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

Piecewise Convex Contouring
of Implicit Functions
Tao Ju Scott Schaefer Joe Warren
Computer Science Department
Rice University
Introduction
• Contouring
– 3D volumetric data
– Zero-contour of scalar field
• Marching Cubes Algorithm [Lorensen and Cline,
1987]
– Voxel-by-voxel contouring
– Table driven algorithm
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.
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
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.
Goal
• Extend table driven contouring to support:
– Fast collision detection.
– Adaptive contouring (no explicit crack
prevention).
Idea: Keep Negative Region Convex
• Generate polygons such that the resulting negative
region is convex inside a voxel.
Non-convex
Convex
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
Construction of Lookup Table
• In 2D, line segments are uniquely determined by
sign configuration.
• In 3D, polygons are NOT uniquely determined by
sign configuration.
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.
Convex Contouring
Examples using Convex Contouring
Beyond Uniform Grids
• Current work: Multi-resolution contouring
– A world of non-uniform grids.
– In 2D: Contouring transition squares between grids of
different resolutions
Beyond Uniform Grids
• Current work: Multi-resolution contouring
– A world of non-uniform grids.
– In 3D: Contouring transition voxels between grids of
different resolutions
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
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.
Examples of Adaptive Convex Contouring
Examples of Multi-resolution Contouring
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 multiresolution convex contouring.
– Topology-preserving adaptive contouring.
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.
temporary