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Mr. Stephane Puechmorel
Ecole nationale de l’aviation civile (ENAC), Université Fédérale de Toulouse, 7 Avenue Edouard Belin, FR-31055 Toulouse CEDEX, France

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Research Keywords & Expertise

0 Differential Geometry
0 Information geometry
0 Statistical Manifold
0 Hessian Manifold
0 foliations

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Information geometry
Statistical Manifold
Hessian Manifold
foliations

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Conference paper
Published: 14 July 2021 in Data and Applications Security and Privacy XXXV
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The sheaf of solutions \(\mathcal {J}_\nabla \) of the Hessian equation on a gauge structure \((M,\nabla )\) is a key ingredient for understanding important properties from the cohomological point of view. In this work, a canonical representation of the group associated by Lie third’s theorem to the Lie algebra formed by the sections of \(\mathcal {J}_\nabla \) is introduced. On the foliation it defines, a characterization of compact hyperbolic leaves is then obtained.

ACS Style

Michel Boyom; Emmanuel Gnandi; Stéphane Puechmorel. Canonical Foliations of Statistical Manifolds with Hyperbolic Compact Leaves. Data and Applications Security and Privacy XXXV 2021, 371 -379.

AMA Style

Michel Boyom, Emmanuel Gnandi, Stéphane Puechmorel. Canonical Foliations of Statistical Manifolds with Hyperbolic Compact Leaves. Data and Applications Security and Privacy XXXV. 2021; ():371-379.

Chicago/Turabian Style

Michel Boyom; Emmanuel Gnandi; Stéphane Puechmorel. 2021. "Canonical Foliations of Statistical Manifolds with Hyperbolic Compact Leaves." Data and Applications Security and Privacy XXXV , no. : 371-379.

Journal article
Published: 21 November 2020 in Mathematics
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Let (M,g) be a Riemannian manifold equipped with a pair of dual connections (∇,∇*). Such a structure is known as a statistical manifold since it was defined in the context of information geometry. This paper aims at defining the complete lift of such a structure to the cotangent bundle T*M using the Riemannian extension of the Levi-Civita connection of M. In the first section, common tensors are associated with pairs of dual connections, emphasizing the cyclic symmetry property of the so-called skewness tensor. In a second section, the complete lift of this tensor is obtained, allowing the definition of dual connections on TT*M with respect to the Riemannian extension. This work was motivated by the general problem of finding the projective limit of a sequence of a finite-dimensional statistical manifold.

ACS Style

Stéphane Puechmorel. Lifting Dual Connections with the Riemann Extension. Mathematics 2020, 8, 2079 .

AMA Style

Stéphane Puechmorel. Lifting Dual Connections with the Riemann Extension. Mathematics. 2020; 8 (11):2079.

Chicago/Turabian Style

Stéphane Puechmorel. 2020. "Lifting Dual Connections with the Riemann Extension." Mathematics 8, no. 11: 2079.

Journal article
Published: 06 November 2020 in Differential Geometry and its Applications
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In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is a Hadamard manifold, i.e. that it is geodesically complete and has everywhere negative sectional curvature. An important consequence for applications is that the Fréchet mean of a set of Dirichlet distributions is uniquely defined in this geometry.

ACS Style

Alice Le Brigant; Stephen C. Preston; Stéphane Puechmorel. Fisher-Rao geometry of Dirichlet distributions. Differential Geometry and its Applications 2020, 74, 101702 .

AMA Style

Alice Le Brigant, Stephen C. Preston, Stéphane Puechmorel. Fisher-Rao geometry of Dirichlet distributions. Differential Geometry and its Applications. 2020; 74 ():101702.

Chicago/Turabian Style

Alice Le Brigant; Stephen C. Preston; Stéphane Puechmorel. 2020. "Fisher-Rao geometry of Dirichlet distributions." Differential Geometry and its Applications 74, no. : 101702.

Journal article
Published: 29 July 2019 in Mathematics
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The Fisher information metric provides a smooth family of probability measures with a Riemannian manifold structure, which is an object in information geometry. The information geometry of the gamma manifold associated with the family of gamma distributions has been well studied. However, only a few results are known for the generalized gamma family that adds an extra shape parameter. The present article gives some new results about the generalized gamma manifold. This paper also introduces an application in medical imaging that is the classification of Alzheimer’s disease population. In the medical field, over the past two decades, a growing number of quantitative image analysis techniques have been developed, including histogram analysis, which is widely used to quantify the diffuse pathological changes of some neurological diseases. This method presents several drawbacks. Indeed, all the information included in the histogram is not used and the histogram is an overly simplistic estimate of a probability distribution. Thus, in this study, we present how using information geometry and the generalized gamma manifold improved the performance of the classification of Alzheimer’s disease population.

ACS Style

Sana Rebbah; Florence Nicol; Stéphane Puechmorel. The Geometry of the Generalized Gamma Manifold and an Application to Medical Imaging. Mathematics 2019, 7, 674 .

AMA Style

Sana Rebbah, Florence Nicol, Stéphane Puechmorel. The Geometry of the Generalized Gamma Manifold and an Application to Medical Imaging. Mathematics. 2019; 7 (8):674.

Chicago/Turabian Style

Sana Rebbah; Florence Nicol; Stéphane Puechmorel. 2019. "The Geometry of the Generalized Gamma Manifold and an Application to Medical Imaging." Mathematics 7, no. 8: 674.

Review
Published: 09 January 2019 in Entropy
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Finding an approximate probability distribution best representing a sample on a measure space is one of the most basic operations in statistics. Many procedures were designed for that purpose when the underlying space is a finite dimensional Euclidean space. In applications, however, such a simple setting may not be adapted and one has to consider data living on a Riemannian manifold. The lack of unique generalizations of the classical distributions, along with theoretical and numerical obstructions require several options to be considered. The present work surveys some possible extensions of well known families of densities to the Riemannian setting, both for parametric and non-parametric estimation.

ACS Style

Alice Le Brigant; Stéphane Puechmorel. Approximation of Densities on Riemannian Manifolds. Entropy 2019, 21, 43 .

AMA Style

Alice Le Brigant, Stéphane Puechmorel. Approximation of Densities on Riemannian Manifolds. Entropy. 2019; 21 (1):43.

Chicago/Turabian Style

Alice Le Brigant; Stéphane Puechmorel. 2019. "Approximation of Densities on Riemannian Manifolds." Entropy 21, no. 1: 43.

Journal article
Published: 10 September 2018 in Mathematical and Computational Applications
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Planning conflict-free trajectories is a long-standing problem in Air Traffic Management. Navigation functions designed specifically to produce flyable trajectories have been previously considered, but lack the robustness to uncertain weather conditions needed for use in an operational context. These uncertainties can be taken into account be modifying the boundary of the domain on which the navigation function is computed. In the following work, we present a method for efficiently taking into account boundary variations, using the Hadamard variation.

ACS Style

Isabelle Santos; Stéphane Puechmorel; Guillaume Dufour. First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World. Mathematical and Computational Applications 2018, 23, 48 .

AMA Style

Isabelle Santos, Stéphane Puechmorel, Guillaume Dufour. First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World. Mathematical and Computational Applications. 2018; 23 (3):48.

Chicago/Turabian Style

Isabelle Santos; Stéphane Puechmorel; Guillaume Dufour. 2018. "First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World." Mathematical and Computational Applications 23, no. 3: 48.

Journal article
Published: 29 August 2018 in Entropy
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In this paper, the problem of clustering rotationally invariant shapes is studied and a solution using Information Geometry tools is provided. Landmarks of a complex shape are defined as probability densities in a statistical manifold. Then, in the setting of shapes clustering through a K-means algorithm, the discriminative power of two different shapes distances are evaluated. The first, derived from Fisher–Rao metric, is related with the minimization of information in the Fisher sense and the other is derived from the Wasserstein distance which measures the minimal transportation cost. A modification of the K-means algorithm is also proposed which allows the variances to vary not only among the landmarks but also among the clusters.

ACS Style

Stefano Antonio Gattone; Angela De Sanctis; Stéphane Puechmorel; Florence Nicol. On the Geodesic Distance in Shapes K-means Clustering. Entropy 2018, 20, 647 .

AMA Style

Stefano Antonio Gattone, Angela De Sanctis, Stéphane Puechmorel, Florence Nicol. On the Geodesic Distance in Shapes K-means Clustering. Entropy. 2018; 20 (9):647.

Chicago/Turabian Style

Stefano Antonio Gattone; Angela De Sanctis; Stéphane Puechmorel; Florence Nicol. 2018. "On the Geodesic Distance in Shapes K-means Clustering." Entropy 20, no. 9: 647.

Preprint
Published: 07 August 2018
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Planning conflict-free trajectories is a long-standing problem in Air Traffic Management. 1 Navigation functions designed specifically to produce flyable trajectories have been previously 2 considered, but lack the robustness to uncertain weather conditions needed for use in an operational 3 context. These uncertainties can be taken into account be modifying the boundary of the domain 4 on which the navigation function is computed. In the following work, we present a method for 5 efficiently taking into account boundary variations, using the Hadamard variation. 6

ACS Style

Isabelle Santos; Stéphane Puechmorel; Guillaume Dufour. First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World. 2018, 1 .

AMA Style

Isabelle Santos, Stéphane Puechmorel, Guillaume Dufour. First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World. . 2018; ():1.

Chicago/Turabian Style

Isabelle Santos; Stéphane Puechmorel; Guillaume Dufour. 2018. "First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World." , no. : 1.

Journal article
Published: 03 May 2018 in Aerospace
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Resilience to uncertainties must be ensured in air traffic management. Unexpected events can either be disruptive, like thunderstorms or the famous volcano ash cloud resulting from the Eyjafjallajökull eruption in Iceland, or simply due to imprecise measurements or incomplete knowledge of the environment. While human operators are able to cope with such situations, it is generally not the case for automated decision support tools. Important examples originate from the numerous attempts made to design algorithms able to solve conflicts between aircraft occurring during flights. The STARGATE (STochastic AppRoach for naviGATion functions in uncertain Environment) project was initiated in order to study the feasibility of inherently robust automated planning algorithms that will not fail when submitted to random perturbations. A mandatory first step is the ability to simulate the usual stochastic phenomenons impairing the system: delays due to airport platforms or air traffic control (ATC) and uncertainties on the wind velocity. The work presented here will detail algorithms suitable for the simulation task.

ACS Style

Stephane Puechmorel; Guillaume Dufour; Romain Fèvre. Simulation of Random Events for Air Traffic Applications. Aerospace 2018, 5, 53 .

AMA Style

Stephane Puechmorel, Guillaume Dufour, Romain Fèvre. Simulation of Random Events for Air Traffic Applications. Aerospace. 2018; 5 (2):53.

Chicago/Turabian Style

Stephane Puechmorel; Guillaume Dufour; Romain Fèvre. 2018. "Simulation of Random Events for Air Traffic Applications." Aerospace 5, no. 2: 53.

Conference paper
Published: 24 October 2017 in Computer Vision
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Directional densities were introduced in the pioneering work of von Mises, with the definition of a rotationally invariant probability distribution on the circle. It was further generalized to more complex objects like the torus or the hyperbolic space. The purpose of the present work is to give a construction of equivalent objects on surfaces with genus larger than or equal to 2, for which an hyperbolic structure exists. Although the directional densities on the torus were introduced by several authors and are closely related to the original von Mises distribution, allowing more than one hole is challenging as one cannot simply add more angular coordinates. The approach taken here is to use a wrapping as in the case of the circular wrapped Gaussian density, but with a summation taken over all the elements of the group that realizes the surface as a quotient of the hyperbolic plane.

ACS Style

Florence Nicol; Stéphane Puechmorel. Von Mises-Like Probability Density Functions on Surfaces. Computer Vision 2017, 701 -708.

AMA Style

Florence Nicol, Stéphane Puechmorel. Von Mises-Like Probability Density Functions on Surfaces. Computer Vision. 2017; ():701-708.

Chicago/Turabian Style

Florence Nicol; Stéphane Puechmorel. 2017. "Von Mises-Like Probability Density Functions on Surfaces." Computer Vision , no. : 701-708.

Journal article
Published: 15 September 2016 in Entropy
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Air traffic management (ATM) aims at providing companies with a safe and ideally optimal aircraft trajectory planning. Air traffic controllers act on flight paths in such a way that no pair of aircraft come closer than the regulatory separation norms. With the increase of traffic, it is expected that the system will reach its limits in the near future: a paradigm change in ATM is planned with the introduction of trajectory-based operations. In this context, sets of well-separated flight paths are computed in advance, tremendously reducing the number of unsafe situations that must be dealt with by controllers. Unfortunately, automated tools used to generate such planning generally issue trajectories not complying with operational practices or even flight dynamics. In this paper, a means of producing realistic air routes from the output of an automated trajectory design tool is investigated. For that purpose, the entropy of a system of curves is first defined, and a mean of iteratively minimizing it is presented. The resulting curves form a route network that is suitable for use in a semi-automated ATM system with human in the loop. The tool introduced in this work is quite versatile and may be applied also to unsupervised classification of curves: an example is given for French traffic.

ACS Style

Stéphane Puechmorel; Florence Nicol. Entropy Minimizing Curves with Application to Flight Path Design and Clustering. Entropy 2016, 18, 337 .

AMA Style

Stéphane Puechmorel, Florence Nicol. Entropy Minimizing Curves with Application to Flight Path Design and Clustering. Entropy. 2016; 18 (9):337.

Chicago/Turabian Style

Stéphane Puechmorel; Florence Nicol. 2016. "Entropy Minimizing Curves with Application to Flight Path Design and Clustering." Entropy 18, no. 9: 337.

Book chapter
Published: 01 January 2015 in Advances in Aerospace Guidance, Navigation and Control
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Multiple aircraft trajectory planning is a central problem in future air traffic management concepts where some part of the separation task, currently assumed by human controllers, will be delegated to on-board automated systems. Several approaches have been taken to address it and fall within two categories: meta-heuristic algorithms or deterministic methods. The framework proposed here models the planning problem as a optimization program in a space of functions with constraints obtained by semi-infinite programming.A specially designed innovative interior point algorithm is used to solve it.

ACS Style

Stephane Puechmorel; Daniel Delahaye. Functional Interior Point Programming Applied to the Aircraft Path Planning Problem. Advances in Aerospace Guidance, Navigation and Control 2015, 521 -529.

AMA Style

Stephane Puechmorel, Daniel Delahaye. Functional Interior Point Programming Applied to the Aircraft Path Planning Problem. Advances in Aerospace Guidance, Navigation and Control. 2015; ():521-529.

Chicago/Turabian Style

Stephane Puechmorel; Daniel Delahaye. 2015. "Functional Interior Point Programming Applied to the Aircraft Path Planning Problem." Advances in Aerospace Guidance, Navigation and Control , no. : 521-529.

Conference paper
Published: 01 January 2015 in Transactions on Petri Nets and Other Models of Concurrency XV
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Air traffic management (ATM) aims at providing companies with a safe and ideally optimal aircraft trajectory planning. Air traffic controllers act on flight paths in such a way that no pair of aircraft come closer than the regulatory separation norm. With the increase of traffic, it is expected that the system will reach its limits in a near future: a paradigm change in ATM is planned with the introduction of trajectory based operations. This paper investigate a mean of producing realistic air routes from the output of an automated trajectory design tool. For that purpose, an entropy associated with a system of curves is defined and a mean of iteratively minimizing it is presented. The network produced is suitable for use in a semi-automated ATM system with human in the loop.

ACS Style

Stephane Puechmorel; Florence Nicol. Entropy Minimizing Curves with Application to Automated Flight Path Design. Transactions on Petri Nets and Other Models of Concurrency XV 2015, 770 -778.

AMA Style

Stephane Puechmorel, Florence Nicol. Entropy Minimizing Curves with Application to Automated Flight Path Design. Transactions on Petri Nets and Other Models of Concurrency XV. 2015; ():770-778.

Chicago/Turabian Style

Stephane Puechmorel; Florence Nicol. 2015. "Entropy Minimizing Curves with Application to Automated Flight Path Design." Transactions on Petri Nets and Other Models of Concurrency XV , no. : 770-778.

Book
Published: 27 June 2013 in Modeling and Optimization of Air Traffic
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ACS Style

Daniel Delahaye; Stephane Puechmorel. Modeling and Optimization of Air Traffic. Modeling and Optimization of Air Traffic 2013, 1 .

AMA Style

Daniel Delahaye, Stephane Puechmorel. Modeling and Optimization of Air Traffic. Modeling and Optimization of Air Traffic. 2013; ():1.

Chicago/Turabian Style

Daniel Delahaye; Stephane Puechmorel. 2013. "Modeling and Optimization of Air Traffic." Modeling and Optimization of Air Traffic , no. : 1.

Conference paper
Published: 01 January 2004 in Computer Vision
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ACS Style

Stephane Puechmorel; Daniel Delahaye. Order Statistics in Artificial Evolution. Computer Vision 2004, 51 -62.

AMA Style

Stephane Puechmorel, Daniel Delahaye. Order Statistics in Artificial Evolution. Computer Vision. 2004; ():51-62.

Chicago/Turabian Style

Stephane Puechmorel; Daniel Delahaye. 2004. "Order Statistics in Artificial Evolution." Computer Vision , no. : 51-62.

Journal article
Published: 01 April 2002 in Air Traffic Control Quarterly
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Field observations and focused interviews of Air Traffic Controllers have been used to generate a list of key complexity factors in Air Traffic Control. The underlying structure of the airspace was identified as relevant in many of the factors. A preliminary investigation has revealed that the structure appears to form the basis for abstractions that reduce the difficulty of maintaining Situational Awareness, particularly the projection of future traffic situations. Three examples of such abstractions were identified: standard flows, groupings, and critical points. Preliminary approaches to developing metrics including these structural considerations are discussed.

ACS Style

Jonathan M. Histon; R. John Hansman; Guillaume Aigoin; Daniel Delahaye; Stephane Puechmorel. Introducing Structural Considerations into Complexity Metrics. Air Traffic Control Quarterly 2002, 10, 115 -130.

AMA Style

Jonathan M. Histon, R. John Hansman, Guillaume Aigoin, Daniel Delahaye, Stephane Puechmorel. Introducing Structural Considerations into Complexity Metrics. Air Traffic Control Quarterly. 2002; 10 (2):115-130.

Chicago/Turabian Style

Jonathan M. Histon; R. John Hansman; Guillaume Aigoin; Daniel Delahaye; Stephane Puechmorel. 2002. "Introducing Structural Considerations into Complexity Metrics." Air Traffic Control Quarterly 10, no. 2: 115-130.