## Vector Field Design on Surfaces (2006)

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### Other Repositories/Bibliography

Venue: | ACM Transactions on Graphics |

Citations: | 51 - 15 self |

### BibTeX

@ARTICLE{Zhang06vectorfield,

author = {Eugene Zhang and Konstantin Mischaikow and Greg Turk},

title = {Vector Field Design on Surfaces},

journal = {ACM Transactions on Graphics},

year = {2006},

volume = {25},

pages = {2006}

}

### OpenURL

### Abstract

Figure 1: This figure shows various vector fields created on surfaces using our vector field design system. The vector field shown at the right was used to guide texture synthesis shown in Figure 12 (right). Vector field design on surfaces is necessary for many graphics applications: example-based texture synthesis, non-photorealistic rendering, and fluid simulation. A vector field design system should allow a user to create a large variety of complex vector fields with relatively little effort. In this paper, we present a vector field design system for surfaces that allows the user to control the number of singularities in the vector field and their placement. Our system combines basis vector fields to make an initial vector field that meets the user’s specifications. The initial vector field often contains unwanted singularities. Such singularities cannot always be eliminated, due to the Poincaré-Hopf index theorem. To reduce the effect caused by these singularities, our system allows a user to move a singularity to a more favorable location or to cancel a pair of singularities. These operations provide topological guarantees for the vector field in that they only affect the user-specified singularities. Other editing operations are also provided so that the user may change the topological and geometric characteristics of the vector field. We demonstrate our vector field design system for several applications: example-based texture synthesis, painterly rendering of images, and pencil sketch illustrations of smooth surfaces.

### Citations

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(Show Context)
Citation Context ...ent-like vector fields, which limit their potential applications. Second, the user should be able to control the the number of singularities in the vector field and their placement. As pointed out in =-=[15, 8]-=-, this is necessary for applications such as example-based texture synthesis and non-photorealistic rendering, in which unwanted singularities cause artifacts in the visual appearance. Third, the syst... |

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(Show Context)
Citation Context ...ing the piecewise linear vector field representation can be found in [20]. We also compute the topological skeleton of the vector field. This is done by following the approach of Helman and Hesselink =-=[6]-=-. Starting from every saddle point, we follow the flow forward in its outgoing directions until the flow is stopped at a singularity or hits the boundary. To trace the trajectories away from a saddle ... |

193 | Painterly rendering with curved brush strokes of multiple sizes
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(Show Context)
Citation Context ...aces, and example-based texture synthesis. There have been numerous published approaches to painterly rendering, and to review them is beyond our scope. In this work, we use the approach by Hertzmann =-=[7]-=- with the following modification: instead of using the image gradient field to guide the brush stroke orientations, the user creates a synthetic vector field with our vector field design system. To ma... |

178 | Mean value coordinates - FLOATER |

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162 | Texture synthesis on surfaces
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(Show Context)
Citation Context ...ting a vector field on a surface. In all the instances that we know, such systems have been created in a quick manner to generate vector fields for a particular application, such as texture synthesis =-=[15, 21, 24]-=-, fluid simulaFigure 2: This figure shows three vector fields and their topological skeletons. The visualization is based on the image-based flow visualization technique by van Wijk [22]. The vector ... |

159 | Lapped textures
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(Show Context)
Citation Context ...ent-like vector fields, which limit their potential applications. Second, the user should be able to control the the number of singularities in the vector field and their placement. As pointed out in =-=[15, 8]-=-, this is necessary for applications such as example-based texture synthesis and non-photorealistic rendering, in which unwanted singularities cause artifacts in the visual appearance. Third, the syst... |

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(Show Context)
Citation Context ...)ρ(Qi+1). Then θi = rφi. For a ray −→ PM inside triangle △QiPQi+1, we have ∠QiPM = r∠ρ(Qi)ρ(P)ρ(M). Note that this construction is similar to the “geodesic polar map” that is used by Welch and Witkin =-=[25]-=- for free-form shape design, with a minor difference: in their setting the parameterization domain is a polygon, not the unit disc as in our case. To transfer Wi to a point K inside triangle QiPQi+1, ... |

146 | Texture synthesis over arbitrary manifold surfaces
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(Show Context)
Citation Context ...ting a vector field on a surface. In all the instances that we know, such systems have been created in a quick manner to generate vector fields for a particular application, such as texture synthesis =-=[15, 21, 24]-=-, fluid simulaFigure 2: This figure shows three vector fields and their topological skeletons. The visualization is based on the image-based flow visualization technique by van Wijk [22]. The vector ... |

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118 | Image based flow visualization
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(Show Context)
Citation Context ...pplications to create many different visual effects by merely using different input vector fields. A vector field design system can also be used to test existing vector field visualization techniques =-=[22, 23]-=-. Vector field design refers to creating a continuous vector field on an input surface based on the user’s specifications or applicationdependent requirements. There are several challenges to the prob... |

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77 | Flows on surfaces of arbitrary topology
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(Show Context)
Citation Context ...es, sinks and saddles are colored in green, red and yellow. Repelling and attracting foci are colored in cyan and magenta. For saddles, their two principle directions are depicted using a cross. tion =-=[17]-=-, or for testing a vector field visualization technique [23]. There are three basic approaches for creating a surface vector field using these systems. In the first approach, a 3D vector field is crea... |

76 | Straightest geodesics on polyhedral surfaces - Polthier, Schmies - 1998 |

73 | Discrete multiscale vector field decomposition - TONG, LOMBEYDA, et al. - 2003 |

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61 | Image based flow visualization for curved surfaces
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(Show Context)
Citation Context ...aluating equations such as 2 and 4 directly (a source and a regular element, respectively) . Let us stress that this is not the only way to create basis vector field. In van Wijk’s visualization tool =-=[23]-=-, an element is translated into a 3D vector field before being projected onto the surface. Constrained optimization is another way to produce an initial vector field with desired behaviors. 6.4 Editin... |

60 | Image and video based painterly animation
- Hays, Essa
- 2004
(Show Context)
Citation Context ...r makes changes to the vector field. Figure 14 shows some painterly rendering results. The final high-quality painterly images in this figure are created off-line using the algorithm of Hays and Essa =-=[5]-=-. We have modified van Wijk’s image based flow visualization technique [23] to create non-photorealistic illustration of surfaces. Figure 13 shows the results of applying this techniques on the feline... |

55 | Identifying vector fields singularities using a discrete hodge decomposition - Polthier, Preuss - 2002 |

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(Show Context)
Citation Context ...inside if any, and the incoming and outgoing directions if the triangle contains a saddle. Details of computing these quantities using the piecewise linear vector field representation can be found in =-=[20]-=-. We also compute the topological skeleton of the vector field. This is done by following the approach of Helman and Hesselink [6]. Starting from every saddle point, we follow the flow forward in its ... |

37 | A Level-Set Method for Flow Visualization - Westermann, Johnson, et al. - 2000 |

35 |
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(Show Context)
Citation Context ... in its outgoing directions until the flow is stopped at a singularity or hits the boundary. To trace the trajectories away from a saddle we use a Runga-Kutta algorithm with adaptive stepsize control =-=[1]-=-. This gives us the two outgoing separatrices. Similarly, we obtain the two incoming separatrices by following the flow backward along the incoming directions of the saddle. Figure 2 shows the topolog... |

30 |
Conley index
- Mischaikow, Mrozek
- 2002
(Show Context)
Citation Context ...larity of positive or negative Poincaré index then one must simultaneously remove a singularity of the opposite sign. To perform these removals we borrow basic ideas from the Conley index theory (see =-=[2, 13, 14]-=- for further details and references) which is a topological generalization of Morse theory. The following concept is the starting point for the Conley index theory. Given a region N ⊂ S, let ∂N denote... |

30 | Designing 2D vector fields of arbitrary topology
- Theisel
(Show Context)
Citation Context ...ting and moving the singularities. This system also lacks the control over vector field topology since the vector field created this way may have unspecified singularities. On the other hand, Theisel =-=[19]-=- proposes a 2D vector field design system in which the users has the complete control over vector field topology. To do so, the user specifies the topological skeleton of the desired vector field and ... |

27 | Topological techniques for efficient rigorous computation
- Mischaikow
(Show Context)
Citation Context ...larity of positive or negative Poincaré index then one must simultaneously remove a singularity of the opposite sign. To perform these removals we borrow basic ideas from the Conley index theory (see =-=[2, 13, 14]-=- for further details and references) which is a topological generalization of Morse theory. The following concept is the starting point for the Conley index theory. Given a region N ⊂ S, let ∂N denote... |

15 | State-of-theart report 2002 in flow visualization - Hauser, Laramee, et al. - 2002 |

14 | Computing singularities of 3d vector fields with geometric algebra
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- 2003
(Show Context)
Citation Context ...e α(C) covers the unit circle is always an integer and this number is the Poincaré index of the singularity p 0 . The index is 0 for any regular point, −1 for saddles, and 1 for sources or sinks (see =-=[12]-=- for further details). For a continuous vector field V defined on a closed orientable manifold S with only isolated singularities, the Poincaré-Hopf Index Theorem tells us that the total indices of al... |

14 | Vector field metrics based on distance measures of first order critical points - Theisel, Weinkauf - 2002 |

13 |
Isolated Invariant Sets and Morse
- Conley
- 1978
(Show Context)
Citation Context ...larity of positive or negative Poincaré index then one must simultaneously remove a singularity of the opposite sign. To perform these removals we borrow basic ideas from the Conley index theory (see =-=[2, 13, 14]-=- for further details and references) which is a topological generalization of Morse theory. The following concept is the starting point for the Conley index theory. Given a region N ⊂ S, let ∂N denote... |

10 |
Computing Geodesic Paths on
- Kimmel, Sethian
- 1998
(Show Context)
Citation Context ...aces, the user can easily regularize these regions by adding elements nearby. The distance component ρ of the polar map can be computed using the fast marching method proposed by Kimmel and Sethian ( =-=[11]-=-). For the points that are near the seed, we obtain the angular component θ by projecting them onto the seed’s tangent plane. We then propagate θ to points that are far away via minimal geodesics. For... |

5 |
Helmut Doleisch. State-of-the-Art Report 2002 in Flow Visualization
- Hauser, Laramee
- 2002
(Show Context)
Citation Context ...rative analysis and editing is repeated until the user is satisfied with the results. 2 PREVIOUS WORK Vector field analysis and visualization have been well studied, and a good survey is available in =-=[4]-=-. However, vector field design is far less explored. There has been some prior work in creating a vector field on a surface. In all the instances that we know, such systems have been created in a quic... |

5 |
A toy vector field based on geometric algebra,” Proceeding Application of Geometric Algebra
- Rockwood, Bunderwala
- 2001
(Show Context)
Citation Context ...gularities than those that the user has specified. Because this system does not provide a way of removing undesired singularities, it lacks control over vector field topology. Rockwood and Bunderwala =-=[16]-=- propose a technique in which the system uses geometric algebra to create a vector field based on user specified singularities locations and types (source, saddle, and etc). The user can interactively... |

3 | Flow-box theorem for lipschitz continuous vector fields - CALCATERRA, BOLDT - 2003 |

1 |
Vector Field Metrics Based on Distance
- Theisel, Weinkauf
- 2002
(Show Context)
Citation Context ... defined on a sphere. The concepts of flow rotation and flow reflection are not new, although we believe that our use of them for singularity movement and cancellation are novel. Theisel and Weinkauf =-=[18]-=- defined four types of vector field operations, including rotation and scaling (including reflection). 5.3 Flow Smoothing The flow smoothing operation is simple yet efficient. Given a vector field V a... |

1 | Illustrating smooth surfaces - Zhang, Mischaikow, et al. - 2000 |