Frédéric Barbaresco

Souriau work on “Structure of Dynamical Systems” and his symplectic model of mechanics and statistical mechanics were elaborated as preamble of his geometric model of quantum mechanics as explained in his interview: “In 1958, I returned to France, to Marseille. And there I found myself confronted with theoretical physicists and with the problems of quantum mechanics which had disturbed me during my studies like all students, I think. I realized that symplectic geometry was an indispensable tool… Voir plus

Souriau work on “Structure of Dynamical Systems” and his symplectic model of mechanics and statistical mechanics were elaborated as preamble of his geometric model of quantum mechanics as explained in his interview: “In 1958, I returned to France, to Marseille. And there I found myself confronted with theoretical physicists and with the problems of quantum mechanics which had disturbed me during my studies like all students, I think. I realized that symplectic geometry was an indispensable tool for quantum mechanics. And that in fact it was even more appropriate to quantum mechanics than it was to classical mechanics. When I wrote my book on the subject I wanted to write a book on quantum mechanics and I realized that I had to present all classical mechanics in detail, as well as statistical mechanics. They were not foreign theories since they were connected by the symplectic structure and by the symmetries. You take two particles that revolve around each other according to Newton's laws, and then you take a hydrogen atom of which you only see the spectrum. These are two objects that have a priori nothing to do with each other; but they have symplectic symmetries in common. A door is ajar.”
This article is a translation of part 2 of French paper [1] by Jean-Marie Souriau "Des principes géométriques pour la mécanique quantique".

"As Landau and Lifchitz have pointed out, the relations between quantum mechanics and classical mechanics are of a very particular type: they coexist instead of succeeding each other. Any analysis of quantum structure is necessarily twofold; especially geometric analysis. We will avoid any ambiguity by putting in the first part, “classic”, everything that can be put there. In particular the study of symmetries; and also, as we shall see, the role of Planck's constant. Then two “classically” defined geometric structures make it possible to determine what the “quantum states” are" Voir moins

We introduce a symplectic bifoliation model of Information Geometry and Heat Theory based on Jean-Marie Souriau's Lie Groups Thermodynamics to describe transverse Poisson structure of metriplectic flow for dissipative phenomena. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, we… Voir plus

We introduce a symplectic bifoliation model of Information Geometry and Heat Theory based on Jean-Marie Souriau's Lie Groups Thermodynamics to describe transverse Poisson structure of metriplectic flow for dissipative phenomena. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, we associate a symplectic bifoliation structure to describe non-dissipative dynamics on symplectic leaves (on level sets of Entropy as constant Casimir function on each leaf), and transversal dissipative dynamics, given by Poisson transverse structure (Entropy production from leaf to leaf). The symplectic foliation orthogonal to the level sets of moment map is the foliation determined by hamiltonian vector fields generated by functions on dual Lie algebra. The orbits of a Hamiltonian action and the level sets of its moment map are polar to each other. The space of Casimir functions on a neighborhood of a point is isomorphic to the space of Casimirs for the transverse Poisson structure. Souriau’s model could be then interpreted by Mademoiselle Paulette Libermann's foliations, clarified as dual to Poisson Γ-structure of Haefliger, which is the maximum extension of the notion of moment in the sense of J.M. Souriau, as introduced by P. Molino, M. Condevaux and P. Dazord in papers of “Séminaire Sud-Rhodanien de Geometrie”. The symplectic duality to a symplectically complete foliation, in the sense of Libermann, associates an orthogonal foliation. Paulette Libermann proved that a Legendre foliation on a contact manifold is complete if and only if the pseudo-orthogonal distribution is completely integrable, and that the contact form is locally equivalent to the Poincaré-Cartan integral invariant. Voir moins

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint… Voir plus

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions . The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. Voir moins

We present in this chapter a new formulation of heat theory and Information Geometry through symplectic and Poisson structures based on Jean-Marie Souriau's symplectic model of statistical mechanics, called “Lie Groups Thermodynamics.” Souriau model was initially described in chapter IV “Statistical Mechanics” of his book “Structure of dynamical systems” published in 1969. This model gives an archetypal, and purely geometric, characterization of Entropy, which appears as an invariant Casimir… Voir plus

We present in this chapter a new formulation of heat theory and Information Geometry through symplectic and Poisson structures based on Jean-Marie Souriau's symplectic model of statistical mechanics, called “Lie Groups Thermodynamics.” Souriau model was initially described in chapter IV “Statistical Mechanics” of his book “Structure of dynamical systems” published in 1969. This model gives an archetypal, and purely geometric, characterization of Entropy, which appears as an invariant Casimir function in coadjoint representation, from which we will deduce a geometric heat equation as Euler-Poincaré equation. Voir moins

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in the framework of Symplectic model of Statistical Mechanics. Based on this model, we will introduce a geometric characterization of Entropy as a generalized Casimir invariant function in coadjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping. The dual space of the Lie algebra foliates into coadjoint orbits that are also the level sets on the entropy that could be… Voir plus

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in the framework of Symplectic model of Statistical Mechanics. Based on this model, we will introduce a geometric characterization of Entropy as a generalized Casimir invariant function in coadjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping. The dual space of the Lie algebra foliates into coadjoint orbits that are also the level sets on the entropy that could be interpreted in the framework of Thermodynamics by the fact that motion remaining on these surfaces is non-dissipative, whereas motion transversal to these surfaces is dissipative. We will also explain the 2nd Principle in thermodynamics by definite positiveness of Souriau tensor extending the Koszul-Fisher metric from Information Geometry, and introduce a new geometric Fourier heat equation with Souriau-Koszul-Fisher tensor. In conclusion, Entropy as Casimir function is characterized by Koszul Poisson Cohomology. Voir moins

The objective of this chapter is to make better known Jean-Marie Souriau works, more particularly his symplectic model of statistical physics, called “Lie groups thermodynamics”. This model was initially described in chapter IV “Statistical Mechanics” of his book “Structure of dynamical systems” published in 1969. We have translated in English some parts of three Souriau’s publications which provide more details about this geometric model of Thermodynamics. Entropy acquires a geometric… Voir plus

The objective of this chapter is to make better known Jean-Marie Souriau works, more particularly his symplectic model of statistical physics, called “Lie groups thermodynamics”. This model was initially described in chapter IV “Statistical Mechanics” of his book “Structure of dynamical systems” published in 1969. We have translated in English some parts of three Souriau’s publications which provide more details about this geometric model of Thermodynamics. Entropy acquires a geometric foundation as a function parameterized by mean of moment map in dual Lie algebra, and in term of foliations. Souriau established the generalized Gibbs laws when the manifold has a symplectic form and a connected Lie group G operates on this manifold by symplectomorphisms. Souriau Entropy is invariant under the action of the group acting on the homogeneous symplectic manifold. As quoted by Souriau, these equations are universal and could be also of great interest in Mathematics. Voir moins

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is… Voir plus

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping. Voir moins

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such… Voir plus

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle. Voir moins

Jean-Marie Souriau extended Urbain Jean Joseph Leverrier algorithm to compute characteristic polynomial of a matrix in 1948. This Souriau algorithm could be used to compute exponential map of a matrix that is a challenge in Lie Group Machine Learning. Main property of Souriau Exponential Map numerical scheme is its scalability with highly parallelization.

Results of UFO Project on Wind Monitoring on Airport with Toulouse Blagnac Airport sensors campaign results

This Special Issue “Differential Geometrical Theory of Statistics” collates selected invited and
contributed talks presented during the conference GSI'15 on “Geometric Science of Information”
which was held at the Ecole Polytechnique, Paris-Saclay Campus, France, in October 2015
(Conference web site: http://www.see.asso.fr/gsi2015).
Let us first start with a short historical review on the birth of the interplay of probability with
geometry and computing, which is rooted in the… Voir plus

This Special Issue “Differential Geometrical Theory of Statistics” collates selected invited and
contributed talks presented during the conference GSI'15 on “Geometric Science of Information”
which was held at the Ecole Polytechnique, Paris-Saclay Campus, France, in October 2015
(Conference web site: http://www.see.asso.fr/gsi2015).
Let us first start with a short historical review on the birth of the interplay of probability with
geometry and computing, which is rooted in the 17th century. Voir moins

This paper studies probability density estimation on the Siegel space. The Siegel space is a generalization of the hyperbolic space. Its Riemannian metric provides an interesting structure to the Toeplitz block Toeplitz matrices that appear in the covariance estimation of radar signals. The main techniques of probability density estimation on Riemannian manifolds are reviewed. For computational reasons, we chose to focus on the kernel density estimation. The main result of the paper is the… Voir plus

This paper studies probability density estimation on the Siegel space. The Siegel space is a generalization of the hyperbolic space. Its Riemannian metric provides an interesting structure to the Toeplitz block Toeplitz matrices that appear in the covariance estimation of radar signals. The main techniques of probability density estimation on Riemannian manifolds are reviewed. For computational reasons, we chose to focus on the kernel density estimation. The main result of the paper is the expression of Pelletier’s kernel density estimator. The computation of the kernels is made possible by the symmetric structure of the Siegel space. The method is applied to density estimation of reflection coefficients from radar observations. Voir moins

Special Issue "Differential Geometrical Theory of Statistics"

Electronic Phased-Array Radars offer new possibilities for Radar Search Pattern Optimization by using variable beam-shapes. Radar Search Pattern Optimization can be approximated by a Set Covering Problem. This allows automatic generation of Search Patterns for non-uniform constrained environments in operational situation. We present a Set Covering Problem approximation for Time-Budget minimization of Search Patterns, in Direction Cosines space, under constraints of range and detection… Voir plus

Electronic Phased-Array Radars offer new possibilities for Radar Search Pattern Optimization by using variable beam-shapes. Radar Search Pattern Optimization can be approximated by a Set Covering Problem. This allows automatic generation of Search Patterns for non-uniform constrained environments in operational situation. We present a Set Covering Problem approximation for Time-Budget minimization of Search Patterns, in Direction Cosines space, under constraints of range and detection probability. Voir moins

This paper explores a new way of monitoring tracking quality based on information geometry methods. Previous methods were not fully satisfying when facing certain maneuvers when both the bias and the variance vary at the same time. The method we propose takes into account all parameters of the movement of the target thanks to the use of an appropriate distance which reflects the mean and the covariance of the distribution we obtain with the Kalman filter. To achieve that, we develop the… Voir plus

This paper explores a new way of monitoring tracking quality based on information geometry methods. Previous methods were not fully satisfying when facing certain maneuvers when both the bias and the variance vary at the same time. The method we propose takes into account all parameters of the movement of the target thanks to the use of an appropriate distance which reflects the mean and the covariance of the distribution we obtain with the Kalman filter. To achieve that, we develop the distance in the manifold of multivariate Gaussians, compute the Fisher-Rao distance, and compare it with bounds. Voir moins

The authors address the estimation of the scatter matrix of a scale mixture of Gaussian stationary autoregressive (AR) vectors. This is equivalent to consider the estimation of a structured scatter matrix of a spherically invariant random vector whose structure comes from an AR modelisation. The Toeplitz structure representative of stationary models is a particular case for the class of structures they consider. For Gaussian AR processes, Burg method is often used in case of stationarity for… Voir plus

The authors address the estimation of the scatter matrix of a scale mixture of Gaussian stationary autoregressive (AR) vectors. This is equivalent to consider the estimation of a structured scatter matrix of a spherically invariant random vector whose structure comes from an AR modelisation. The Toeplitz structure representative of stationary models is a particular case for the class of structures they consider. For Gaussian AR processes, Burg method is often used in case of stationarity for its efficiency when few samples are available. Unfortunately, if they directly apply these methods to estimate the common scatter matrix of N vectors coming from a non-Gaussian distribution, their efficiency will strongly decrease. They propose then to adapt these methods to scale mixtures of AR vectors by changing the energy functional minimised in the Burg algorithm. Moreover, they study several approaches of robust modification of the introduced Burg algorithms, based on Fréchet medians defined for the Euclidean or the Poincaré metric, in presence of outliers or contaminating distributions. The considered structured modelisation is motivated by radar applications, the performances of their methods will then be compared with the very popular fixed point (FP) estimator and OS-CFAR detector through radar simulated scenarios. Voir moins

Runway operation is the limiting factor for the overall throughput of airports. Today the International Civil Aviation Organization (ICAO) imposes wake vortex separation minima between following aircrafts that are based on simple pair-wise rules. However, the lifetime of wake vortices results from a much broader basis of factors, that ranges from a large set of aircraft parameters to meteorological conditions and traffic mix. In particular atmospheric conditions can significantly reduce wake… Voir plus

Runway operation is the limiting factor for the overall throughput of airports. Today the International Civil Aviation Organization (ICAO) imposes wake vortex separation minima between following aircrafts that are based on simple pair-wise rules. However, the lifetime of wake vortices results from a much broader basis of factors, that ranges from a large set of aircraft parameters to meteorological conditions and traffic mix. In particular atmospheric conditions can significantly reduce wake hazard, for instance, in case of strong turbulence or crosswinds. While such situations could allow a reduction of the separation minima, safety reasons and the current technical challenges of detecting and managing such scenarios leads to the strict application of the ICAO standards. With the aid of accurate wind data and precise measurements of wake vortices, more efficient intervals could be set, particularly when weather conditions turn favourable. Depending on traffic intensity, these adjustments could enhance airport capacity, and generate major commercial benefits. This study deals with recent development in the radar technology to attain such goals. It presents (i) the trials of an electronic scanning radar to be used in a future wake turbulence advisory system and (ii) theoretical and numerical analysis of the radar response in clear air and in rainy weather. Part of this work has been achieved with the support of the European ATM research program SESAR. Voir moins

At airports, runway operation is the limiting factor for the overall throughput; specifically the fixed and overly conservative ICAO wake turbulence separation minima. The wake turbulence hazardous flows can dissipate quicker because of decay due to air turbulence or be transported out of the way on oncoming traffic by cross-wind, yet wake turbulence separation minima do not take into account wind conditions. Indeed, for safety reasons, most airports assume a worst-case scenario and use… Voir plus

At airports, runway operation is the limiting factor for the overall throughput; specifically the fixed and overly conservative ICAO wake turbulence separation minima. The wake turbulence hazardous flows can dissipate quicker because of decay due to air turbulence or be transported out of the way on oncoming traffic by cross-wind, yet wake turbulence separation minima do not take into account wind conditions. Indeed, for safety reasons, most airports assume a worst-case scenario and use conservative separations. However, with the aid of accurate EDR (Eddy Dissipation Rate) retrieval by Ultra-Fast High-Range Resolution X-band Electronic-Scanning radar sensors, more efficient intervals can be set, particularly when atmosphere is unstable and turbulent, accelerating Wake-Vortex decay. Depending on traffic volume, these adjustments can generate capacity gains, which have major commercial benefits. This paper presents Electronic scanning radar trials at Toulouse-Blagnac Airport for UFO (Ultra-Fast wind sensOrs for wake-vortex hazards mitigation) project, funded by European FP7 program, on Radar EDR retrieval & Calibration. Voir moins

Classical Radar processing for non-stationary signal, corresponding to fast time variation of Doppler Spectrum in one burst, is no longer optimal. This phenomenon could be observed for high speed or abrupt Doppler variations of clutter or target signal but also in case of target migration during the burst duration due to high range resolution. We propose new Radar Doppler processing assuming that each non-stationary signal in one burst can be split into several short signals with less Doppler… Voir plus

Classical Radar processing for non-stationary signal, corresponding to fast time variation of Doppler Spectrum in one burst, is no longer optimal. This phenomenon could be observed for high speed or abrupt Doppler variations of clutter or target signal but also in case of target migration during the burst duration due to high range resolution. We propose new Radar Doppler processing assuming that each non-stationary signal in one burst can be split into several short signals with less Doppler resolution but locally stationary, represented by time series of Toeplitz covariance matrices. In Information Geometry (IG) framework, these time series could be defined as a geodesic path (or geodesic polygon in discrete case) on covariance Toeplitz Hermitian Positive Definite matrix manifold. For this micro-Doppler analysis, we generalize the Fréchet distance between two curves in the plane to geodesic paths in abstract IG metric spaces of covariance matrix manifold. This approach is used for robust detection of target in case of non-stationary Time-Doppler spectrum (NS-OS-HDR-CFAR). Voir moins

ATALA objective is to break TRL4/5 Wall to speed up insertion of innovative algorithms in Radar Sensors Products. ATALA is also a derisking tool for Engineering Teams & Customers. We will illustrate validation of ATALA with 2 Use-Cases results: monitoring of commercial aircraft wake-vortex, detection of small/slow synthetic targets on relevant clutter environment. ATALA is based on two main principles: SDR4U (Software Defined Radar for You), PAR4U (Automatic Parallelization for You).

This paper presents the radar tasks scheduling problem for a set of tasks called Hard Time Constraint tasks which cannot delay their processing by more than a positive constant α and for a set of tasks called Soft Time Constraint tasks which can be processed at any time. For this problem, we present a method composed of an algorithm which sequence HTC tasks; composed of an heuristic which sequence STC tasks and composed of a selection rule in order to determine which task has to be scheduled… Voir plus

This paper presents the radar tasks scheduling problem for a set of tasks called Hard Time Constraint tasks which cannot delay their processing by more than a positive constant α and for a set of tasks called Soft Time Constraint tasks which can be processed at any time. For this problem, we present a method composed of an algorithm which sequence HTC tasks; composed of an heuristic which sequence STC tasks and composed of a selection rule in order to determine which task has to be scheduled between the first task of both sequences. Voir moins

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from “Characteristic Functions”, was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be… Voir plus

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from “Characteristic Functions”, was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of “Information Geometry” theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean “Moment map” by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the exclusive Cartan “Inner Product”. Interpreting Legendre transform as Fourier transform in (Min,+) algebra, we conclude with a definition of Entropy given by a relation mixing Fourier/Laplace transforms: Entropy = (minus) Fourier(Min,+) o Log o Laplace(+,X). Voir moins

At airports, surface operation on the runway is the limiting factor for the overall throughput; specifically the fixed and overly conservative ICAO wake turbulence separation minima. The wake turbulence hazardous flows can dissipate quicker because of decay due to air turbulence or be transported out of the way on oncoming traffic by cross-wind, yet wake turbulence separation minima do not take into account wind conditions. Indeed, for safety reasons, most airports assume a worst-case scenario… Voir plus

At airports, surface operation on the runway is the limiting factor for the overall throughput; specifically the fixed and overly conservative ICAO wake turbulence separation minima. The wake turbulence hazardous flows can dissipate quicker because of decay due to air turbulence or be transported out of the way on oncoming traffic by cross-wind, yet wake turbulence separation minima do not take into account wind conditions. Indeed, for safety reasons, most airports assume a worst-case scenario and use conservative separations; the interval between aircraft taking off or landing therefore often amounts to several minutes. However, with the aid of accurate wind data and precise measurements of wake vortex by radar sensors, more efficient intervals can be set, particularly when weather conditions are stable. Depending on traffic volume, these adjustments can generate capacity gains, which have major commercial benefits. This paper presents the developments of a wake turbulence system supporting increased throughput as part of the European ATM research program SESAR. Voir moins

The recent development of the full digital array technology paves the way to the design of multistatic RADAR systems relying on agile waveforms at emission. This new paradigm fits perfectly with the concept of phase conjugation (or time reversal if applied to large-band data). This technique allows indeed to adaptively build a wave focusing onto a target, leading to an improvement in detection range or in search time as compared to classical approaches. The DORT method, issued from phase… Voir plus

The recent development of the full digital array technology paves the way to the design of multistatic RADAR systems relying on agile waveforms at emission. This new paradigm fits perfectly with the concept of phase conjugation (or time reversal if applied to large-band data). This technique allows indeed to adaptively build a wave focusing onto a target, leading to an improvement in detection range or in search time as compared to classical approaches. The DORT method, issued from phase conjugation, even permits to detect multiple targets from the knowledge of the multistatic matrix of the antenna array. Based on both a theoretical analysis and an experimental proof provided here, it appears that these methods appear as very promising, especially for cueing and ultra-fast reacquisition modes where the phase conjugation/DORT SNR requirements are more easily fulfilled. Voir moins

We propose to use new shortest path computation methods based on Front propagation with Level Set approach for a radar application. This new radar function consists in computing most threatening trajectories & corridors in the radar coverage in order to adapt radar modes for detection optimization. This Radar problem may be declined as a variationnal problem solved by calculus of variations and front propagation based on an adaptation of Fermat's principle of least time with an Hamilton-Jacobi… Voir plus

We propose to use new shortest path computation methods based on Front propagation with Level Set approach for a radar application. This new radar function consists in computing most threatening trajectories & corridors in the radar coverage in order to adapt radar modes for detection optimization. This Radar problem may be declined as a variationnal problem solved by calculus of variations and front propagation based on an adaptation of Fermat's principle of least time with an Hamilton-Jacobi formulation. A partial differential equation PDE drives the temporal evolution of contours of constant action (level lines of the manifold defined by the minimal potential surface given by the integration of a local function of the detection probability along every potential trajectories). The orthogonality between geodesics (shortest path) and curves of iso-action provides a simple numerical scheme for geodesics computation based on a steepest gradient descent algorithm (backtracking on the level-lines of iso-action). We underline the analogy of this radar problem with Feynman/Schwinger's principle that states close connexion between variational principle and quantum theory. Finally, we have extended the problem to anisotropic constraint induced by Radar Cross Section. Voir moins

Information Geometry has been introduced by Rao, and axiomatized by Chentsov, to define a distance between statistical distributions that is invariant to non-singular parameterization transformations. For Doppler/Array/STAP Radar Processing, Information Geometry Approach will give key role to Homogenous Symmetric bounded domains geometry. For Radar, we will observe that Information Geometry metric could be related to Kähler metric, given by Hessian of Kähler potential (Entropy of Radar Signal… Voir plus

Information Geometry has been introduced by Rao, and axiomatized by Chentsov, to define a distance between statistical distributions that is invariant to non-singular parameterization transformations. For Doppler/Array/STAP Radar Processing, Information Geometry Approach will give key role to Homogenous Symmetric bounded domains geometry. For Radar, we will observe that Information Geometry metric could be related to Kähler metric, given by Hessian of Kähler potential (Entropy of Radar Signal given by −log[det(R)] ). To take into account Toeplitz structure of Time/Space Covariance Matrix or Toeplitz-Block-Toeplitz structure of Space-Time Covariance matrix, Parameterization known as Partial Iwasawa Decomposition could be applied through Complex Autoregressive Model or Multi-channel Autoregressive Model. Then, Hyperbolic Geometry of Poincaré Unit Disk or Symplectic Geometry of Siegel Unit Disk will be used as natural space to compute “p-mean” ( p=2 for “mean”, p=1 for “median”) of covariance matrices via Karcher flow derived from Weiszfeld algorithm extension on Cartan-Hadamard manifold. This new mathematical framework will allow development of Ordered Statistic (OS) concept for Hermitian Positive Definite Covariance Space/Time Toeplitz matrices or for Space-Time Toeplitz-Block-Toeplitz matrices. We will define Ordered Statistic High Doppler Resolution CFAR (OS-HDR-CFAR) and Ordered Statistic Space-Time Adaptive Processing (OS-STAP). Voir moins

A method for modeling and tracking convective clouds within radar images is presented. An object modeling approach is used, based on the extraction of either morphological or grayscale skeletons from 2-dimensionnal cross-section of 3-dimensional radar data. Grayscale skeletons are appropriate shape descriptors for non-rigid and heterogeneous objects, in which gray-level local maxima correspond to regions of interest. The modeling scheme is enhanced by meta-data linked to some chosen points of… Voir plus

A method for modeling and tracking convective clouds within radar images is presented. An object modeling approach is used, based on the extraction of either morphological or grayscale skeletons from 2-dimensionnal cross-section of 3-dimensional radar data. Grayscale skeletons are appropriate shape descriptors for non-rigid and heterogeneous objects, in which gray-level local maxima correspond to regions of interest. The modeling scheme is enhanced by meta-data linked to some chosen points of the skeleton; this provides a good representation of the weather scene in terms of hazards for an aircraft. Skeletons are stored within a graph structure and tracked among successive pictures by means of relaxation labeling processes. The deduced advection field is used to nowcast the clouds evolution. Satisfying results are obtained concerning advection forecasts. Convective activity forecasts are promising, even if they must be carefully interpreted. Voir moins

The capabilities of modern multifunction/mission Radar can be only fully realized by using new sensor control strategies and the most obvious sensor management imperative is the development of optimal realtime waveform scheduling algorithms. For this purpose, Thales is studying Intelligent Radar Time Resources management for Multi-mission extended air Defence radar used for both Air and Ballistic Missile Defence, based on innovative Active Electronically Steered Antenna technology. This paper… Voir plus

The capabilities of modern multifunction/mission Radar can be only fully realized by using new sensor control strategies and the most obvious sensor management imperative is the development of optimal realtime waveform scheduling algorithms. For this purpose, Thales is studying Intelligent Radar Time Resources management for Multi-mission extended air Defence radar used for both Air and Ballistic Missile Defence, based on innovative Active Electronically Steered Antenna technology. This paper describes the functional architecture of radar resources management used for adaptive time budget optimization and key enablers for advanced cognition, agility and autonomy capabilities. We conclude with simulation in the ASTRAD framework. Voir moins

The capabilities of modern military multifunction/multi-mission/multi-role radars cannot be fully realised by using older sensor control strategies, due to the advance in radar technology and the complexity of the tactical environment. The most obvious radar management imperative is the real-time optimisation of radar functionality & radar resource management to meet the changing operational mission, according to an assessment of the current environment conditions and tactical requests from an… Voir plus

The capabilities of modern military multifunction/multi-mission/multi-role radars cannot be fully realised by using older sensor control strategies, due to the advance in radar technology and the complexity of the tactical environment. The most obvious radar management imperative is the real-time optimisation of radar functionality & radar resource management to meet the changing operational mission, according to an assessment of the current environment conditions and tactical requests from an external centre. The objective of the Tutorial is to present: • Main principles of Intelligent Radar Resources Management; • Strategies for Radar Time Budget Regulation; • Algorithms for: threat assessment, dynamic dwell priority allocation, real-time dwell planning/scheduling, Load Handling/Control, … • New advanced tools for Radar Management Simulation on Complex scenarios. The Tutorial will be illustrated with simulation and campaign results of advanced Thales Multi-function/Mission Radars (e.g. MASTER A radar). Voir moins

We are interested in non-stationary targets, and represent their time/Doppler spectra in the form of curves tracing the evolution of the non-stationarity of the radar signal. We explain how to use this representation for Non Cooperative Target Recognition (NCTR) of helicopter signatures. The signature of reference used to recognize a specific type of helicopter is represented by the average curve over multiple simulations where we slightly vary certain parameters, such as the rotation speed of… Voir plus

We are interested in non-stationary targets, and represent their time/Doppler spectra in the form of curves tracing the evolution of the non-stationarity of the radar signal. We explain how to use this representation for Non Cooperative Target Recognition (NCTR) of helicopter signatures. The signature of reference used to recognize a specific type of helicopter is represented by the average curve over multiple simulations where we slightly vary certain parameters, such as the rotation speed of the blades, to model the hazards of real situations. The curves that we consider lie in the statistical manifold of centered stationary Gaussian distributions. In order to compare or average different signatures, the space of such curves is equipped with a metric and seen as a Riemannian manifold. We give algorithms to effectively compute distances and mean curves using this metric. Voir moins