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Piecewise Convex Contouring of Implicit Functions

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Title: Piecewise Convex Contouring of Implicit Functions


1
Piecewise Convex Contouring of Implicit Functions
  • Tao Ju Scott Schaefer Joe Warren
  • Computer Science Department
  • Rice University

2
Introduction
  • Contouring
  • 3D volumetric data
  • Zero-contour of scalar field
  • Marching Cubes Algorithm Lorensen and Cline,
    1987
  • Voxel-by-voxel contouring
  • Table driven algorithm

3
2D Marching Cubes
  • Generate line segments that connect zero-value
    points on the edges of the square.
  • Partition the square into positive and negative
    regions.
  • Connected with contours of neighboring squares.

4
3D Marching Cubes
  • Generate polygons that connect zero-value points
    on the edges of the voxel.
  • Partition the voxel into positive and negative
    regions.
  • Connected with contours of neighboring voxels

5
Key Idea Table Driven Contouring
  • Structure of the lookup table
  • Indexed by signs at the corners of the voxel.
  • Each entry is a list of polygons whose vertices
    lie on edges of the voxel.
  • Exact locations of vertices (zero-value points)
    are calculated from the magnitude of scalar
    values at the corners of the voxel.

6
Goal
  • Extend table driven contouring to support
  • Fast collision detection.
  • Adaptive contouring (no explicit crack
    prevention).

7
Idea Keep Negative Region Convex
  • Generate polygons such that the resulting
    negative region is convex inside a voxel.

Non-convex
Convex
8
Fast Point Classification
  • Bound the point to its enclosing voxel.
  • Build extended planes for each polygon on the
    contour inside the voxel.
  • Test the point against those extended planes.

Inside negative region
Outside negative region
9
Construction of Lookup Table
  • In 2D, line segments are uniquely determined by
    sign configuration.
  • In 3D, polygons are NOT uniquely determined by
    sign configuration.

10
Algorithm Convex Contouring
  • In 3D, line segments on the faces of the voxel
    connecting zero-value points are uniquely
    determined by sign configuration (table lookup).
  • Contouring algorithm
  • Lookup cycles of line segments on faces of the
    voxel.
  • Compute positions of zero-value points on the
    edges.
  • Convex triangulation of cycles.

11
Convex Contouring
12
Examples using Convex Contouring
13
Beyond Uniform Grids
  • Current work Multi-resolution contouring
  • A world of non-uniform grids.
  • In 2D Contouring transition squares between
    grids of different resolutions

14
Beyond Uniform Grids
  • Current work Multi-resolution contouring
  • A world of non-uniform grids.
  • In 3D Contouring transition voxels between grids
    of different resolutions

15
Strategy Adaptive Convex Contouring
  • Build expanded lookup table for transitional
    voxels with extra vertices.
  • Polygons connected with contours from neighboring
    voxels.

Transition Voxel 1
Transition Voxel 2
16
Benefits of Adaptive Convex Contouring
  • Automatic method for computing table
  • Fast contouring using table lookup
  • Crack prevention
  • Contours are consistent across the transitional
    face/edge. No crack-filling is necessary.

17
Examples of Adaptive Convex Contouring
18
Examples of Multi-resolution Contouring
19
Conclusion
  • Convex contouring algorithm.
  • Fast Collision Detection.
  • Crack-free adaptive contouring.
  • Real-time contouring with lookup table.
  • Future work
  • Real applications, such as games, using
    multi-resolution convex contouring.
  • Topology-preserving adaptive contouring.

20
Acknowledgements
  • Special thanks to Scott Schaefer for
    implementation of the multi-resolution contouring
    program.
  • Special thanks to the Stanford Graphics
    Laboratory for models of the bunny.

21
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