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

• 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.

2D Marching Cubes

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 f

rom 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 insi

de 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 di

fferent resolutions

Strategy: Adaptive Convex Contouring• Build expanded lookup table for transitional voxels w

ith extra vertices.• Polygons connected with contours from neighboring

voxels.

Transition Voxel 1 Transition Voxel 2

Benefits of Adaptive Convex Contouring

• Crack prevention– Contours are consistent across the transitional

face/edge. No crack-filling is necessary.

• Automatic method for computing table• Fast contouring using table lookup

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 multi-

resolution 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.

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