Piecewise Convex Contouring of Implicit Functions Tao Ju Scott Schaefer Joe Warren Computer Science...
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![Page 1: Piecewise Convex Contouring of Implicit Functions Tao Ju Scott Schaefer Joe Warren Computer Science Department Rice University.](https://reader038.fdocuments.us/reader038/viewer/2022110206/56649cf95503460f949c9eb2/html5/thumbnails/1.jpg)
Piecewise Convex Contouring of Implicit Functions
Tao Ju Scott Schaefer Joe Warren
Computer Science Department
Rice University
![Page 2: Piecewise Convex Contouring of Implicit Functions Tao Ju Scott Schaefer Joe Warren Computer Science Department Rice University.](https://reader038.fdocuments.us/reader038/viewer/2022110206/56649cf95503460f949c9eb2/html5/thumbnails/2.jpg)
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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• 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
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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
![Page 5: Piecewise Convex Contouring of Implicit Functions Tao Ju Scott Schaefer Joe Warren Computer Science Department Rice University.](https://reader038.fdocuments.us/reader038/viewer/2022110206/56649cf95503460f949c9eb2/html5/thumbnails/5.jpg)
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
![Page 12: Piecewise Convex Contouring of Implicit Functions Tao Ju Scott Schaefer Joe Warren Computer Science Department Rice University.](https://reader038.fdocuments.us/reader038/viewer/2022110206/56649cf95503460f949c9eb2/html5/thumbnails/12.jpg)
Examples using Convex Contouring
![Page 13: Piecewise Convex Contouring of Implicit Functions Tao Ju Scott Schaefer Joe Warren Computer Science Department Rice University.](https://reader038.fdocuments.us/reader038/viewer/2022110206/56649cf95503460f949c9eb2/html5/thumbnails/13.jpg)
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 w
ith extra vertices.• Polygons connected with contours from neighboring
voxels.
Transition Voxel 1 Transition Voxel 2
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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
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Examples of Adaptive Convex Contouring
![Page 18: Piecewise Convex Contouring of Implicit Functions Tao Ju Scott Schaefer Joe Warren Computer Science Department Rice University.](https://reader038.fdocuments.us/reader038/viewer/2022110206/56649cf95503460f949c9eb2/html5/thumbnails/18.jpg)
Examples of Multi-resolution Contouring
![Page 19: Piecewise Convex Contouring of Implicit Functions Tao Ju Scott Schaefer Joe Warren Computer Science Department Rice University.](https://reader038.fdocuments.us/reader038/viewer/2022110206/56649cf95503460f949c9eb2/html5/thumbnails/19.jpg)
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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