Piecewise Convex Contouring of Implicit Functions

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Piecewise Convex Contouring of Implicit Functions

• Tao Ju Scott Schaefer Joe Warren
• Computer Science Department
• Rice University

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Introduction

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

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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.

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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

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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.

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Goal

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

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Idea Keep Negative Region Convex

• Generate polygons such that the resulting
  negative region is convex inside a voxel.

Non-convex
Convex
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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
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Construction of Lookup Table

• In 2D, line segments are uniquely determined by
  sign configuration.
• In 3D, polygons are NOT uniquely determined by
  sign configuration.

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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.

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Convex Contouring
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Examples using Convex Contouring
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Beyond Uniform Grids

• Current work Multi-resolution contouring
• A world of non-uniform grids.
• In 2D Contouring transition squares between
  grids of different resolutions

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Beyond Uniform Grids

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

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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
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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.

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Examples of Adaptive Convex Contouring
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Examples of Multi-resolution Contouring
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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.

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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.

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temporary