Special Issue Article

The Visual Computer

, Volume 24, Issue 12, pp 1013-1023

First online:

Twofold adaptive partition of unity implicits

  • J. P. GoisAffiliated withUniversidade de São Paulo Email author 
  • , V. Polizelli-JuniorAffiliated withUniversidade de São Paulo
  • , T. EtieneAffiliated withUniversidade de São Paulo
  • , E. TejadaAffiliated withUniversität Stuttgart
  • , A. CasteloAffiliated withUniversidade de São Paulo
  • , L. G. NonatoAffiliated withUniversidade de São Paulo
  • , T. ErtlAffiliated withUniversität Stuttgart

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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.


Algebraic triangulation Partition of unity implicits Orthogonal polynomials