The Experts below are selected from a list of 249 Experts worldwide ranked by ideXlab platform
Nicolas Ray - One of the best experts on this subject based on the ideXlab platform.
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Robust Polylines Tracing for N-Symmetry Direction Field on Triangulated Surfaces
ACM Transactions on Graphics, 2014Co-Authors: Nicolas Ray, Dmitry SokolovAbstract:We are proposing an algorithm for tracing polylines that are oriented by a Direction field defined on a triangle mesh. The challenge is to ensure that two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is impossible to enforce with existing algorithms. The core of our contribution is to determine how polylines cross each triangle. Our solution is inspired by EdgeMaps where each triangle boundary is decomposed into inflow and outflow intervals such that each inflow interval is mapped onto an outflow interval. To cross a triangle, we find the inflow interval that contains the entry point, and link it to the corresponding outflow interval, with the same barycentric coordinate. To ensure that polylines cannot merge or cross, we introduce a new Direction field representation, we resolve the inflow/outflow interval pairing with a guaranteed combinatorial algorithm, and propagate the barycentric positions with arbitrary precision number representation. Using these techniques, two streamlines crossing the same triangle cannot merge or cross, but only locally overlap when all streamline extremities are located on the same edge. Cross-free and merge-free polylines can be traced on the mesh by iteratively crossing triangles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled.
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Tracing cross-free polylines oriented by a N-Symmetry Direction field on triangulated surfaces
ACM Transactions on Graphics, 2014Co-Authors: Nicolas Ray, Dmitry SokolovAbstract:We propose an algorithm for tracing polylines on a triangle mesh such that: they are aligned with a N-Symmetry Direction field, and two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is very difficult to enforce with numerical integration of vector fields. We propose an alternative solution based on "stream-mesh", a new combinatorial data structure that defines, for each point of a triangle edge, where the corresponding polyline leaves the triangle. It makes it possible to trace polylines by iteratively crossing triangles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled. The polylines inherits the cross-free property of the stream-mesh, except inside triangles where avoiding local overlaps would require higher order polycurves.
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N-Symmetry Direction Field Design
ACM Transactions on Graphics, 2008Co-Authors: Nicolas Ray, Bruno Vallet, Bruno LévyAbstract:Many algorithms in computer graphics and geometry processing use two orthogonal smooth Direction fields (unit tangent vector fields) defined over a surface. For instance, these Direction fields are used in texture synthesis, in geometry processing or in non-photorealistic rendering to distribute and orient elements on the surface. Such Direction fields can be designed in fundamentally different ways, according to the Symmetry requested: inverting a Direction or swapping two Directions may be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized Direction fields. As a consequence, existing Direction field design algorithms are limited to use non-optimum local relaxation procedures. In this paper, we formalize N-Symmetry Direction fields, a generalization of classical Direction fields. We give a new definition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the Poincare-Hopf theorem in the case of N-Symmetry Direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-Symmetry Direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user defined singularities and Directions.
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N-Symmetry Direction field design
ACM Transactions on Graphics, 2008Co-Authors: Nicolas Ray, Bruno Vallet, Bruno LévyAbstract:Many algorithms in computer graphics and geometry processing use two orthogonal smooth Direction fields (unit tangent vector fields) defined over a surface. For instance, these Direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and orient elements on the surface. Such Direction fields can be designed in fundamentally different ways, according to the Symmetry requested: inverting a Direction or swapping two Directions might be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized Direction fields. As a consequence, existing Direction field design algorithms are limited to using nonoptimum local relaxation procedures. In this article, we formalize N-Symmetry Direction fields, a generalization of classical Direction fields. We give a new definition of their singularities to explain how they relate to the topology of the surface. Specifically, we provide an accessible demonstration of the Poincare-Hopf theorem in the case of N-Symmetry Direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-Symmetry Direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user-defined singularities and Directions.
Bruno Lévy - One of the best experts on this subject based on the ideXlab platform.
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N-Symmetry Direction Field Design
ACM Transactions on Graphics, 2008Co-Authors: Nicolas Ray, Bruno Vallet, Bruno LévyAbstract:Many algorithms in computer graphics and geometry processing use two orthogonal smooth Direction fields (unit tangent vector fields) defined over a surface. For instance, these Direction fields are used in texture synthesis, in geometry processing or in non-photorealistic rendering to distribute and orient elements on the surface. Such Direction fields can be designed in fundamentally different ways, according to the Symmetry requested: inverting a Direction or swapping two Directions may be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized Direction fields. As a consequence, existing Direction field design algorithms are limited to use non-optimum local relaxation procedures. In this paper, we formalize N-Symmetry Direction fields, a generalization of classical Direction fields. We give a new definition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the Poincare-Hopf theorem in the case of N-Symmetry Direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-Symmetry Direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user defined singularities and Directions.
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N-Symmetry Direction field design
ACM Transactions on Graphics, 2008Co-Authors: Nicolas Ray, Bruno Vallet, Bruno LévyAbstract:Many algorithms in computer graphics and geometry processing use two orthogonal smooth Direction fields (unit tangent vector fields) defined over a surface. For instance, these Direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and orient elements on the surface. Such Direction fields can be designed in fundamentally different ways, according to the Symmetry requested: inverting a Direction or swapping two Directions might be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized Direction fields. As a consequence, existing Direction field design algorithms are limited to using nonoptimum local relaxation procedures. In this article, we formalize N-Symmetry Direction fields, a generalization of classical Direction fields. We give a new definition of their singularities to explain how they relate to the topology of the surface. Specifically, we provide an accessible demonstration of the Poincare-Hopf theorem in the case of N-Symmetry Direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-Symmetry Direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user-defined singularities and Directions.
Dmitry Sokolov - One of the best experts on this subject based on the ideXlab platform.
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Robust Polylines Tracing for N-Symmetry Direction Field on Triangulated Surfaces
ACM Transactions on Graphics, 2014Co-Authors: Nicolas Ray, Dmitry SokolovAbstract:We are proposing an algorithm for tracing polylines that are oriented by a Direction field defined on a triangle mesh. The challenge is to ensure that two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is impossible to enforce with existing algorithms. The core of our contribution is to determine how polylines cross each triangle. Our solution is inspired by EdgeMaps where each triangle boundary is decomposed into inflow and outflow intervals such that each inflow interval is mapped onto an outflow interval. To cross a triangle, we find the inflow interval that contains the entry point, and link it to the corresponding outflow interval, with the same barycentric coordinate. To ensure that polylines cannot merge or cross, we introduce a new Direction field representation, we resolve the inflow/outflow interval pairing with a guaranteed combinatorial algorithm, and propagate the barycentric positions with arbitrary precision number representation. Using these techniques, two streamlines crossing the same triangle cannot merge or cross, but only locally overlap when all streamline extremities are located on the same edge. Cross-free and merge-free polylines can be traced on the mesh by iteratively crossing triangles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled.
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Tracing cross-free polylines oriented by a N-Symmetry Direction field on triangulated surfaces
ACM Transactions on Graphics, 2014Co-Authors: Nicolas Ray, Dmitry SokolovAbstract:We propose an algorithm for tracing polylines on a triangle mesh such that: they are aligned with a N-Symmetry Direction field, and two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is very difficult to enforce with numerical integration of vector fields. We propose an alternative solution based on "stream-mesh", a new combinatorial data structure that defines, for each point of a triangle edge, where the corresponding polyline leaves the triangle. It makes it possible to trace polylines by iteratively crossing triangles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled. The polylines inherits the cross-free property of the stream-mesh, except inside triangles where avoiding local overlaps would require higher order polycurves.
Bruno Vallet - One of the best experts on this subject based on the ideXlab platform.
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N-Symmetry Direction Field Design
ACM Transactions on Graphics, 2008Co-Authors: Nicolas Ray, Bruno Vallet, Bruno LévyAbstract:Many algorithms in computer graphics and geometry processing use two orthogonal smooth Direction fields (unit tangent vector fields) defined over a surface. For instance, these Direction fields are used in texture synthesis, in geometry processing or in non-photorealistic rendering to distribute and orient elements on the surface. Such Direction fields can be designed in fundamentally different ways, according to the Symmetry requested: inverting a Direction or swapping two Directions may be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized Direction fields. As a consequence, existing Direction field design algorithms are limited to use non-optimum local relaxation procedures. In this paper, we formalize N-Symmetry Direction fields, a generalization of classical Direction fields. We give a new definition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the Poincare-Hopf theorem in the case of N-Symmetry Direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-Symmetry Direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user defined singularities and Directions.
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N-Symmetry Direction field design
ACM Transactions on Graphics, 2008Co-Authors: Nicolas Ray, Bruno Vallet, Bruno LévyAbstract:Many algorithms in computer graphics and geometry processing use two orthogonal smooth Direction fields (unit tangent vector fields) defined over a surface. For instance, these Direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and orient elements on the surface. Such Direction fields can be designed in fundamentally different ways, according to the Symmetry requested: inverting a Direction or swapping two Directions might be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized Direction fields. As a consequence, existing Direction field design algorithms are limited to using nonoptimum local relaxation procedures. In this article, we formalize N-Symmetry Direction fields, a generalization of classical Direction fields. We give a new definition of their singularities to explain how they relate to the topology of the surface. Specifically, we provide an accessible demonstration of the Poincare-Hopf theorem in the case of N-Symmetry Direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-Symmetry Direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user-defined singularities and Directions.
C.w. Rowley - One of the best experts on this subject based on the ideXlab platform.
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Template-Based Stabilization of Relative Equilibria in Systems with Continuous Symmetry
Journal of Nonlinear Science, 2007Co-Authors: S. Ahuja, Ioannis G. Kevrekidis, C.w. RowleyAbstract:We present an approach to the design of feedback control laws that stabilize relative equilibria of general nonlinear systems with continuous Symmetry. Using a template-based method, we factor out the dynamics associated with the Symmetry variables and obtain evolution equations in a reduced frame that evolves in the Symmetry Direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. We present two different approaches of control design. The first approach assumes that the closed loop system is affine in the control and that the actuation is equivariant. We derive feedback laws for the reduced system that minimize a quadratic cost function. The second approach is more general; here the actuation need not be equivariant, but the actuators can be translated in the Symmetry Direction. The controller resulting from this approach leaves the dynamics associated with the Symmetry variable unchanged. Both approaches are simple to implement, as they use standard tools available from linear control theory. We illustrate the approaches on three examples: a rotationally invariant planar ODE, an inverted pendulum on a cart, and the Kuramoto-Sivashinsky equation with periodic boundary conditions.
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Template-based stabilization of relative equilibria
2006 American Control Conference, 2006Co-Authors: S. Ahuja, Ioannis G. Kevrekidis, C.w. RowleyAbstract:We present an approach to the design of feedback control laws that stabilize the relative equilibria of general nonlinear systems with continuous Symmetry. Using a template-based method, we factor out the dynamics associated with the Symmetry variables and obtain evolution equations in a reduced frame that evolves in the Symmetry Direction. The relative equilibria of the original system are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. Assuming equivariant actuation, we derive feedback laws for the reduced system that are optimal in the sense that they minimize a quadratic cost function. We illustrate the method by stabilizing unstable traveling waves of a dissipative PDE possessing translational invariance.
