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 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-convexConvex

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 regionOutside 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 1Transition 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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