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39
Mixedinteger quadrangulation
 ACM Trans. Graph
, 2009
"... the input mesh by some simple heuristic or by the user. (b) In a global optimization procedure a cross field is generated on the mesh which interpolates the given constraints and is as smooth as possible elsewhere. The optimization includes the automatic generation and placement of singularities. (c ..."
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Cited by 50 (9 self)
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the input mesh by some simple heuristic or by the user. (b) In a global optimization procedure a cross field is generated on the mesh which interpolates the given constraints and is as smooth as possible elsewhere. The optimization includes the automatic generation and placement of singularities. (c) A globally smooth parametrization is computed on the surface whose isoparameter lines follow the cross field directions and singularities lie at integer locations. (d) Finally, a consistent, feature aligned quadmesh can be extracted. We present a novel method for quadrangulating a given triangle mesh. After constructing an as smooth as possible symmetric cross field satisfying a sparse set of directional constraints (to capture the geometric structure of the surface), the mesh is cut open in order to enable a low distortion unfolding. Then a seamless globally smooth parametrization is computed whose isoparameter lines follow the cross field directions. In contrast to previous methods, sparsely distributed directional constraints are sufficient to automatically determine the appropriate number, type and position of singularities in the quadrangulation. Both steps of the algorithm (cross field and parametrization) can be formulated as a mixedinteger problem which we solve very efficiently by an adaptive greedy solver. We show several complex examples where high quality quad meshes are generated in a fully automatic manner.
Interactive tensor field design and visualization on surfaces
 Online]. Available: http://eecs.oregonstate.edu/library/files/2005106/tenflddesn.pdf
, 2005
"... Abstract—Designing tensor fields in the plane and on surfaces is a necessary task in many graphics applications, such as painterly rendering, penandink sketching of smooth surfaces, and anisotropic remeshing. In this article, we present an interactive design system that allows a user to create a w ..."
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Cited by 30 (10 self)
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Abstract—Designing tensor fields in the plane and on surfaces is a necessary task in many graphics applications, such as painterly rendering, penandink sketching of smooth surfaces, and anisotropic remeshing. In this article, we present an interactive design system that allows a user to create a wide variety of symmetric tensor fields over 3D surfaces either from scratch or by modifying a meaningful input tensor field such as the curvature tensor. Our system converts each user specification into a basis tensor field and combines them with the input field to make an initial tensor field. However, such a field often contains unwanted degenerate points which cannot always be eliminated due to topological constraints of the underlying surface. To reduce the artifacts caused by these degenerate points, our system allows the user to move a degenerate point or to cancel a pair of degenerate points that have opposite tensor indices. These operations provide control over the number and location of the degenerate points in the field. We observe that a tensor field can be locally converted into a vector field so that there is a onetoone correspondence between the set of degenerate points in the tensor field and the set of singularities in the vector field. This conversion allows us to effectively perform degenerate point pair cancellation and movement by using similar operations for vector fields. In addition, we adapt the imagebased flow visualization technique to tensor fields, therefore allowing interactive display of tensor fields on surfaces. We demonstrate the capabilities of our tensor field design system with painterly rendering, penandink sketching of surfaces, and anisotropic remeshing. Index Terms—Tensor field design, tensor field visualization, nonphotorealistic rendering, surfaces, remeshing, tensor field topology. 1
Rotational Symmetry Field Design on Surfaces
"... tensor smoothed as a 4RoSy field, (c) topological editing operations applied to (b), and (d) more global smoothing performed on (b). Notice that treating the curvature tensor as a 4RoSy field (b) leads to fewer unnatural singularities and therefore less visual artifacts than as a 2RoSy field (a). ..."
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Cited by 30 (5 self)
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tensor smoothed as a 4RoSy field, (c) topological editing operations applied to (b), and (d) more global smoothing performed on (b). Notice that treating the curvature tensor as a 4RoSy field (b) leads to fewer unnatural singularities and therefore less visual artifacts than as a 2RoSy field (a). In addition, both topological editing (c) and global smoothing (d) can be used to remove more singularities from (b). However, topological editing (c) provides local control while excessive global smoothing (d) can cause hatch directions to deviate from their natural orientations (neck and chest). Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and penandink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces. As the foundation of our system, we provide comprehensive analysis for rotational symmetry fields on surfaces and present efficient algorithms to identify singularities and separatrices. We also describe design operations that allow a rotational symmetry field to be created and modified in an intuitive fashion by using the idea of basis fields and relaxation. In particular, we provide control over the topology of a rotational symmetry field by allowing the user to remove singularities from the field or to move them to more desirable locations.
Vector field editing and periodic orbit extraction using morse decomposition
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 2007
"... 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 pa ..."
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Cited by 27 (13 self)
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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 streamlinebased technique that allows vector field topology to be easily identified.
Efficient Morse Decompositions of Vector Fields
"... Abstract — Existing topologybased vector field analysis techniques rely on the ability to extract the individual trajectories such as fixed points, periodic orbits and separatrices which are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field an ..."
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Cited by 20 (8 self)
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Abstract — Existing topologybased vector field analysis techniques rely on the ability to extract the individual trajectories such as fixed points, periodic orbits and separatrices which are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field analysis unsuitable for rigorous interpretations. We advocate the use of Morse decompositions, which are robust with respect to perturbations, to encode the topological structures of a vector field in the form of a directed graph, called a Morse connection graph (MCG). While an MCG exists for every vector field, it need not be unique. Previous techniques for computing MCG’s, while fast, are overly conservative and usually results in MCG’s that are too coarse to be useful for the applications. To address this issue, we present a new technique for performing Morse decomposition based on the concept of τmaps, which typically provides finer MCG’s than existing techniques. Furthermore, the choice of τ provides a natural tradeoff between the fineness of the MCG’s and the computational costs. We provide efficient implementations of Morse decomposition based on τmaps, which include the use of forward and backward mapping techniques and an adaptive approach in constructing better approximations of the images of the triangles in the meshes used for simulation.Furthermore, we propose the use of spatial τmaps in addition to the original temporal τmaps. These techniques provide additional tradeoffs between the quality of the MCG’s and the speed of computation. We demonstrate the utility of our technique with various examples in plane and on surfaces including engine simulation datasets. Index Terms — Vector field topology, Morse decomposition, τmaps, Morse connection graph, flow combinatorialization.
Asymmetric Tensor Analysis for Flow Visualization
 IEEE Trans. on Visualization and Computer Graphics
"... Abstract—The gradient of a velocity vector field is an asymmetric tensor field which can provide critical insight that is difficult to infer from traditional trajectorybased vector field visualization techniques. We describe the structures in the eigenvalue and eigenvector fields of the gradient te ..."
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Cited by 12 (7 self)
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Abstract—The gradient of a velocity vector field is an asymmetric tensor field which can provide critical insight that is difficult to infer from traditional trajectorybased vector field visualization techniques. We describe the structures in the eigenvalue and eigenvector fields of the gradient tensor and how these structures can be used to infer the behaviors of the velocity field that can represent either a 2D compressible flow or the projection of a 3D compressible or incompressible flow onto a 2D manifold. To illustrate the structures in asymmetric tensor fields, we introduce the notions of eigenvalue manifold and eigenvector manifold. These concepts afford a number of theoretical results that clarify the connections between symmetric and antisymmetric components in tensor fields. In addition, these manifolds naturally lead to partitions of tensor fields, which we use to design effective visualization strategies. Moreover, we extend eigenvectors continuously into the complex domains which we refer to as pseudoeigenvectors. We make use of evenly spaced tensor lines following pseudoeigenvectors to illustrate the local linearization of tensors everywhere inside complex domains simultaneously. Both eigenvalue manifold and eigenvector manifold are supported by a tensor reparameterization with physical meaning. This allows us to relate our tensor analysis to physical quantities such as rotation, angular deformation, and dilation, which provide a physical interpretation of our tensordriven vector field analysis in the context of fluid mechanics. To demonstrate the utility of our approach, we have applied our visualization techniques and interpretation to the study of the Sullivan Vortex as well as computational fluid dynamics simulation data. Index Terms—Tensor field visualization, flow analysis, asymmetric tensors, flow segmentation, tensor field topology, surfaces. Ç 1
Stable Morse Decompositions for Piecewise Constant Vector Fields on Surfaces,” Computer Graphics Forum, to appear
, 2011
"... Abstract—In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. ..."
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Cited by 10 (6 self)
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Abstract—In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. To overcome the intrinsic difficulty in PC vector fields (in particular, discontinuity along mesh edges), we borrow results from the theory of differential inclusions. The input vector field and its PC variant have similar Morse decompositions. We introduce a robust and efficient algorithm to compute Morse decompositions of a PC vector field. Our approach provides subtriangle precision for Morse sets. In addition, we describe a Morse set classification framework which we use to color code the Morse sets in order to enhance the visualization. We demonstrate the benefits of our approach with three wellknown simulation data sets, for which our method has produced Morse decompositions that are similar to or finer than those obtained using existing techniques, and is over an order of magnitude faster. Index Terms—Morse decomposition, vector field topology. Ç 1
Representing higherorder singularities in vector fields on piecewise linear surfaces
 IEEE Transactions on Visualization and Computer Graphics
, 2006
"... Abstract—Accurately representing higherorder singularities of vector fields defined on piecewise linear surfaces is a nontrivial problem. In this work, we introduce a concise yet complete interpolation scheme of vector fields on arbitrary triangulated surfaces. The scheme enables arbitrary singula ..."
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Cited by 9 (4 self)
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Abstract—Accurately representing higherorder singularities of vector fields defined on piecewise linear surfaces is a nontrivial problem. In this work, we introduce a concise yet complete interpolation scheme of vector fields on arbitrary triangulated surfaces. The scheme enables arbitrary singularities to be represented at vertices. The representation can be considered as a facetbased “encoding ” of vector fields on piecewise linear surfaces. The vector field is described in polar coordinates over each facet, with a facet edge being chosen as the reference to define the angle. An integer called the period jump is associated to each edge of the triangulation to remove the ambiguity when interpolating the direction of the vector field between two facets that share an edge. To interpolate the vector field, we first linearly interpolate the angle of rotation of the vectors along the edges of the facet graph. Then, we use a variant of Nielson’s sidevertex scheme to interpolate the vector field over the entire surface. With our representation, we remove the bound imposed on the complexity of singularities that a vertex can represent by its connectivity. This bound is a limitation generally exists in vertexbased linear schemes. Furthermore, using our data structure, the index of a vertex of a vector field can be combinatorily determined. We show the simplicity of the interpolation scheme with a GPUaccelerated algorithm for a LICbased visualization of the sodefined vector fields, operating in image space. We demonstrate the algorithm applied to various vector fields on curved surfaces. Index Terms—vector field visualization, higherorder singularities, line integral convolution, GPU. 1
Video Painting with SpaceTimeVarying Style Parameters
"... Abstract—Artists use different means of stylization to control the focus on different objects in the scene. This allows them to portray complex meaning and achieve certain artistic effects. Most prior work on painterly rendering of videos, however, uses only a single painting style, with fixed globa ..."
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Cited by 9 (0 self)
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Abstract—Artists use different means of stylization to control the focus on different objects in the scene. This allows them to portray complex meaning and achieve certain artistic effects. Most prior work on painterly rendering of videos, however, uses only a single painting style, with fixed global parameters, irrespective of objects and their layout in the images. This often leads to inadequate artistic control. Moreover, brush stroke orientation is typically assumed to follow an everywhere continuous directional field. In this article, we propose a video painting system that accounts for the spatial support of objects in the images or video, and uses this information to specify style parameters and stroke orientation for painterly rendering. Since objects occupy distinct image locations and move relatively smoothly from one video frame to another, our objectbased painterly rendering approach is characterized by style parameters that coherently vary in space and time. Spacetimevarying style parameters enable more artistic freedom, such as emphasis/deemphasis, increase or decrease of contrast, exaggeration or abstraction of different objects in the scene in a temporally coherent fashion. Index Terms—Nonphotorealistic rendering, video painting, multistyle painting, tensor field design 1
Trivial Connections on Discrete Surfaces
 SGP 2010 / COMPUTER GRAPHICS FORUM
, 2010
"... This paper presents a straightforward algorithm for constructing connections on discrete surfaces that are as smooth as possible everywhere but on a set of isolated singularities with given index. We compute these connections by solving a single linear system built from standard operators. The solut ..."
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Cited by 7 (1 self)
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This paper presents a straightforward algorithm for constructing connections on discrete surfaces that are as smooth as possible everywhere but on a set of isolated singularities with given index. We compute these connections by solving a single linear system built from standard operators. The solution can be used to design rotationally symmetric direction fields with userspecified singularities and directional constraints.