Edge subdivision schemes and the construction of smooth vector fields

Wang, Ke; Weiwei, Weiwei; Tong, Yiying; Desbrun, Mathieu; Schröder, Peter

doi:10.1145/1141911.1141991

Cited by 43 publications

(15 citation statements)

References 27 publications

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“…Extending this scheme to N-RoSy fields is straightforward, and it has been done when N = 2 ]. Recently, Wang et al [2006] have proposed another scheme based on edge subdivision and discrete differential forms. Adapting their scheme to N-RoSy fields is also promising.…”

Section: For Each Separatrix Direction W We Perform Streamline Tracingmentioning

confidence: 99%

Rotational symmetry field design on surfaces

Palacios

Zhang

2007

ACM SIGGRAPH 2007 Papers

113

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At the core of our analysis and design implementations is the observation that N-way rotational symmetries can be described by symmetric N-th order tensors, which allows an efficient vector-based representation that not only supports coherent definitions of arithmetic operations on rotational symmetries but also enables many analysis and design operations for vector fields to be adapted to rotational symmetry fields.To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing.

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Section: For Each Separatrix Direction W We Perform Streamline Tracingmentioning

confidence: 99%

Rotational symmetry field design on surfaces

Palacios

Zhang

2007

ACM SIGGRAPH 2007 Papers

113

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“…In [Tricoche et al 2003], a method is presented to simplify the topology of symmetric, second order 2D direction fields. Recent work on vector field design include [Wang et al 2006] where subdivision schemes are used to define bases for discrete differential 0-and 2-forms, and [Fisher et al 2007] that uses Whitney forms and the Hodge decomposition. Both approaches allow to design vector fields under user constraints, but do not provide any guarantee that new singularities will not appear, nor tackle the problem of symmetric fields.…”

mentioning

confidence: 99%

N-symmetry direction field design

Ray¹,

Vallet²,

Li³

et al. 2008

ACM Trans. Graph.

194

190

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Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction elds (unit tangent vector elds) dened over a surface. For instance, these direction elds are used in texture synthesis, in geometry processing or in non-photorealistic rendering to distribute and orient elements on the surface. Such direction elds can be designed in fundamentally dierent 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 unied formalism is still lacking for the mathematical object that characterizes these generalized direction elds. As a consequence, existing direction eld design algorithms are limited to use non-optimum local relaxation procedures.In this paper, we formalize N-symmetry direction elds, a generalization of classical direction elds. We give a new denition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the Poincaré-Hopf theorem in the case of N-symmetry direction elds on 2-manifolds. Based on this theorem, we explain how to control the topology of N-symmetry direction elds on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly ecient algorithm to design a smooth eld interpolating user dened singularities and directions.

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“…The edge-based approach to represent vector fields, introduced in graphics by [Wang et al 2006;Fisher et al 2007], has its mathematical foundations in Discrete Exterior Calculus (DEC) [Hirani 2003;Desbrun et al 2006;Crane et al 2013]. DEC defines discrete differential k-forms (here k = 0, 1, 2) on triangle meshes and expresses relevant operators such as divergence, curl, gradient, and Laplacian, as simple sparse matrices acting on intrinsic (coordinate-free) coefficients "living" on vertices, edges, and triangles.…”

Section: Edge-based Vector Field Representationmentioning

confidence: 99%

“…While these are usually minor in most graphics applications, having higher-order 1-form bases would definitely remove all discontinuities, and thus allow for accurate approximations of higher order derivatives. One simple construction of higher-order Whitney bases for k-forms was proposed through subdivision schemes [Wang et al 2006], where 0-form bases are found using the traditional Loop scheme, and 2-form bases use half-box splines. In particular, it was shown how to derive the corresponding 1-form bases such that the "formule de commutation" is exactly enforced-ensuring that the reconstructed k-form of the discrete exterior derivative d of a (k−1)-form matches the continuous exterior derivative d of a reconstructed (k −1)-form.…”

Section: Higher-order Whitney Basesmentioning

confidence: 99%

Vector field processing on triangle meshes

Goes¹,

Desbrun

Tong

2016

ACM SIGGRAPH 2016 Courses

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While scalar fields on surfaces have been staples of geometry processing, the use of tangent vector fields has steadily grown in geometry processing over the last two decades: they are crucial to encoding directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different tradeoffs among simplicity, efficiency, and accuracy depending on the targeted application.This course reviews the three main families of discretizations used to design computational tools for vector field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, we go over a large body of recent developments in vector field processing in the area of discrete differential geometry. We also discuss the theoretical and practical limitations of each type of discretization, and cover increasingly-common extensions such as n-direction and n-vector fields.While the course will focus on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finitedimensional vector fields, important differential geometric notions will also be covered: as often in Discrete Differential Geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians. PrefaceT hese course notes provide a review of the various representations of tangent vector fields on triangulated surfaces. Over the past decades, a number of approaches to express, design, and analyze vector fields on arbitrary meshes have been proposed in geometry processing, targeting a series of applications varying from rendering to animation. Some describe vector fields through values on vertices, some on edges, and some on faces; some treat vector fields as piecewise constant, some as piecewise linear, and some even involve non-linear finite elements. This lack of a unified approach to vector field representation and manipulation can be confusing for practitioners, and rare are the references that discuss the pros and cons of these various representations. The chapters in this document were written with this fact in mind, as a way to offer an introduction to the motivations, benefits, and shortcomings of the most common discretizations of vector fields in graphics-along with their associated discrete differential operators (curl, divergence, Laplacian, etc). Emphasis is placed on the tradeoffs between simplicity, efficiency, and accuracy of these geometry processing tools for applications ranging from texture synthesis to meshing.Course rationale. In many ways, this course is a continuation of ACM SIGGRAPH courses on Discrete Differential Geometry (SIGGRAPH '05 and '06, and SIGGRAPH Asia '09) and Discrete Exte...

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Edge subdivision schemes and the construction of smooth vector fields

Cited by 43 publications

References 27 publications

Rotational symmetry field design on surfaces

Rotational symmetry field design on surfaces

N-symmetry direction field design

Vector field processing on triangle meshes

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