Fast Computation of Generalized Voronoi Diagrams Using Graphics Hardware

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• Fast Computation of Generalized Voronoi Diagrams
  Using Graphics Hardware
• Kenneth E. Hoff III, Tim Culver, John Keyser,
• Ming Lin, and Dinesh Manocha
• University of North Carolina at Chapel Hill
• SIGGRAPH 99

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What is a Voronoi Diagram?

• Given a collection of geometric primitives, it is
  a subdivision of space into cells such that all
  points in a cell are closer to one primitive than
  to any other

Voronoi Site
Voronoi Region
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• Ordinary
• Point sites
• Nearest Euclidean distance

• Generalized
• Higher-order site geometry
• Varying distance metrics

Higher-orderSites
2.0
0.5
Weighted Distances
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Why Should We Compute Them?

• It is a fundamental concept
• Descartes Astronomy 1644 Heavens
• Dirichlet Math 1850 Dirichlet tesselation
• Voronoi Math 1908 Voronoi diagram
• Boldyrev Geology 1909 area of influence
  polygons
• Thiessen Meteorology 1911 Theissen polygons
• Niggli Crystallography 1927 domains of action
• Wigner Seitz Physics 1933 Wigner-Seitz
  regions
• Frank Casper Physics 1958 atom domains
• Brown Ecology 1965 areas potentially available
• Mead Ecology 1966 plant polygons
• Hoofd et al. Anatomy 1985 capillary domains
• Icke Astronomy 1987 Voronoi diagram

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Why Should We Compute Them?

• Useful in a wide variety of applications
• Collision Detection
• Surface Reconstruction
• Robot Motion Planning
• Non-Photorealistic Rendering
• Surface Simplification
• Mesh Generation
• Shape Analysis

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What Makes Them Useful?

• Ultimate Proximity Information

Nearest Site
Maximally Clear Path
Density Estimation
Nearest Neighbors
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Outline

• Generalized Voronoi Diagram Computation
• Exact and Approximate Algorithms
• Previous Work
• Our Goal
• Basic Idea
• Our Approach
• Basic Queries
• Applications
• Conclusion

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Generalized Voronoi Diagram ComputationExact
Algorithms

• Previous work
• Lee82
• Chiang92
• Okabe92
• Dutta93
• Milenkovic93
• Hoffmann94
• Sherbrooke95
• Held97
• Culver99

Computes Analytic Boundary
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Previous Work Exact Algorithms

• Compute analytic boundaries
• but...
• Boundaries composed of high-degree curves and
  surfaces and their intersections
• Complex and difficult to implement
• Robustness and accuracy problems

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Generalized Voronoi Diagram Computation
Approximate Algorithms
Exact Algorithm
Analytic Boundary
Discretize Sites
Discretize Space
Previous work Lavender92, Sheehy95, Vleugels 95
96, Teichmann97
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Previous Work Approximate Algorithms

• Provide practical solutions
• but...
• Difficult to error-bound
• Restricted to static geometry
• Relatively slow

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Our Goal
Approximate generalized Voronoi diagram
computation that is

• Simple to understand and implement
• Easily generalized
• Efficient and practical

with all sources of error fully enumerated
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Outline

• Generalized Voronoi Diagram Computation
• Basic Idea
• Brute-force Algorithm
• Cone Drawing
• Graphics Hardware Acceleration
• Our Approach
• Basic Queries
• Applications
• Conclusion

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Brute-force Algorithm
Record ID of the closest site to each sample point
Coarsepoint-samplingresult
Finerpoint-samplingresult
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Cone Drawing

• To visualize Voronoi diagram for points in 2D

Perspective, 3/4 view
Parallel, top view
Dirichlet 1850 Voronoi 1908
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Graphics Hardware Acceleration
Our 2-part discrete Voronoidiagram representation
Simply rasterize the cones using graphics hardware
Depth Buffer
Color Buffer
Distance
Site IDs
Haeberli90, Woo97
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Outline

• Generalized Voronoi Diagram Computation
• Basic Idea
• Our Approach
• Meshing Distance Function
• Generalizations
• 3D
• Sources of Error
• Basic Queries
• Applications
• Conclusion

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The Distance Function
Evaluate distance at each pixel for all
sitesAccelerate using graphics hardware
Point
Line
Triangle
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Approximating the Distance Function

• Avoid per-pixel distance evaluationPoint-sample
  the distance functionReconstruct by rendering
  polygonal mesh

Point
Line
Triangle
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The Error Bound
Error bound is determined by the pixel
resolution ? ? farthest distance a point can be
from a pixel sample point
Close-up of pixel grid
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Meshing the Distance Function
Shape of distance function for a 2D point is a
cone
Need a bounded-error tessellation of the cone
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Shape of Distance Functions
Sweep apex of cone along higher-order site to
obtain the shape of the distance function
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Example Distance Meshes
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Curves
Tessellate curve into a polylineTessellation
error is added to meshing error
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Weighted and Farthest Distance
Nearest
Farthest
Weighted
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3D Voronoi Diagrams
Graphics hardware can generate one 2D slice at a
time
Point sites
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3D Voronoi Diagrams
Slices of the distance function for a 3D point
site
Distance meshes used to approximate slices
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3D Voronoi Diagrams
Point
Line segment
Triangle
1 sheet of a hyperboloid
Elliptical cone
Plane
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3D Voronoi Diagrams
Points and a triangle
Polygonal model
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Sources of Error

• Distance Error
• Meshing
• Tessellation
• Hardware Precision
• Combinatorial Error
• Distance
• Pixel Resolution
• Z-buffer Precision

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Adaptive Resolution
Zoom in to reduce resolution error...
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Outline

• Generalized Voronoi Diagram Computation
• Basic Idea
• Our Approach
• Basic Queries
• Nearest Site
• Boundary Finding
• Nearest-Neighbor Finding
• Maximally Clear Point
• Applications
• Conclusion

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Nearest Site

• Table lookup on query point

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Boundary Finding

• Isosurface extraction boundary walking
• Offset difference image to mark boundary pixels

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Nearest-Neighbor Finding

• Which colors touch in the image?
• Walk the boundaries (like boundary finding)

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Maximally Clear Point

• Point with the largest depth value (greatest
  distance)

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Outline

• Generalized Voronoi Diagram Computation
• Basic Idea
• Our Approach
• Basic Operations
• Applications
• Motion Planning
• Medial Axis Computation
• Dynamic Mosaics
• Conclusion

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Real-time Motion Planning Static Scene
Plan motion of piano (arrow) through 100K
triangle model
Distance buffer of floorplan used as potential
field
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Real-time Motion Planning Dynamic Scene
Plan motion of music stand around moving furniture
Distance buffer of floor-plan used as potential
field
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Medial Axis Computation
Per-featureVoronoi diagram
Per-feature-colordistance mesh
Internal Voronoi diagram
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Dynamic Mosaics
1000 moving points
Source image
Dynamic Mosaic Tiling
Static mosaics by Paul Haeberli in SIGGRAPH 90
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Outline

• Generalized Voronoi Diagram Computation
• Basic Idea
• Our Approach
• Basic Operations
• Applications
• Conclusion

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Conclusion

• Meshing Distance Functions
• Graphics Hardware Acceleration
• Brute-force Approach

Fast and Simple, Approximate Generalized Voronoi
Diagrams Bounded Error
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Future Work

• Improve distance meshing for 3D primitives
• More applications
• Motion planning with more degrees of freedom
• Accelerate exact Voronoi diagram computation
• Surface reconstruction
• Medical imaging segmentation and registration
• Finite-element mesh generation

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Acknowledgements

• Sarah Hoff
• Chris Weigle
• Stefan Gottschalk
• Mark Peercy
• UNC Computer Science
• SGI Advanced Graphics
• Berkeley Walkthrough Group

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Acknowledgements

• Army Research Office
• National Institute of Health
• National Science Foundation
• Office of Naval Research
• Intel

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