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Generalized Hermitian Radial Basis Functions Implicits from polygonal mesh constraints

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Abstract

In this work we investigate a generalized interpolation approach using radial basis functions to reconstruct implicit surfaces from polygonal meshes. With this method, the user can define with great flexibility three sets of constraint interpolants: points, normals, and tangents; allowing to balance computational complexity, precision, and feature modeling. Furthermore, this flexibility makes possible to avoid untrustworthy information, such as normals estimated on triangles with bad aspect ratio. We present results of the method for applications related to the problem of modeling 2D curves from polygons and 3D surfaces from polygonal meshes. We also apply the method to problems involving subdivision surfaces and front-tracking of moving boundaries. Finally, as our technique generalizes the recently proposed HRBF Implicits technique, comparisons with this approach are also conducted.

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Acknowledgements

The authors thank to FAPESP—Fundação de Amparo à Pesquisa do Estado de São Paulo, CNPq—National Council for Scientific and Technological Development and CAPES—Coordenação de Aperfeiçoamento de Pessoal de Nível Superior. Homer and Pig models are provided courtesy of INRIA by the AIM@SHAPE Shape Repository.

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Authors and Affiliations

  1. Universidade Federal do ABC, Santo André, Brazil

    João Paulo Gois, Diogo Fernando Trevisan & Harlen Costa Batagelo

  2. University of British Columbia, Vancouver, Canada

    Ives Macêdo

Authors
  1. João Paulo Gois
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  2. Diogo Fernando Trevisan
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  3. Harlen Costa Batagelo
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  4. Ives Macêdo
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Corresponding author

Correspondence to João Paulo Gois.

Appendix: Formulas for radial functions

Appendix: Formulas for radial functions

In this appendix we provide the gradients and Hessians for the radial functions employed in this work to simplify the computational implementation.

Polyharmonic Spline

The Polyharmonic Spline is the globally supported radial function:

$$\phi(\mathbf {x}) = \|\mathbf {x}\|^3 $$

and conditionally positive definite of order 2 [13], which makes necessary the use of a first-order polynomial to ensure unicity in the solution. Its gradient reads

$$\nabla\phi(\mathbf {x}) = 3\mathbf {x}\|\mathbf {x}\|, $$

and its Hessian is

where is the identity matrix. Posing Hϕ(0):=0 3×3 ensures Hessian’s continuity [5].

Compactly supported Wendland’s function

Wendland provides a family of compactly supported positive definite radial functions [38]. The Wendland Compactly Supported RBFs are in the form ϕ d,k with various degrees of continuity (\(\mathcal{C}^{k}\)) and dimensionality d [25, 38]. In this work we use the function:

$$\phi_{3,1}(\mathbf {x}) = \bigl(1-\|\mathbf {x}\| \bigr)_+^4 \bigl(4\|\mathbf {x}\|+1 \bigr). $$

The gradient of this Wendland function reads

$$\nabla\phi_{3,1}(\mathbf {x}) = -20 \bigl(1 - \|\mathbf {x}\| \bigr)_+^3\mathbf {x}$$

whereas its Hessian is

and posing ensures Hessian continuity. The size of the support for Wendland’s radial functions can be changed by posing: \(\phi (\mathbf {y}) = \phi (\frac{\mathbf {x}}{h} )\), where h is the user-tuned support radius parameter.

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Gois, J.P., Trevisan, D.F., Batagelo, H.C. et al. Generalized Hermitian Radial Basis Functions Implicits from polygonal mesh constraints. Vis Comput 29, 651–661 (2013). https://doi.org/10.1007/s00371-013-0802-8

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