## Vector field editing and periodic orbit extraction using morse decomposition (2007)

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Venue: | IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS |

Citations: | 25 - 12 self |

### BibTeX

@ARTICLE{Chen07vectorfield,

author = {Guoning Chen and Konstantin Mischaikow and Robert S. Laramee and Paweł Pilarczyk and Eugene Zhang},

title = {Vector field editing and periodic orbit extraction using morse decomposition},

journal = {IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS},

year = {2007},

volume = {13},

number = {4},

pages = {769--785}

}

### Years of Citing Articles

### OpenURL

### Abstract

Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper, we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory, which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithms to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating data sets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.

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Citation Context ... other features (separatrices and periodic orbits) are often not well-preserved. In Figure 18, a vector field with three periodic orbits is depicted using IBFV [18] (a), and evenly-placed streamlines =-=[45]-=- (b). Notice that it is difficult to see periodic orbits and separatrices using texturebased methods such as IBFV. Streamline-based methods can better illustrate trajectories. However, most existing m... |

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Citation Context ...ased (NTB) [6]. Existing NTB techniques are usually based on performing Laplacian smoothing on the potential of a vector field inside the specified region. One example of these work is by Tong et al. =-=[29]-=-, who decompose a vector field using Hodge-decomposition and then smooth each-component independently before summing them. TB techniques simplify the topology of a vector field explicitly. Tricoche et... |

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Citation Context ...so propose efficient algorithms to extract vector field topology. Following their footsteps, much research has been conducted in topological analysis of vector fields. For example, Scheuermann et al. =-=[24]-=- use clifford algebra to study the non-linear fixed points of a vector field and propose an efficient algorithm to merge nearby first-order fixed points. Tricoche et al. [1] and Polthier and Preuβ [26... |

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Citation Context ... velocity field extracted from experiments or numerical simulations of fluids. In both cases, there are occasions in which one wishes to simplify the dynamic structure in a coherent admissible manner =-=[1]-=-. This latter step requires the ability to edit the underlying vector field. Furthermore, there are problems where the construction and modification of a vector field represents a preliminary step tow... |

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Citation Context ...-based flow visualization technique (IBFV). The basic idea of this system is the use of basis vector fields that correspond to various types of fixed points. This system is later extended to surfaces =-=[19]-=-, [20]. None of these 2smethods provide explicit control over the number and location of fixed points since unspecified fixed points may appear. Theisel [21] proposes a planar vector field design syst... |

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Citation Context ... flow visualization technique (IBFV). The basic idea of this system is the use of basis vector fields that correspond to various types of fixed points. This system is later extended to surfaces [19], =-=[20]-=-. None of these 2smethods provide explicit control over the number and location of fixed points since unspecified fixed points may appear. Theisel [21] proposes a planar vector field design system in ... |

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Citation Context ...24] use clifford algebra to study the non-linear fixed points of a vector field and propose an efficient algorithm to merge nearby first-order fixed points. Tricoche et al. [1] and Polthier and Preuβ =-=[26]-=- give efficient methods to locate fixed points in a vector field. Wischgoll and Scheuermann [27] develop a method to extract closed streamlines in a 2D vector field defined on a triangle mesh. Note th... |

52 | A.: A flow-guided streamline seeding strategy
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Citation Context ...ts and separatrices using texturebased methods such as IBFV. Streamline-based methods can better illustrate trajectories. However, most existing methods such as Jobard and Lefer [45] and Verma et al. =-=[46]-=- do not take into account periodic orbits or separatrices in seed placement and streamline termination criteria. This causes visual discontinuity in periodic orbits and missing separatrices. Several r... |

51 | The parallel vectors operator - a vector field visualization primitive
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Citation Context ...nd to be close to periodic orbits, a rigorous mathematical study on the subject is needed. Furthermore, other methods for extracting separation and attachment points, such as that of Peikert and Roth =-=[49]-=-, may lead to more numerically stable results. Second, our current MCG and ECG construction methods assume closed surfaces. We are investigating means to extend them to handle surfaces with boundaries... |

48 |
The Topology of Symmetric, Second-Order Tensor Fields
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Citation Context ...at are connected by a separatrix. They refer to this operation as pair annihilation. A similar operation, named pair cancellation, has been used to remove a wedge and trisector pair in a tensor field =-=[30]-=-. We will follow this convention and refer to such an operation as fixed point pair cancellation. Zhang et al. [6] provide a fixed point pair cancellation method based on Conley theory. They also exte... |

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Citation Context ...effects [5]. There is substantial literature on the subject of vector field topology extraction and simplification, with considerable focus on the identification and manipulation of fixed points (see =-=[6]-=- and references therein). On the other hand, periodic orbits are essential structures of non-gradient vector fields, such as those in electromagnetism, chemical reactions, fluid dynamics, locomotion c... |

45 |
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Citation Context ...x, the vjs are the adjacent vertices that are either in the interior or on the boundary of N, and V represents the vector field. The weights !ijs are determined using Floater’s mean-value coordinates =-=[36]-=-. Equation (4) is a sparse linear system, which we solve by using a conjugate gradient method [37]. For convenience, we refer to a vertex v as being fixed if the vector value at v is part of the const... |

42 |
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Citation Context ... past twenty years. Much excellent work exists, and to review it all is beyond the scope of this paper. Here, we only refer to the most relevant work. Interested readers can find a complete survey in =-=[14]-=-, [15], [16]. A. Vector Field Design There has been some work in creating vector fields on the plane and surfaces, most of which is for graphics applications such as texture synthesis [2], [3], [4] an... |

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Citation Context ...ndamental need to be able to incorporate them into the subject of vector field visualization and design. For example, Figure 1 shows the swirl motion of fluid in a combustion chamber using simulation =-=[7]-=-. Periodic orbits appear in some planar slices along the main axis of the chamber (middle) as well as the boundary geometry (Figure 13). The existence and locations of the periodic orbits provide clue... |

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Citation Context ...n of two new types of design elements: attachment elements and separation elements. Before describing these elements, we briefly review the concepts of attachment and separation points from Kenwright =-=[25]-=-. Given a vector field V and a point p0 in the plane, we consider the following two values: e1 ×u and e2 ×u, where u is the vector value at p0 and e1 and e2 are the major and minor eigenvectors of the... |

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Citation Context ... This system is later extended to surfaces [19], [20]. None of these 2smethods provide explicit control over the number and location of fixed points since unspecified fixed points may appear. Theisel =-=[21]-=- proposes a planar vector field design system in which the user has complete control over fixed points and separatrices. However, this requires the user to provide the complete topological skeleton of... |

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Citation Context ...e and use linear interpolation to locate attachment and separation points. This formulation assumes that an eigenvector field can be treated as a vector field. However, as pointed out by Zhang et al. =-=[42]-=-, treating an eigenvector field as a vector field will lead to discontinuities in the vector field and cause visual artifacts in tensor field visualization and non-photorealistic rendering. We have ob... |

29 |
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Citation Context ...Jacobians at the vertices. This efficiently removes the need to carefully assign directions to eigenvectors at the two vertices of an edge. To perform tracing on surfaces, we use a Runge-Kutta scheme =-=[43]-=- that has been adapted to surfaces with a piecewise interpolation scheme that guarantees vector field continuity across vertices and edges [6]. B. ECG Construction and Display Since the construction o... |

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Citation Context ...at the topological methods of Conley theory have been used to obtain computer assisted, but mathematically rigorous proofs concerning the structure of a wide variety nonlinear dynamical systems [34], =-=[35]-=-. Thus, our confidence level in the validity of the visualized structures and modifications is higher for those objects identified with the MCG than the ECG. B. Vector Field Simplification on Surfaces... |

27 |
Conley index
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Citation Context ...ts. Isolated invariant sets posses two essential properties. First, there are efficient algorithms for identifying isolating neighborhoods [11]. Second, there exists an index, called the Conley index =-=[31]-=-, that identifies the types of modifications to the structure of the invariant set that are topologically permissible. For example, the Conley index of the vector field shown in Fig. 3a inside the whi... |

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Citation Context ...out that the topological methods of Conley theory have been used to obtain computer assisted, but mathematically rigorous proofs concerning the structure of a wide variety nonlinear dynamical systems =-=[34]-=-, [35]. Thus, our confidence level in the validity of the visualized structures and modifications is higher for those objects identified with the MCG than the ECG. B. Vector Field Simplification on Su... |

22 |
Exploring Global Dynamics: A Numerical Algorithm Based on the Conley Index Theory
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Citation Context ...7. To be able to detect periodic orbits even when they are not approached by any fixed point, we have developed a new periodic orbit detection method that has drawn ideas from the Morse decomposition =-=[41]-=- and separation and attachment lines [25]. 7sFig. 7. An example of our periodic orbit detection algorithm. First, we compute strongly connected components and only consider components where periodic o... |

20 | T.: Grid-independent detection of closed stream lines in 2D vector fields
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Citation Context ... [27] develop a method to extract closed streamlines in a 2D vector field defined on a triangle mesh. Note that closed streamlines are in fact attracting and repelling periodic orbits. Theisel et al. =-=[28]-=- propose a mesh-independent periodic orbit detection method for planar domains. In contrast to these approaches, our automatic detection algorithm is extended to surfaces. Furthermore, this is the fir... |

17 | Extracting higher order critical points and topological simplification of 3d vector fields
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Citation Context ...d streamline termination criteria. This causes visual discontinuity in periodic orbits and missing separatrices. Several researchers have incorporated vector field topology into texture-based methods =-=[47]-=-. Most of the figures in this paper are created in that fashion. On the other hand, streamlinebased methods can better illustrate individual streamlines, which makes it an attractive approach when int... |

13 |
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Citation Context ...n arbitrary system is impossible. For example, Hilbert’s 16th problem, that of bounding the number of isolated periodic orbits for a polynomial vector field on the plane, remains essentially unsolved =-=[10]-=-. Second, the existence of noise reduces the importance of objects such as fixed points and periodic orbits. In this paper, we develop a vector field visualization and design system that extracts and ... |

13 | Topological construction and visualization of higher order 3D vector fields. Computer Graphics Forum (Eurographics 2004
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Citation Context ...ble automatic extraction and visualization of periodic orbits on surfaces. We also introduce topology simplification operations for periodic orbits. There has also been recent work by Weinkauf et al. =-=[22]-=- on the design of 3D vector fields. B. Vector Field Topology and Analysis Helman and Hesselink [23] introduce vector field topology for the visualization of vector fields. They also propose efficient ... |

12 | An Algorithmic Approach to Chain Recurrence
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Citation Context ... and/or gradient-like behavior, i.e. separatrices. This involves the construction of Morse decompositions. A theoretical computational foundation for the types of algorithms we employ can be found in =-=[12]-=-, [13]. The second is to identify the type of dynamics occuring in these regions, i.e. the existence of fixed points, periodic orbits, and separatrices. This is done using numerical methods and the Co... |

12 |
DOLEISCH H.: The State of the Art
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Citation Context ...twenty years. Much excellent work exists, and to review it all is beyond the scope of this paper. Here, we only refer to the most relevant work. Interested readers can find a complete survey in [14], =-=[15]-=-, [16]. A. Vector Field Design There has been some work in creating vector fields on the plane and surfaces, most of which is for graphics applications such as texture synthesis [2], [3], [4] and flui... |

11 |
Graduate Texts in Mathematics
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Citation Context ..., and T (X) denote the collections of vertices, edges, and triangles which are contained in X. Let C0(X), C1(X), and C2(X) be the free abelian groups on the sets V (X), E (X), and T (X), respectively =-=[48]-=- and set C−1(X) = 0. Recall, that if (N,L) is an index pair then L ⊂ ∂N. Since, N is given as the union of a set of triangles, V (L) ⊂ V (N), E (L) ⊂ E (N), and T (L) = /0. Thus, the quotient groups C... |

8 | A computational approach to conley’s decomposition theorem
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Citation Context ...st stage is conducted purely on the graph level. Let R = {R} � {Si} and A = {A} � {Si}. Note when R and A are directly connected, the set of {Si} is empty. It should also be noted that Kalies and Ban =-=[13]-=- provide a dimension independent algorithm for determining intervals in a Morse decomposition. In the second stage, we consider the minimal set of triangles in the domain that contain R. We then grow ... |

7 |
Detection and visualization of planar closed streamline
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(Show Context)
Citation Context ...fficient algorithm to merge nearby first-order fixed points. Tricoche et al. [1] and Polthier and Preuβ [26] give efficient methods to locate fixed points in a vector field. Wischgoll and Scheuermann =-=[27]-=- develop a method to extract closed streamlines in a 2D vector field defined on a triangle mesh. Note that closed streamlines are in fact attracting and repelling periodic orbits. Theisel et al. [28] ... |

7 |
Vizualization of Intricate Flow Structures for Vortex Breakdown Analysis
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(Show Context)
Citation Context ... swirl motion. However, other less ideal singularities are found near the perimeter of the geometry. The method we use here is similar to the moving cutting plane topology approach of Tricoche et al. =-=[44]-=-. We note that caution must be used when interpreting these results since the vector field has been projected onto 2D slices. On the other hand, the engineers involved are very familiar with the simul... |

6 |
Conley index. In: Handbook of dynamical systems 2
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(Show Context)
Citation Context ...ts. Isolated invariant sets posses two essential properties. First, there are efficient algorithms for identifying isolating neighborhoods [12]. Second, there exists an index, called the Conley index =-=[32]-=-, that identifies the types of modifications to the structure of the invariant set that are topologically permissible. For example, the Conley index of the vector field shown in Figure 2 (upper-left) ... |