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“Dual Domain Extrapolation”
Bruno Lévy
ACM Transactions on Graphics (SIGGRAPH conference proceedings) - SIGGRAPH 2003 Session \,: Parameterization, 2003

Abstract: Shape optimization and surface fairing for polygon meshes have been active research areas for the last few years. Existing approaches either require the border of the surface to be fixed, or are only applicable to closed surfaces. In this paper, we propose a new approach, that computes natural boundaries. This makes it possible not only to smooth an existing geometry, but also to extrapolate its shape beyond the existing border. Our approach is based on a global parameterization of the surface and on a minimization of the squared curvatures, discretized on the edges of the surface. The so-constructed surface is an approximation of a minimal energy surface (MES). Using a global parameterization makes it possible to completely decouple the outer fairness (surface smoothness) from the inner fairness (mesh quality). In addition, the parameter space provides the user with a new means of controlling the shape of the surface. When used as a geometry filter, our approach computes a smoothed mesh that is discrete conformal to the original one. This allows smoothing textured meshes without introducing distortions.

BibTex reference 

@ARTICLE{levy:DDE:2003,
   AUTHOR     = "Lévy, Bruno",
   TITLE      = "Dual Domain Extrapolation",
   JOURNAL    = "ACM Transactions on Graphics (SIGGRAPH conference proceedings)",
   YEAR       = "2003",
   VOLUME     = "22",
   NUMBER     = "3",
   PAGES      = "364-369",
   MONTH      = "Jul",
   NOTE       = "SIGGRAPH 2003 Session \,: Parameterization",
}

Supplemental material, links, hindsight ...

Dual Domain Extrapolation

Repairing a scanned mesh.

  • Repairing a scanned mesh. This surface has a highly complex border and many holes, due to shadows when the face was scanned. A parameterization of the surface is computed, and extrapolated in parameter space thanks to a constrained Delaunay triangulation. The new vertices are placed in 3D space by minimizing our discrete MES criterion (minimal energy surfaces).

Shape extrapolation and editing in parameter space.

  • Shape extrapolation. This animation demonstrates how the discrete MES criterion optimizes the shape of the object, without requiring any boundary condition.
  • Shape editing in parameter space. In addition, the parameter space provides the user with a new way of designing the shape of a surface.
  • Result. Drawing the border in parameter space means trimming a virtual surface, extrapolated from the initial patches.

Mesh fairing.

  • outer and inner fairing. The smoothed mesh computed by DDE is discrete conformal to the original one, making it possible to smooth texture mapped meshes without introducing distortions. Mesh optimization can be applied in parameter space, and by relaunching DDE, inner fairness can be optimized. The parameterization decouples outer and inner fairness.
  • repairing a statue. This tool can be used to repair geometries.

Archaelogical modeling.

  • A multi-chart data set. This type of data sets are difficult to repair using conventional tools.
  • Placing the patches in parameter space. In parameter space, placing the patches is intuitive and easy. Then, a constrained Delaunay triangulation is computed.
  • Result. This video shows how DDE optimizes the location of the new vertices. Note that the border may be cropped by the user if needed.