Rapid Deformation of Isocontours by Interactive Editing of Implicit Functions

in: Proc. 25th Gocad Meeting, Nancy

Abstract

In this article we present a method to deform iso-contour shapes - like geological horizons - rapidly in an interactive way. We define an implicit function j on the nodes of a simplicial complex. An isovalue contour is a shape where j takes the constant value w so j −w = 0 is true. Let us consider an isovalue surface Sw of j(x,y, z) that is defined on the nodes of a tetrahedral mesh in the 3D Euclidean space. Editing the implicit function j(x,y, z) automatically leads to a deformation of Sw. In our approach we use the Discrete Smooth Interpolation (DSI) method to interpolate j(x,y, z) on a three dimensional Euclidean domain. Changing the location of one or more property control points and a successive re-interpolation of j(x,y, z) results in a modified implicit function j∗(x,y, z) and consequently to a deformed isovalue surface S∗ w. To achieve immediate model update the re-interpolation has to be performed as fast as possible. Therefore we use a matrix version of DSI and only update the altered coefficients. Further the new solution j∗(x,y, z) for this interpolation problem lies very close to the initial solution j0(x,y, z), so a numerical method like Conjugate Gradient converges very fast. Restrictions of the model editing on local regions additionally speed up our algorithm. By this we can deform isovalue surfaces like geological horizons in real-time with immediate graphical user feedback. Topological constraints like the non-intersection of horizons is granted automatically by this method.

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    BibTeX Reference

    @INPROCEEDINGS{Frank05GM,
        author = { Frank, Tobias and Mallet, Jean-Laurent },
         title = { Rapid Deformation of Isocontours by Interactive Editing of Implicit Functions },
     booktitle = { Proc. 25th Gocad Meeting, Nancy },
          year = { 2005 },
      abstract = { In this article we present a method to deform iso-contour shapes - like geological horizons - rapidly in an
    interactive way. We define an implicit function j on the nodes of a simplicial complex. An isovalue contour
    is a shape where j takes the constant value w so j −w = 0 is true. Let us consider an isovalue surface Sw
    of j(x,y, z) that is defined on the nodes of a tetrahedral mesh in the 3D Euclidean space. Editing the implicit
    function j(x,y, z) automatically leads to a deformation of Sw.
    In our approach we use the Discrete Smooth Interpolation (DSI) method to interpolate j(x,y, z) on a
    three dimensional Euclidean domain. Changing the location of one or more property control points and a
    successive re-interpolation of j(x,y, z) results in a modified implicit function j∗(x,y, z) and consequently
    to a deformed isovalue surface S∗
    w. To achieve immediate model update the re-interpolation has to be
    performed as fast as possible. Therefore we use a matrix version of DSI and only update the altered coefficients.
    Further the new solution j∗(x,y, z) for this interpolation problem lies very close to the initial solution
    j0(x,y, z), so a numerical method like Conjugate Gradient converges very fast. Restrictions of the model
    editing on local regions additionally speed up our algorithm. By this we can deform isovalue surfaces like
    geological horizons in real-time with immediate graphical user feedback. Topological constraints like the
    non-intersection of horizons is granted automatically by this method. }
    }