## Design of tangent vector fields (2007)

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Venue: | ACM Trans. Graph |

Citations: | 44 - 4 self |

### BibTeX

@ARTICLE{Fisher07designof,

author = {Matthew Fisher and Peter Schröder},

title = {Design of tangent vector fields},

journal = {ACM Trans. Graph},

year = {2007},

volume = {26},

pages = {56}

}

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### Abstract

Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1-forms), we obtain an intrinsic, coordinatefree formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.

### Citations

423 |
Imaging vector fields using line integral convolution
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(Show Context)
Citation Context ...f surfaces. Examples include anisotropic shading [Schlick 1994], texture synthesis [Turk 2001; Wei and Levoy 2001], non-photorealistic rendering [Hertzmann and Zorin 2000], line integral convolution [=-=Cabral and Leedom 1993-=-], and spot noise [van Wijk 1991] among many others. In some settings one starts with a given vector field, while in others the user must first design a vector field. We present a method for designing... |

348 | Discrete Differential-Geometry Operators for Triangulated 2-Manifolds
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Citation Context ...may even “fold back” on themselves. Nonetheless, there is a simple approach to compute these areas correctly by iterating over all triangles and accumulating signed elementary areas into each vertex [=-=Meyer et al. 2002-=-, Section 3.3]. Consider pi jk together with its circumcentric dual vertex p∗ i jk . Decompose pi jk into 6 smaller triangles each formed by a corner (e.g., pi), the midpoint of an adjacent edge (i.e.... |

237 | Illustrating smooth surfaces
- Hertzmann, Zorin
- 2000
(Show Context)
Citation Context ...ear in many applications that control the appearance of surfaces. Examples include anisotropic shading [Schlick 1994], texture synthesis [Turk 2001; Wei and Levoy 2001], non-photorealistic rendering [=-=Hertzmann and Zorin 2000-=-], line integral convolution [Cabral and Leedom 1993], and spot noise [van Wijk 1991] among many others. In some settings one starts with a given vector field, while in others the user must first desi... |

159 |
Geometric integration theory
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- 1957
(Show Context)
Citation Context ...ve us access to the curl-free (dc v) and divergence-free (δc t) parts of a vector field (see Section 3.1). 2.4 Reconstruction Discrete k-forms can be PL interpolated with the aid of Whitney elements [=-=Whitney 1957-=-]. For 0-forms these are the standard PL hat functions φ v = {φ i|v i ∈ V }. Appropriate 1-form interpolators φ e = {φ i j|e i j ∈ E} follow as φ i j = φ idφ j − φ jdφ i. The φi j are supported on the... |

154 | Lapped textures - Praun, Finkelstein, et al. - 2000 |

153 | Texture synthesis on surfaces - Turk - 2001 |

144 | Spot noise: Texture synthesis for data visualization - Wijk - 1991 |

139 | Texture synthesis over arbitrary manifold surfaces
- Wei, Levoy
- 2001
(Show Context)
Citation Context ...oduction Smoothly varying tangent vector fields appear in many applications that control the appearance of surfaces. Examples include anisotropic shading [Schlick 1994], texture synthesis [Turk 2001; =-=Wei and Levoy 2001-=-], non-photorealistic rendering [Hertzmann and Zorin 2000], line integral convolution [Cabral and Leedom 1993], and spot noise [van Wijk 1991] among many others. In some settings one starts with a giv... |

114 | Global conformal surface parameterization
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(Show Context)
Citation Context ...ngles. It also gives us discrete versions of such important tools as the Hodge decomposition. Previously this machinery has been used successfully in settings such as surface parameterization (e.g., [=-=Gu and Yau 2003-=-; Gortler et al. 2006; Tong et al. 2006]), physical simulation [Elcott et al. 2007], and vector field decomposition and smoothing [Tong et al. 2003; Polthier and Preuß 2003]. A rigorous treatment of t... |

89 | KOBBELT L.: An intuitive framework for real-time freeform modeling
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Citation Context ...constraints incorporated through weighted least squares (wLSQ). When applied to 0-forms, i.e., functions given as values at vertices, such approaches are well known in geometric modeling and editing [=-=Botsch and Kobbelt 2004-=-; Sorkine et al. 2005], where they are based on the Laplace and higher-order Laplace operators. In essence the same ideas work for 1-form data using the corresponding Laplace operators, as we demonstr... |

88 | J.E.: Discrete Exterior Calculus
- Desbrun, Hirani, et al.
- 2011
(Show Context)
Citation Context ...ntext of interactive, anisotropic texture synthesis on arbitrary-topology surface meshes. 2 Mathematical Tools Our approach has its mathematical foundation in Discrete Exterior Calculus [Mercat 2001; =-=Hirani 2003-=-; Desbrun et al. 2006], which 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 ... |

73 | An inexpensive brdf model for physicallybased rendering
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Citation Context ...roblems for 1-forms, texture synthesis 1 Introduction Smoothly varying tangent vector fields appear in many applications that control the appearance of surfaces. Examples include anisotropic shading [=-=Schlick 1994-=-], texture synthesis [Turk 2001; Wei and Levoy 2001], non-photorealistic rendering [Hertzmann and Zorin 2000], line integral convolution [Cabral and Leedom 1993], and spot noise [van Wijk 1991] among ... |

72 |
Designing Quadrangulations with Discrete Harmonic Forms
- Tong, Alliez, et al.
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(Show Context)
Citation Context ...ons of such important tools as the Hodge decomposition. Previously this machinery has been used successfully in settings such as surface parameterization (e.g., [Gu and Yau 2003; Gortler et al. 2006; =-=Tong et al. 2006-=-]), physical simulation [Elcott et al. 2007], and vector field decomposition and smoothing [Tong et al. 2003; Polthier and Preuß 2003]. A rigorous treatment of the connection between discrete and cont... |

69 | Discrete multiscale vector field decomposition
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- 2003
(Show Context)
Citation Context ... in settings such as surface parameterization (e.g., [Gu and Yau 2003; Gortler et al. 2006; Tong et al. 2006]), physical simulation [Elcott et al. 2007], and vector field decomposition and smoothing [=-=Tong et al. 2003-=-; Polthier and Preuß 2003]. A rigorous treatment of the connection between discrete and continuous settings can be found in the survey of Arnold et al. [2006], where finite element techniques are used... |

55 | Appearance-space texture synthesis - LEFEBVRE, HOPPE - 2006 |

55 | Discrete Riemann surfaces and the Ising model - Mercat |

53 |
K.: Decorating implicit surfaces
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- 1995
(Show Context)
Citation Context ...s functions to form this interpolation, while Turk [2001] uses hierarchical low pass filtering. One may also perform such interpolation or alignment optimizations directly in the (2D) tangent spaces [=-=Pedersen 1995-=-; Hertzmann and Zorin 2000; Wei and Levoy 2001], though coordinate transformations between different tangent spaces must be accounted for carefully. All these approaches rely on explicit coordinate fr... |

53 | Identifying vector fields singularities using a discrete hodge decomposition - Polthier, Preu - 2003 |

47 |
Discrete Differential Forms for Computational Modeling
- DESBRUN, KANSO, et al.
- 2008
(Show Context)
Citation Context ...are able to formulate vector field design as a linear problem by using an intrinsic, coordinate-free approach based on discrete differential forms and the associated Discrete Exterior Calculus (DEC) [=-=Desbrun et al. 2006-=-]. The central concept in this approach is the representation of fields through measurements on cells: a 0-form represents a scalar function through its values at vertices (0-dim cells), while a 2-for... |

44 | WARDETZKY M.: On the convergence of metric and geometric properties of polyhedral surfaces
- HILDEBRANDT, POLTHIER
(Show Context)
Citation Context ...bitrarily well so long as the mesh is suitably fine. More generally, Dodziuk et al. [1976] show that Whitney 1-forms can approximate any smooth vector field arbitrarily well in the L∞ norm (see also [=-=Hildebrandt et al. 2006-=-; Arnold et al. 2006]). For very coarse triangulations, however, it is likely better to use either the nonlinear interpolants of Zhang et al. or smoother DEC bases [Wang et al. 2006]. The latter are a... |

42 | Modifying a sparse Cholesky factorization
- Davis, Hager
- 1999
(Show Context)
Citation Context ...f constraints |Z| =: k will be far smaller than |E|, the dimension of M. This favors a Cholesky pre-factorization strategy for M = CC T coupled with incremental modification of the Cholesky factor C [=-=Davis and Hager 1999-=-; Davis and Hager 2001]. For general Z i this can be quite involved and may even change the sparsity structure of the Cholesky factor. We are in the fortunate setting that all our constraints are very... |

37 | Efficient linear system solvers for mesh processing
- BOTSCH, BOMMES, et al.
- 2005
(Show Context)
Citation Context ...metric positive (semi-)definite and sparse. In the case of 0-form Laplace (as well as bi-Laplace) systems, approaches based on Cholesky factorization have been shown to exhibit excellent performance [=-=Botsch et al. 2005-=-], and we chose such a (black-box) solver as well (TAUCS [Toledo 2003] with a modification supplied by Olga Sorkine). One could factor the entire left hand side of Eq. (6) as one monolithic matrix. Ch... |

36 | A.: A discrete Laplace-Beltrami operator for simplicial surfaces - BOBENKO, SPRINGBORN - 2005 |

30 | Designing 2d vector fields of arbitrary topology - Theisel - 2002 |

27 | Discontinuous Galerkin methods for elliptic problems
- Arnold, Brezzi, et al.
- 1999
(Show Context)
Citation Context ...stay in a linear framework, taking advantage of the fact that the edge coefficients arise from the solution of a discrete Poisson problem for which we, in effect, use a discontinuous Galerkin method [=-=Arnold et al. 2000-=-]. This connection with finite element theory ensures that the Whitney-interpolated fields approximate the underlying smooth solution arbitrarily well so long as the mesh is suitably fine. More genera... |

25 | Discrete one-forms on meshes and applications to 3D mesh parameterization
- Gortler, Gotsman, et al.
- 2005
(Show Context)
Citation Context ...ves us discrete versions of such important tools as the Hodge decomposition. Previously this machinery has been used successfully in settings such as surface parameterization (e.g., [Gu and Yau 2003; =-=Gortler et al. 2006-=-; Tong et al. 2006]), physical simulation [Elcott et al. 2007], and vector field decomposition and smoothing [Tong et al. 2003; Polthier and Preuß 2003]. A rigorous treatment of the connection between... |

22 |
Farthest Point Seeding for Efficient Placement of Streamlines
- Mebarki, Alliez, et al.
- 2005
(Show Context)
Citation Context ...nal contribution of 1/2( fi j + fil) to the above sum. Figure 7: Comparison of tangential, normal, and free boundary conditions with two different strokes as constraints, rendered with the method of [=-=Mebarki et al. 2005-=-] (inset shows mesh resolution). 4.2 Assembly of Boundary Modified Laplacian As indicated earlier, M ∇× is assembled without modification. M ∇· is assembled as the product of d T 0 ⋆ 1 with its transp... |

20 | Riemannian structures and triangulations of manifolds - Dodziuk, Patodi - 1977 |

17 | S.: Geometry-aware bases for shape approximation
- SORKINE, COHEN-OR, et al.
(Show Context)
Citation Context ...through weighted least squares (wLSQ). When applied to 0-forms, i.e., functions given as values at vertices, such approaches are well known in geometric modeling and editing [Botsch and Kobbelt 2004; =-=Sorkine et al. 2005-=-], where they are based on the Laplace and higher-order Laplace operators. In essence the same ideas work for 1-form data using the corresponding Laplace operators, as we demonstrate here for the firs... |

15 | W.W.: Row modifications of a sparse cholesky factorization
- Davis, Hager
- 2005
(Show Context)
Citation Context ... will be far smaller than |E|, the dimension of M. This favors a Cholesky pre-factorization strategy for M = CC T coupled with incremental modification of the Cholesky factor C [Davis and Hager 1999; =-=Davis and Hager 2001-=-]. For general Z i this can be quite involved and may even change the sparsity structure of the Cholesky factor. We are in the fortunate setting that all our constraints are very simple: each Z i is a... |

14 | Building your own DEC at home - Elcott, Schröder - 2006 |

2 |
Finite Element Exterior Calculus
- Arnold, Falk, et al.
- 2006
(Show Context)
Citation Context ... the mesh is suitably fine. More generally, Dodziuk et al. [1976] show that Whitney 1-forms can approximate any smooth vector field arbitrarily well in the L∞ norm (see also [Hildebrandt et al. 2006; =-=Arnold et al. 2006-=-]). For very coarse triangulations, however, it is likely better to use either the nonlinear interpolants of Zhang et al. or smoother DEC bases [Wang et al. 2006]. The latter are always continuous and... |

2 |
Yee-Schemes on a Tetrahedral Mesh, with Diagonal Lumping
- Bossavit, Kettunen
- 1999
(Show Context)
Citation Context ... |e ∗ |/|e|, (⋆ 2) t t = |t ∗ |/|t|, are sufficient (the superscript ∗ denotes the Voronoi dual of a given cell). These so called diagonal Hodge-star matrices may be regarded as lumped mass matrices [=-=Bossavit and Kettunen 1999-=-] and are nothing more than ratios of dual to primal intrinsic volumes. (We will use the symbol ⋆ to denote either ⋆ 0, ⋆ 1, or ⋆ 2 depending on the type of its argument.) We can now consider the L 2-... |

2 | An Algorithm for the Construction of Intrinsic Delaunay Triangulations with Applications to Digital Geometry Processing - al - 2006 |