The Experts below are selected from a list of 19380 Experts worldwide ranked by ideXlab platform
Remco Duits - One of the best experts on this subject based on the ideXlab platform.
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Left-Invariant evolutions of wavelet transforms on the similitude group
Applied and Computational Harmonic Analysis, 2015Co-Authors: Upanshu Sharma, Remco DuitsAbstract:Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) Invariant operators on images to Left-Invariant operators on multiple-scale orientation scores. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at Left-Invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on Left-Invariant diffusions on orientation scores defined on Euclidean motion group. Keywords: Continuous wavelet transform; Left-Invariant vector fields; Similitude group; Evolution equations; Diffusions on Lie groups; Medical imaging
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Left-Invariant evolutions of wavelet transforms on the Similitude Group
2013Co-Authors: Upanshu Sharma, Remco DuitsAbstract:Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) Invariant operators on images to Left-Invariant operators on multiple-scale orientation scores. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at Left-Invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on Left-Invariant diffusions on orientation scores defined on Euclidean motion group.
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Mathematical Methods for Signal and Image Analysis and Representation - Left Invariant evolution equations on Gabor transforms
2012Co-Authors: Remco Duits, Hartmut Führ, Bart JanssenAbstract:By means of the unitary Gabor transform one can relate operators on signals to operators on the space of Gabor transforms. In order to obtain a translation and modulation Invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be Left Invariant, i.e. it should commute with the Left regular action of the reduced Heisenberg group H r . By using the Left Invariant vector fields on H r and the corresponding Left-Invariant vector fields on phase space in the generators of our transport and diffusion equations on Gabor transforms we naturally employ the essential group structure on the domain of a Gabor transform. Here we mainly restrict ourselves to non-linear adaptive Left-Invariant convection (reassignment), while maintaining the original signal.
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Left Invariant evolution equations on Gabor transforms
Computational Imaging and Vision, 2011Co-Authors: Remco Duits, Hartmut Führ, Bart JanssenAbstract:By means of the unitary Gabor transform one can relate operators on signals to operators on the space of Gabor transforms. In order to obtain a translation and modulation Invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be Left Invariant, i.e. it should commute with the Left regular action of the reduced Heisenberg group Hr. By using the Left Invariant vector fields on Hr and the corresponding Left-Invariant vector fields on phase space in the generators of our transport and diffusion equations on Gabor transforms we naturally employ the essential group structure on the domain of a Gabor transform. Here we mainly restrict ourselves to non-linear adaptive Left-Invariant convection (reassignment), while maintaining the original signal.
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Left-Invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part I: Linear Left-Invariant diffusion equations on SE(2)
Quarterly of Applied Mathematics, 2010Co-Authors: Remco Duits, Erik FrankenAbstract:We provide the explicit solutions of linear, Left-Invariant, diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2) = R^2 x T. These parabolic equations are forward Kolmogorov equations for well-known stochastic processes for contour enhancement and contour completion. The solutions are given by group convolution with the corresponding Green's functions. In earlier work we have solved the forward Kolmogorov equations (or Fokker-Planck equations) for stochastic processes on contour completion. Here we mainly focus on the forward Kolmogorov equations for contour enhancement processes which do not include convection. We derive explicit formulas for the Green's functions (i.e., the heat kernels on SE(2)) of the Left-Invariant partial differential equations related to the contour enhancement process. By applying a contraction we approximate the Left-Invariant vector fields on SE(2) by Left-Invariant generators of a Heisenberg group, and we derive suitable approximations of the Green's functions. The exact Green's functions are used in so-called collision distributions on SE(2), which are the product of two Left-Invariant resolvent diffusions given an initial distribution on SE(2). We use the Left-Invariant evolution processes for automated contour enhancement in noisy medical image data using a so-called orientation score, which is obtained from a grey-value image by means of a special type of unitary wavelet transformation. Here the real part of the (invertible) orientation score serves as an initial condition in the collision distribution.
Hiroshi Tamaru - One of the best experts on this subject based on the ideXlab platform.
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A classification of Left-Invariant symplectic structures on some Lie groups.
arXiv: Differential Geometry, 2020Co-Authors: Luis Pedro Castellanos Moscoso, Hiroshi TamaruAbstract:We are interested in the classification of Left-Invariant symplectic structures on Lie groups. Some classifications are known, especially in low dimensions. In this paper we establish a new approach to classify (up to automorphism and scale) Left-Invariant symplectic structures on Lie groups. The procedure is based on the moduli space of Left-Invariant nondegenerate $2$-forms. Then we apply our procedure for two particular Lie groups of dimension $2n$ and give classifications of Left-Invariant symplectic structures on them.
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A classification of Left-Invariant Lorentzian metrics on some nilpotent Lie groups.
arXiv: Differential Geometry, 2020Co-Authors: Yuji Kondo, Hiroshi TamaruAbstract:It has been known that there exist exactly three Left-Invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify Left-Invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n \geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each Left-Invariant metric can be identified with an orbit of a certain group action on some symmetric space.
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on the moduli spaces of Left Invariant pseudo riemannian metrics on lie groups
Hiroshima Mathematical Journal, 2016Co-Authors: Akira Kubo, Kensuke Onda, Yuichiro Taketomi, Hiroshi TamaruAbstract:The moduli space of Left-Invariant pseudo-Riemannian metrics on a given Lie group is defined as the orbit space of a certain isometric action on some pseudo- Riemannian symmetric space. In terms of the moduli space, we formulate a procedure to obtain a generalization of Milnor frames for Left-Invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on Left-Invariant Riemannian metrics. In this paper, we describe the orbit space of the action of a particular parabolic subgroup, and then apply it to obtain a generalization of Milnor frames for so-called the Lie groups of real hyperbolic spaces, and also for the three-dimensional Heisenberg group. As a corollary we show that all Left-Invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.
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The Space of Left-Invariant Riemannian Metrics
Springer Proceedings in Mathematics & Statistics, 2016Co-Authors: Hiroshi TamaruAbstract:Geometry of Left-Invariant Riemannian metrics on Lie groups has been studied very actively. We have proposed a new framework for studying this topic from the viewpoint of the space of Left-Invariant metrics. In this expository paper, we introduce our framework, and mention two results. One is a generalization of Milnor frames, and another is a characterization of solvsolitons of dimension three in terms of submanifold geometry.
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On the moduli spaces of Left-Invariant pseudo-Riemannian metrics on Lie groups
arXiv: Differential Geometry, 2015Co-Authors: Akira Kubo, Kensuke Onda, Yuichiro Taketomi, Hiroshi TamaruAbstract:In this paper, we formulate a procedure to obtain a generalization of Milnor frames for Left-Invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on Left-Invariant Riemannian metrics, and is based on the moduli space of Left-Invariant pseudo-Riemannian metrics. As one of applications, we show that any Left-Invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.
Upanshu Sharma - One of the best experts on this subject based on the ideXlab platform.
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Left-Invariant evolutions of wavelet transforms on the similitude group
Applied and Computational Harmonic Analysis, 2015Co-Authors: Upanshu Sharma, Remco DuitsAbstract:Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) Invariant operators on images to Left-Invariant operators on multiple-scale orientation scores. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at Left-Invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on Left-Invariant diffusions on orientation scores defined on Euclidean motion group. Keywords: Continuous wavelet transform; Left-Invariant vector fields; Similitude group; Evolution equations; Diffusions on Lie groups; Medical imaging
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Left-Invariant evolutions of wavelet transforms on the Similitude Group
2013Co-Authors: Upanshu Sharma, Remco DuitsAbstract:Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) Invariant operators on images to Left-Invariant operators on multiple-scale orientation scores. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at Left-Invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on Left-Invariant diffusions on orientation scores defined on Euclidean motion group.
Hamid Reza Salimi Moghaddam - One of the best experts on this subject based on the ideXlab platform.
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The Relation Between Automorphism Group and Isometry Group of Left Invariant $ (\alpha,\beta)$-metrics
arXiv: Differential Geometry, 2019Co-Authors: Masumeh Nejadahmad, Hamid Reza Salimi MoghaddamAbstract:This work generalizes the results of an earlier paper by the second author, from Randers metrics to $(\alpha,\beta)$-metrics. Let $F$ be an $(\alpha,\beta)$-metric which is defined by a Left Invariant vector field and a Left Invariant Riemannian metric on a simply connected real Lie group $G$. We consider the automorphism and isometry groups of the Finsler manifold $(G,F)$ and their intersection. We prove that for an arbitrary Left Invariant vector field $X$ and any compact subgroup $K$ of automorphisms which $X$ is Invariant under them, there exists an $(\alpha,\beta)$-metric such that $K$ is a subgroup of its isometry group.
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Left Invariant lifted $(\alpha,\beta)$-metrics of Douglas type on tangent Lie groups
arXiv: Differential Geometry, 2019Co-Authors: Masumeh Nejadahmad, Hamid Reza Salimi MoghaddamAbstract:In this paper we study lifted Left Invariant $(\alpha,\beta)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a Left Invariant $(\alpha,\beta)$-metric of Douglas type $F$, induced by a Left Invariant Riemannian metric $g$. Using vertical and complete lifts, we construct the vertical and complete lifted $(\alpha,\beta)$-metrics $F^v$ and $F^c$ on the tangent Lie group $TG$ and give necessary and sufficient conditions for them to be of Douglas type. Then, the flag curvature of these metrics are studied. Finally, as some special cases, the flag curvatures of $F^v$ and $F^c$ in the cases of Randers metrics of Douglas type, and Kropina and Matsumoto metrics of Berwald type are given.
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Left Invariant Randers metrics of Berwald type on tangent Lie groups
International Journal of Geometric Methods in Modern Physics, 2017Co-Authors: Farhad Asgari, Hamid Reza Salimi MoghaddamAbstract:Let G be a Lie group equipped with a Left Invariant Randers metric of Berward type F, with underlying Left Invariant Riemannian metric g. Suppose that F and g are lifted Randers and Riemannian metrics arising from F and g on the tangent Lie group TG by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold (TG,F) and the sectional curvature of the Riemannian manifold (G,g) when F is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.
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On the Left Invariant $(\alpha,\beta)$-metrics on some Lie groups
arXiv: Differential Geometry, 2016Co-Authors: Masoumeh Hosseini, Hamid Reza Salimi MoghaddamAbstract:We give the explicit formulas of the flag curvatures of Left Invariant Matsumoto and Kropina metrics of Berwald type. We can see these formulas are different from previous results given recently. Using these formulas, we prove that at any point of an arbitrary connected non-commutative nilpotent Lie group, the flag curvature of any Left Invariant Matsumoto and Kropina metrics of Berwald type admits zero, positive and negative values, this is a generalization of Wolf's theorem. Then we study $(\alpha,\beta)$-metrics of Berwald type and also Randers metrics of Douglas type on two interesting families of Lie groups considered by Milnor and Kaiser, containing Heisenberg Lie groups. On these spaces, we present some necessary and sufficient conditions for $(\alpha,\beta)$-metrics to be of Berwald type and also some necessary and sufficient conditions for Randers metrics to be of Douglas type. All Left Invariant non-Berwaldian Randers metrics of Douglas type are given and the flag curvatures are computed.
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on the Left Invariant alpha beta metrics on some lie groups
arXiv: Differential Geometry, 2016Co-Authors: Masoumeh Hosseini, Hamid Reza Salimi MoghaddamAbstract:We give the explicit formulas of the flag curvatures of Left Invariant Matsumoto and Kropina metrics of Berwald type. We can see these formulas are different from previous results given recently. Using these formulas, we prove that at any point of an arbitrary connected non-commutative nilpotent Lie group, the flag curvature of any Left Invariant Matsumoto and Kropina metrics of Berwald type admits zero, positive and negative values, this is a generalization of Wolf's theorem. Then we study $(\alpha,\beta)$-metrics of Berwald type and also Randers metrics of Douglas type on two interesting families of Lie groups considered by Milnor and Kaiser, containing Heisenberg Lie groups. On these spaces, we present some necessary and sufficient conditions for $(\alpha,\beta)$-metrics to be of Berwald type and also some necessary and sufficient conditions for Randers metrics to be of Douglas type. All Left Invariant non-Berwaldian Randers metrics of Douglas type are given and the flag curvatures are computed.
Alberto Medina - One of the best experts on this subject based on the ideXlab platform.
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Left Invariant semi Riemannian metrics on quadratic Lie groups
arXiv: Differential Geometry, 2011Co-Authors: Shirley Bromberg, Alberto MedinaAbstract:To determine the Lie groups that admit a flat (eventually complete) Left Invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of Left Invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-Invariant semi Riemannian metric. We give a useful necessary and sufficient condition that guaranties the flatness of a Left Invariant semi Riemannian metric defined on a quadratic Lie group. All these semi Riemannian metrics are complete. We show that there are no Riemannian or Lorentzian flat Left Invariant metrics on non Abelian quadratic Lie groups, and that every quadratic 3 step nilpotent Lie group admits a flat Left Invariant semi Riemannian metric. The case of quadratic 2 step nilpotent Lie groups is also addressed.
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classical yang baxter equation and Left Invariant affine geometry on lie groups
Manuscripta Mathematica, 2004Co-Authors: Andre Diatta, Alberto MedinaAbstract:Let G be a Lie group, T*G=Lie(G)*⋊G its cotangent bundle considered as a Lie group, where G acts on Lie(G)* via the coadjoint action. Each solution r of the Classical Yang Baxter Equation on G, corresponds to a connected Lie subgroup H of T*G such that Lie(H) is a Lagrangian graph in Lie(G)⊕Lie(G)* and H carries a Left Invariant affine structure. If r is invertible, the Poisson Lie tensor π given by r on G is polynomial of degree at most 2 and every double Lie group of (G,π) is endowed with an affine and a complex structures ∇ and J, both Left Invariant and given by r, such that ∇J=0.
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Classical Yang-Baxter Equation and Left Invariant Affine Geometry on Lie Groups
2002Co-Authors: Andre Diatta, Alberto MedinaAbstract:Let G be a Lie group with Lie algebra $ \Cal G: = T_\epsilon G$ and $T^*G = \Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G acts on $\Cal G^*$ via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutions $r\in \wedge^2 \Cal G$ of the Classical Yang-Baxter Equation in G, and the set of connected Lie subgroups of $T^*G$ which carry a Left Invariant affine structure and whose Lie algebras are lagrangian graphs in $ \Cal G \oplus \Cal G^*$. An invertible solution r endows G with a Left Invariant symplectic structure and hence a Left Invariant affine structure. In this case we prove that the Poisson Lie tensor $\pi := r^+ - r^-$ is polynomial of degree at most 2 and the double Lie groups of $(G,\pi)$ also carry a canonical Left Invariant affine structure. In the general case of (non necessarly invertible) solutions r, we supply a necessary and suffisant condition to the geodesic completness of the associated affine structure