Piecewise Convex Contouring of Implicit Functions Tao Ju

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

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

2 D 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.

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

Algorithm: Convex Contouring • In 3 D, 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 2 D: Contouring transition squares between grids of different resolutions

Beyond Uniform Grids • Current work: Multi-resolution contouring – A world of non-uniform grids. – In 3 D: 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 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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