differential equation as in the observation model). The following results are obtained. First, it is shown that antidevelopment reduces the original filtering problem to a classical one, i.e., an additive white noise model. Second, a new form of the Zakai and filtering equations is derived which has a structure very similar to that of classical equations. Currently, there are few well understood general numerical solution methods for filtering problems with observation in a manifold. Reduction to classical filtering problems seems de...

Decompounding on Compact Lie Groups

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 the physical inverse problem of multiple scattering.

Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices

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 noise or faulty measurements. The present paper tackles this challenge by introducing new probability distributions, called Riemannian Laplace distributions on the space P m. First, it shows that these distributions provide a statistical foundation for the concept of the Riemannian median, which offers improved robustness in dealing with outliers (in comparison to the more popular concept of the Riemannian center of mass). Second, it describes an original expectation-maximization algorithm, for estimating mixtures of Riemannian Laplace distributions. This algorithm is applied to the problem of texture classification, in computer vision, which is considered in the presence of outliers. It is shown to give significantly better performance with respect to other recently-proposed approaches.

Gaussian Distributions on Riemannian Symmetric Spaces: Statistical Learning With Structured Covariance Matrices

The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing challenge is to develop Riemannian-geometric tools which are adapted to structured covariance matrices. This paper proposes to meet this challenge by introducing a new class of probability distributions, Gaussian distributions of structured covariance matrices. These are Riemannian analogs of Gaussian distributions, which only sample from covariance matrices having a preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The usefulness of these distributions stems from three features: 1) they are completely tractable, analytically, or numerically, when dealing with large covariance matrices; 2) they provide a statistical foundation to the concept of structured Riemannian barycentre (i.e., Fréchet or geometric mean); and 3) they lead to efficient statistical learning algorithms, which realise, among others, density estimation and classification of structured covariance matrices. This paper starts from the observation that several spaces of structured covariance matrices, considered from a geometric point of view, are Riemannian symmetric spaces. Accordingly, it develops an original theory of Gaussian distributions on Riemannian symmetric spaces, of their statistical inference, and of their relationship to the concept of Riemannian barycentre. Then, it uses this original theory to give a detailed description of Gaussian distributions of three kinds of structured covariance matrices, complex, Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density estimation and classification of structured covariance matrices, based on Gaussian distribution mixture models.

Texture Classification Using Rao’s Distance on the Space of Covariance Matrices

The current paper introduces new prior distributions on the zero-mean multivariate Gaussian model, with the aim of applying them to the classification of covariance matrices populations. These new prior distributions are entirely based on the Riemannian geometry of the multivariate Gaussian model. More precisely, the proposed Riemannian Gaussian distribution has two parameters, the centre of mass \(\bar{Y}\) and the dispersion parameter \(\sigma \). Its density with respect to Riemannian volume is proportional to \(\exp (-d^2(Y; \bar{Y}))\), where \(d^2(Y; \bar{Y})\) is the square of Rao’s Riemannian distance. We derive its maximum likelihood estimators and propose an experiment on the VisTex database for the classification of texture images.

New Riemannian Priors on the Univariate Normal Model

The current paper introduces new prior distributions on the univariate normal model, with the aim of applying them to the classification of univariate normal populations. These new prior distributions are entirely based on the Riemannian geometry of the univariate normal model, so that they can be thought of as “Riemannian priors”. Precisely, if {pθ ; θ ∈ Θ} is any parametrization of the univariate normal model, the paper considers prior distributions G( θ - , γ) with hyperparameters θ - ∈ Θ and γ > 0, whose density with respect to Riemannian volume is proportional to exp(−d2(θ, θ - )/2γ2), where d2(θ, θ - ) is the square of Rao’s Riemannian distance. The distributions G( θ - , γ) are termed Gaussian distributions on the univariate normal model. The motivation for considering a distribution G( θ - , γ) is that this distribution gives a geometric representation of a class or cluster of univariate normal populations. Indeed, G( θ - , γ) has a unique mode θ - (precisely, θ - is the unique Riemannian center of mass of G( θ - , γ), as shown in the paper), and its dispersion away from θ - is given by γ. Therefore, one thinks of members of the class represented by G( θ - , γ) as being centered around θ - and lying within a typical distance determined by γ. The paper defines rigorously the Gaussian distributions G( θ - , γ) and describes an algorithm for computing maximum likelihood estimates of their hyperparameters. Based on this algorithm and on the Laplace approximation, it describes how the distributions G( θ - , γ) can be used as prior distributions for Bayesian classification of large univariate normal populations. In a concrete application to texture image classification, it is shown that this leads to an improvement in performance over the use of conjugate priors.

Extrinsic Mean of Brownian Distributions on Compact Lie Groups

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 transform in a nice way under group operations and satisfy an extension of the central limit theorem. Brownian distributions on a compact Lie group <i>G</i> belong to one of two parametric families <i>N</i>_L(<i>g</i>,<i>C</i>) and <i>N</i>_R(<i>g</i>,<i>C</i>)-<i>g</i> ∈ <i>G</i> and <i>C</i> a positive-definite symmetric matrix. In particular, the parameter <i>g</i> appears as a location parameter. An approach based on the extrinsic mean for estimation of the parameters <i>g</i> and <i>C</i> is studied in detail. It is shown that <i>g</i> is the unique extrinsic mean for a Brownian distribution <i>N</i>_L(<i>g</i>,<i>C</i>) or <i>N</i>_R(<i>g</i>,<i>C</i>). Resulting estimates are proved to be consistent and asymptotically normal. While they may also be used to simultaneously estimate <i>g</i> and <i>C</i>, it is seen this requires that <i>G</i> be embedded into a higher dimensional matrix Lie group. Going beyond Brownian distributions, it is shown the extrinsic mean can be used to recover the location parameter for a wider class of distributions arising more generally from Lévy processes. The compact Lie group structure places limitations on the analogy between normal distributions and Brownian distributions. This is illustrated by the study of multivariate Brownian distributions. These are introduced as Brownian distributions on some product group-e.g., <i>G</i> ×<i>G</i>. This paper describes their covariance structure and considers its transformation under group operations.

Structure Tensor Riemannian Statistical Models for CBIR and Classification of Remote Sensing Images

This paper deals with parametric techniques for the description of texture on very high resolution (VHR) remote sensing images. These techniques focus on the property of anisotropy as described by the local structure tensor (LST). The novelty of this paper consists in proposing several comprehensive statistical frameworks to handle LST fields for rotation-invariant texture discrimination tasks. These frameworks are all based on probability models defined on the Riemannian manifold of positive definite matrices: a recent Riemannian Gaussian model on the affine-invariant metric space and a multivariate Gaussian distribution on the Log-Euclidean space. A thorough comparison of the proposed methods is performed with respect to some state-of-the-art texture analysis methods. Three experimental protocols are considered based on VHR remote sensing data. The first one consists of a content-based image retrieval (CBIR) protocol for browsing oyster field patches. The second one concerns a supervised classification protocol for grouping maritime pine forest stands in different age classes. The third one is, again, a CBIR protocol performed on the UC Merced land use/land cover patch collection. Tensor-based approaches show similar or even better results than the state-of-the-art texture analysis methods considered for comparison in all the experimental contexts.

Parameters estimate of Riemannian Gaussian distribution in the manifold of covariance matrices

The study of Pm, the manifold of m × m symmetric positive definite matrices, has recently become widely popular in many engineering applications, like radar signal processing, mechanics, computer vision, image processing, and medical imaging. A large body of literature is devoted to the barycentre of a set of points in Pm and the concept of barycentre has become essential to many applications and procedures, for instance classification of SPD matrices. However this concept is often used alone in order to define and characterize a set of points. Less attention is paid to the characterization of the shape of samples in the manifold, or to the definition of a probabilistic model, to represent the statistical variability of data in Pm. Here we consider Gaussian distributions and mixtures of Gaussian distributions on Pm. In particular we deal with parameter estimation of such distributions. This problem, while it is simple in the manifold P2, becomes harder for higher dimensions, since there are some quantities involved whose analytic expression is difficult to derive. In this paper we introduce a smooth estimate of these quantities using convex cubic splines, and we show that in this case the parameters estimate is coherent with theoretical results. We also present some simulations and a real EEG data analysis.