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 HillSIGGRAPH ‘99

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

Ordinary • Point sites• Nearest Euclidean distance

Generalized• Higher-order site geometry• Varying distance metrics

Weighted Distances

Higher-orderSites

2.0

0.5

Why Should We Compute Them?It is a fundamental concept

Descartes Astronomy 1644 “Heavens”Dirichlet Math 1850 Dirichlet tesselationVoronoi Math 1908 Voronoi diagramBoldyrev Geology 1909 area of influence polygonsThiessen Meteorology 1911 Theissen polygonsNiggli Crystallography 1927 domains of actionWigner & Seitz Physics 1933 Wigner-Seitz regionsFrank & Casper Physics 1958 atom domainsBrown Ecology 1965 areas potentially availableMead Ecology 1966 plant polygonsHoofd et al. Anatomy 1985 capillary domainsIcke Astronomy 1987 Voronoi diagram

Why Should We Compute Them?Useful in a wide variety of applications

Collision DetectionSurface ReconstructionRobot Motion Planning

Non-Photorealistic RenderingSurface Simplification

Mesh GenerationShape Analysis

What Makes Them Useful?“Ultimate” Proximity Information

Nearest Site Maximally Clear Path

Density Estimation Nearest Neighbors

Outline

• Generalized Voronoi Diagram Computation– Exact and Approximate Algorithms– Previous Work– Our Goal

• Basic Idea• Our Approach• Basic Queries• Applications• Conclusion

Generalized Voronoi Diagram Computation“Exact” Algorithms

Computes Analytic Boundary

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

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

Generalized Voronoi Diagram Computation

Analytic Boundary Discretize Sites Discretize Space

Exact Algorithm Approximate Algorithms

Previous workLavender92, Sheehy95, Vleugels 95 & 96, Teichmann97

Previous Work: Approximate Algorithms

• Provide practical solutions

but...

• Difficult to error-bound• Restricted to static geometry• Relatively slow

Our Goal

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

Approximate generalized Voronoi diagram computation that is:

with all sources of error fully enumerated

Outline

• Generalized Voronoi Diagram Computation• Basic Idea

– Brute-force Algorithm– Cone Drawing– Graphics Hardware Acceleration

• Our Approach• Basic Queries• Applications• Conclusion

Brute-force Algorithm

Record ID of the closest site to each sample

point

Coarsepoint-sampling

result

Finerpoint-sampling

result

Cone Drawing

To visualize Voronoi diagram for points in 2D…

Perspective, 3/4 view Parallel, top view

Dirichlet 1850 & Voronoi 1908

Graphics Hardware Acceleration

Our 2-part discrete Voronoidiagram representation

Distance

Depth Buffer

Site IDs

Color Buffer

Simply rasterize the cones using

graphics hardware

Haeberli90, Woo97

Outline

• Generalized Voronoi Diagram Computation• Basic Idea• Our Approach

– Meshing Distance Function– Generalizations– 3D– Sources of Error

• Basic Queries• Applications• Conclusion

The Distance Function

Point Line Triangle

Evaluate distance at each pixel for all sitesAccelerate using graphics hardware

Approximating the Distance Function

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

Point Line Triangle

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

Meshing the Distance Function

Shape of distance function for a 2D point is a cone

Need a bounded-error tessellation of the cone

Shape of Distance Functions

Sweep apex of cone along higher-order site to obtain the shape of the distance function

Example Distance Meshes

Curves

Tessellate curve into a polylineTessellation error is added to meshing error

Weighted and Farthest Distance

Nearest FarthestWeighted

3D Voronoi Diagrams

Graphics hardware can generate one 2D slice at a time

Point sites

3D Voronoi Diagrams

Slices of the distance function for a 3D point site

Distance meshes used to approximate slices

3D Voronoi DiagramsPoint Line segment Triangle

1 sheet of a hyperboloid

Elliptical cone Plane

3D Voronoi Diagrams

Points and a triangle Polygonal model

Sources of Error

• Distance Error– Meshing– Tessellation– Hardware Precision

• Combinatorial Error– Distance– Pixel Resolution– Z-buffer Precision

Adaptive Resolution

Zoom in to reduce resolution error...

Outline

• Generalized Voronoi Diagram Computation• Basic Idea• Our Approach• Basic Queries

– Nearest Site– Boundary Finding– Nearest-Neighbor Finding– Maximally Clear Point

• Applications• Conclusion

Nearest Site

Table lookup on query point

Boundary Finding

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

Nearest-Neighbor Finding

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

Maximally Clear Point

Point with the largest depth value (greatest distance)

Outline

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

– Motion Planning– Medial Axis Computation– Dynamic Mosaics

• Conclusion

Real-time Motion Planning : Static Scene

Plan motion of piano (arrow) through 100K triangle model

Distance buffer of floorplan used as potential field

Real-time Motion Planning : Dynamic Scene

Plan motion of music stand around moving furniture

Distance buffer of floor-plan used as potential field

Medial Axis Computation

Per-featureVoronoi diagram

Per-feature-colordistance mesh

Internal Voronoi diagram

Dynamic Mosaics

Static mosaics by Paul Haeberli in SIGGRAPH ‘90

1000 moving points

Source image

Dynamic Mosaic Tiling

Outline

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

Conclusion

Meshing Distance FunctionsGraphics Hardware AccelerationBrute-force Approach

Fast and Simple, ApproximateGeneralized Voronoi DiagramsBounded Error

+

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

Acknowledgements

Sarah HoffChris Weigle

Stefan GottschalkMark Peercy

UNC Computer ScienceSGI Advanced Graphics

Berkeley Walkthrough Group

Acknowledgements

Army Research OfficeNational Institute of Health

National Science FoundationOffice of Naval Research

Intel

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