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Twofold adaptive partition of unity implicits

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Abstract

Partition of Unity Implicits (PUI) has been recently introduced for surface reconstruction from point clouds. In this work, we propose a PUI method that employs a set of well-observed solutions in order to produce geometrically pleasant results without requiring time consuming or mathematically overloaded computations. One feature of our technique is the use of multivariate orthogonal polynomials in the least-squares approximation, which allows the recursive refinement of the local fittings in terms of the degree of the polynomial. However, since the use of high-order approximations based only on the number of available points is not reliable, we introduce the concept of coverage domain. In addition, the method relies on the use of an algebraically defined triangulation to handle two important tasks in PUI: the spatial decomposition and an adaptive polygonization. As the spatial subdivision is based on tetrahedra, the generated mesh may present poorly-shaped triangles that are improved in this work by means a specific vertex displacement technique. Furthermore, we also address sharp features and raw data treatment. A further contribution is based on the PUI locality property that leads to an intuitive scheme for improving or repairing the surface by means of editing local functions.

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References

  1. Ohtake, Y., Belyaev, A., Alexa, M., Turk, G., Seidel, H.P.: Multi-level partition of unity implicits. ACM Trans. Graph. 22(3), 463–470 (2003)

    Article  Google Scholar 

  2. Gois, J.P., Polizelli-Junior, V., Etiene, T., Tejada, E., Castelo, A., Ertl, T., Nonato, L.G.: Robust and adaptive surface reconstruction using partition of unity implicits. In: Brazilian Symposium on Computer Graphics and Image Processing, pp. 95–102 (2007)

  3. Bloomenthal, J.: An implicit surface polygonizer. In: Graphics Gems IV, pp. 324–349. Academic Press, San Diego (1994). citeseer.ist.psu.edu/bloomenthal94implicit.html

    Google Scholar 

  4. Ohtake, Y., Belyaev, A., Seidel, H.P.: Sparse surface reconstruction with adaptive partition of unity and radial basis functions. Graph. Models 68(1), 15–24 (2006)

    Article  MATH  Google Scholar 

  5. Mederos, B., Arouca, S., Lage, M., Lopes, H., Velho, L.: Improved partition of unity implicit surface reconstruction. Technical Report TR-0406, IMPA, Brazil (2006)

  6. Chen, Y.L., Lai, S.H.: A partition-of-unity based algorithm for implicit surface reconstruction using belief propagation. In: IEEE International Conference on Shape Modeling and Applications, pp. 147–155 (2007)

  7. Xia, Q., Wang, M.Y., Wu, X.: Orthogonal least squares in partition of unity surface reconstruction with radial basis function. In: Conference on Geometric Modeling and Imaging, pp. 28–33 (2006)

  8. Tobor, I., Reuter, P., Schlick, C.: Reconstructing multi-scale variational partition of unity implicit surfaces with attributes. Graph. Models 68(1), 25–41 (2006)

    Article  MATH  Google Scholar 

  9. Kazhdan, M.: Reconstruction of solid models from oriented point sets. In: Eurographics Symposium on Geometry Processing, pp. 73–82 (2005)

  10. Kazhdan, M., Bolitho, M., Hoppe, H.: Poisson surface reconstruction. In: Eurographics Symposium on Geometry Processing, pp. 61–70 (2006)

  11. Bolitho, M., Kazhdan, M., Burns, R., Hoppe, H.: Multilevel streaming for out-of-core surface reconstruction. In: Eurographics Symposium on Geometry Processing, pp. 69–78 (2007)

  12. Bloomenthal, J.: Polygonization of implicit surfaces. Comput. Aided Geom. Des. 5(4), 341–355 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hall, M., Warren, J.: Adaptive polygonalization of implicitly defined surfaces. IEEE Comput. Graph. Appl. 10(6), 33–42 (1990)

    Article  Google Scholar 

  14. Paiva, A., Lopes, H., Lewiner, T., de Figueiredo, L.H.: Robust adaptive meshes for implicit surfaces. In: Brazilian Symposium on Computer Graphics and Image Processing, pp. 205–212 (2006)

  15. Kazhdan, M., Klein, A., Dalal, K., Hoppe, H.: Unconstrained isosurface extraction on arbitrary octrees. In: Eurographics Symposium on Geometry Processing, pp. 125–133 (2007)

  16. Castelo, A., Nonato, L.G., Siqueira, M., Minghim, R., Tavares, G.: The j1a triangulation: An adaptive triangulation in any dimension. Comput. Graph. 30(5), 737–753 (2006)

    Article  Google Scholar 

  17. de Figueiredo, L.H., Gomes, J.M., Terzopoulos, D., Velho, L.: Physically-based methods for polygonization of implicit surfaces. In: Conference on Graphics Interface, pp. 250–257 (1992)

  18. Schreiner, J., Scheidegger, C., Silva, C.: High-quality extraction of isosurfaces from regular and irregular grids. IEEE Trans. Vis. Comput. Graph. 12(5), 1205–1212 (2006)

    Article  Google Scholar 

  19. Bartels, R.H., Jezioranski, J.J.: Least-squares fitting using orthogonal multinomials. ACM Trans. Math. Softw. 11(3), 201–217 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  20. Amenta, N., Kil, Y.J.: The domain of a point set surfaces. Eurographics Symp. Point-based Graph. 1(1), 139–147 (2004)

    Google Scholar 

  21. Kobbelt, L.P., Botsch, M., Schwanecke, U., Seidel, H.P.: Feature sensitive surface extraction from volume data. In: SIGGRAPH’01, pp. 57–66 (2001)

  22. PovRay: Persistence of vision. http://www.porvay.org (2007)

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Authors and Affiliations

  1. Universidade de São Paulo, São Carlos, Brazil

    J. P. Gois, V. Polizelli-Junior, T. Etiene, A. Castelo & L. G. Nonato

  2. Universität Stuttgart, Stuttgart, Germany

    E. Tejada & T. Ertl

Authors
  1. J. P. Gois
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  2. V. Polizelli-Junior
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  3. T. Etiene
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  4. E. Tejada
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  5. A. Castelo
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  6. L. G. Nonato
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  7. T. Ertl
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Correspondence to J. P. Gois.

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Gois, J.P., Polizelli-Junior, V., Etiene, T. et al. Twofold adaptive partition of unity implicits. TVC 24, 1013–1023 (2008). https://doi.org/10.1007/s00371-008-0297-x

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