Reeb graph
2016-03-04 11:00
1271 查看
From Wikipedia, the free encyclopedia
Reeb graph of the height function on the torus.
A Reeb graph (named after Georges
Reeb) is a mathematical object reflecting the evolution of the level
sets of a real-valued function on a manifold.[1] Originally
introduced as a tool inMorse theory,[2] Reeb
graphs found a wide variety of applications in computational geometry and computer
graphics, including computer aided geometric
design, topology-based shape
matching,[3]topological
simplification and cleaning, surface segmentation and parametrization, and efficient computation of level sets. In a special case of a function on a flat space, the Reeb graph forms a polytree and
is also called a contour tree.[4]
Given a topological
space X and a continuous function f: X → R,
define an equivalence relation ∼ on X where p∼q whenever p and q belong
to the same connected component of a single level
set f−1(c) for some real c. The Reeb graph is the quotient
space X /∼ endowed with the quotient topology.
If f is a Morse
function with distinct critical values, the Reeb graph can be described more explicitly.
Its nodes, or vertices, correspond to the critical level sets f−1(c). The pattern in which the arcs, or edges, meet at the nodes/vertices reflects the change in topology of the level set f−1(t)
as t passes through the critical value c. For example, if c is a minimum or a maximum of f, a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree 1.
If c is a saddle point of index 1 and two components of f−1(t)
merge at t = c as t increases, the corresponding vertex of the Reeb graph has degree 3 and looks like the letter "Y"; the same reasoning applies if the index of c is dim X−1 and a component of f−1(c)
splits into two.
Jump
up^ Harish Doraiswamy, Vijay Natarajan, Efficient
algorithms for computing Reeb graphs, Computational Geometry 42 (2009) 606–616
Jump
up^ G. Reeb, Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique, C. R. Acad. Sci. Paris 222 (1946) 847–849
Jump
up^ Tung, Tony; Schmitt, Francis (2005). "The
Augmented Multiresolution Reeb Graph Approach for Content-Based Retrieval of 3D Shapes". International Journal of Shape Modeling (IJSM) 11 (1): 91–120.
Jump
up^ Carr, Hamish; Snoeyink, Jack; Axen, Ulrike (2000), "Computing contour trees in all dimensions", Proc.
11th ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 918–926.
From Wikipedia, the free encyclopedia
Reeb graph of the height function on the torus.
A Reeb graph (named after Georges
Reeb) is a mathematical object reflecting the evolution of the level
sets of a real-valued function on a manifold.[1] Originally
introduced as a tool inMorse theory,[2] Reeb
graphs found a wide variety of applications in computational geometry and computer
graphics, including computer aided geometric
design, topology-based shape
matching,[3]topological
simplification and cleaning, surface segmentation and parametrization, and efficient computation of level sets. In a special case of a function on a flat space, the Reeb graph forms a polytree and
is also called a contour tree.[4]
Given a topological
space X and a continuous function f: X → R,
define an equivalence relation ∼ on X where p∼q whenever p and q belong
to the same connected component of a single level
set f−1(c) for some real c. The Reeb graph is the quotient
space X /∼ endowed with the quotient topology.
If f is a Morse
function with distinct critical values, the Reeb graph can be described more explicitly.
Its nodes, or vertices, correspond to the critical level sets f−1(c). The pattern in which the arcs, or edges, meet at the nodes/vertices reflects the change in topology of the level set f−1(t)
as t passes through the critical value c. For example, if c is a minimum or a maximum of f, a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree 1.
If c is a saddle point of index 1 and two components of f−1(t)
merge at t = c as t increases, the corresponding vertex of the Reeb graph has degree 3 and looks like the letter "Y"; the same reasoning applies if the index of c is dim X−1 and a component of f−1(c)
splits into two.
Jump
up^ Harish Doraiswamy, Vijay Natarajan, Efficient
algorithms for computing Reeb graphs, Computational Geometry 42 (2009) 606–616
Jump
up^ G. Reeb, Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique, C. R. Acad. Sci. Paris 222 (1946) 847–849
Jump
up^ Tung, Tony; Schmitt, Francis (2005). "The
Augmented Multiresolution Reeb Graph Approach for Content-Based Retrieval of 3D Shapes". International Journal of Shape Modeling (IJSM) 11 (1): 91–120.
Jump
up^ Carr, Hamish; Snoeyink, Jack; Axen, Ulrike (2000), "Computing contour trees in all dimensions", Proc.
11th ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 918–926.
Reeb graph of the height function on the torus.
A Reeb graph (named after Georges
Reeb) is a mathematical object reflecting the evolution of the level
sets of a real-valued function on a manifold.[1] Originally
introduced as a tool inMorse theory,[2] Reeb
graphs found a wide variety of applications in computational geometry and computer
graphics, including computer aided geometric
design, topology-based shape
matching,[3]topological
simplification and cleaning, surface segmentation and parametrization, and efficient computation of level sets. In a special case of a function on a flat space, the Reeb graph forms a polytree and
is also called a contour tree.[4]
Formal definition[edit]
Given a topologicalspace X and a continuous function f: X → R,
define an equivalence relation ∼ on X where p∼q whenever p and q belong
to the same connected component of a single level
set f−1(c) for some real c. The Reeb graph is the quotient
space X /∼ endowed with the quotient topology.
Description for Morse functions[edit]
If f is a Morsefunction with distinct critical values, the Reeb graph can be described more explicitly.
Its nodes, or vertices, correspond to the critical level sets f−1(c). The pattern in which the arcs, or edges, meet at the nodes/vertices reflects the change in topology of the level set f−1(t)
as t passes through the critical value c. For example, if c is a minimum or a maximum of f, a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree 1.
If c is a saddle point of index 1 and two components of f−1(t)
merge at t = c as t increases, the corresponding vertex of the Reeb graph has degree 3 and looks like the letter "Y"; the same reasoning applies if the index of c is dim X−1 and a component of f−1(c)
splits into two.
References[edit]
Jumpup^ Harish Doraiswamy, Vijay Natarajan, Efficient
algorithms for computing Reeb graphs, Computational Geometry 42 (2009) 606–616
Jump
up^ G. Reeb, Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique, C. R. Acad. Sci. Paris 222 (1946) 847–849
Jump
up^ Tung, Tony; Schmitt, Francis (2005). "The
Augmented Multiresolution Reeb Graph Approach for Content-Based Retrieval of 3D Shapes". International Journal of Shape Modeling (IJSM) 11 (1): 91–120.
Jump
up^ Carr, Hamish; Snoeyink, Jack; Axen, Ulrike (2000), "Computing contour trees in all dimensions", Proc.
11th ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 918–926.
From Wikipedia, the free encyclopedia
Reeb graph of the height function on the torus.
A Reeb graph (named after Georges
Reeb) is a mathematical object reflecting the evolution of the level
sets of a real-valued function on a manifold.[1] Originally
introduced as a tool inMorse theory,[2] Reeb
graphs found a wide variety of applications in computational geometry and computer
graphics, including computer aided geometric
design, topology-based shape
matching,[3]topological
simplification and cleaning, surface segmentation and parametrization, and efficient computation of level sets. In a special case of a function on a flat space, the Reeb graph forms a polytree and
is also called a contour tree.[4]
Formal definition[edit]
Given a topologicalspace X and a continuous function f: X → R,
define an equivalence relation ∼ on X where p∼q whenever p and q belong
to the same connected component of a single level
set f−1(c) for some real c. The Reeb graph is the quotient
space X /∼ endowed with the quotient topology.
Description for Morse functions[edit]
If f is a Morsefunction with distinct critical values, the Reeb graph can be described more explicitly.
Its nodes, or vertices, correspond to the critical level sets f−1(c). The pattern in which the arcs, or edges, meet at the nodes/vertices reflects the change in topology of the level set f−1(t)
as t passes through the critical value c. For example, if c is a minimum or a maximum of f, a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree 1.
If c is a saddle point of index 1 and two components of f−1(t)
merge at t = c as t increases, the corresponding vertex of the Reeb graph has degree 3 and looks like the letter "Y"; the same reasoning applies if the index of c is dim X−1 and a component of f−1(c)
splits into two.
References[edit]
Jumpup^ Harish Doraiswamy, Vijay Natarajan, Efficient
algorithms for computing Reeb graphs, Computational Geometry 42 (2009) 606–616
Jump
up^ G. Reeb, Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique, C. R. Acad. Sci. Paris 222 (1946) 847–849
Jump
up^ Tung, Tony; Schmitt, Francis (2005). "The
Augmented Multiresolution Reeb Graph Approach for Content-Based Retrieval of 3D Shapes". International Journal of Shape Modeling (IJSM) 11 (1): 91–120.
Jump
up^ Carr, Hamish; Snoeyink, Jack; Axen, Ulrike (2000), "Computing contour trees in all dimensions", Proc.
11th ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 918–926.
相关文章推荐
- 使用winpcap编写sniffer,遇到的乖离
- eltiTnmuloCteehSlecxE.168
- Android开发之蓝牙详解(二)
- NanoPi 2 Fire 连接使用USB WiFi
- FreeRTOS系列第18篇---FreeRTOS队列API函数
- 各种ESB产品比较(转)
- Raid 5 故障恢复
- 二分查找真的有你想象中那么简单吗?
- tableviewcell自适应cell高度
- windows下安装memcache
- servlet中常用方法
- 单独的一个模块如何安装
- Reeb 图
- Fragment(碎片)
- Android兼容包Support v4.v7.v13区别与应用场景
- mac 启动 进度条卡在一半处
- 深入浅出nodejs学习记录
- 关于Myeclipse修改默认编码,而content types中update不能更新问题
- spring filter的targetFilterLifecycle作用
- 英语学习