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Polygonal Meshes
Thomas Funkhouser
Princeton University
COS 526, Fall 2010
Digital Geometry Processing
Processing of 3D surfaceso Creation, acquisitiono Storage, transmissiono Editing, animation, simulationo Manufacture
Applicationso Movies, gameso Computer-aided designo Medicine, biologyo Art, history
Sweldens
Digital Geometry Processing
Many possible surface representationso Polygonal mesheso Parametric surfaceso Subdivision surfaceso Implicit surfaceso etc.
Digital Geometry Processing
Let’s focus on 3D polygonal mesheso Simple, common representationo Rendering with hardware supporto Output of many acquisition toolso Input to many simulation/analysis tools
Stanford Graphics Lab
3D Polygonal Meshes
Set of polygonal faces representing a 2D surface embedded in 3D
Zorin & Schroeder, SIGGRAPH 99, Course Notes
3D Polygonal Meshes
Set of polygonal faces representing a 2D surface embedded in 3D
Zorin & Schroeder, SIGGRAPH 99, Course Notes
Face
Vertex
Edge
2
Outline
Acquisition
Processing
Representation
Outline
Acquisition
Processing
Representation
Polygonal Mesh Acquisition
Interactive modelingo Polygon editorso Interchange formats
Scannerso Laser range scannerso CAT, MRI, etc. (isosurfaces)
Simulationso Physical processes
Polygonal Mesh Acquisition
Interactive modelingØ Polygon editorso Interchange formats
Scannerso Laser range scannerso CAT, MRI, etc. (isosurfaces)
Simulationso Physical processes
Sketchup
Blender
Polygonal Mesh Acquisition
Interactive modelingo Polygon editorsØ Interchange formats
Scannerso Laser range scannerso CAT, MRI, etc. (isosurfaces)
Simulationso Physical processes
Princeton Shape Benchmark
Polygonal Mesh Acquisition
Interactive modelingo Polygon editorso Interchange formats
ScannersØ Laser range scannerso CAT, MRI, etc. (isosurfaces)
Simulationso Physical processes
Digital Michelangelo ProjectStanford
3
Polygonal Mesh Acquisition
Interactive modelingo Polygon editorso Interchange formats
Scannerso Laser range scannerso CAT, MRI, etc. (isosurfaces)
Simulationso Physical processes
Large Geometric Model RepositoryGeorgia Tech
Polygonal Mesh Acquisition
Interactive modelingo Polygon editorso Interchange formats
Scannerso Laser range scannerso CAT, MRI, etc. (isosurfaces)
SimulationsØ Physical processes
MIT
Outline
Acquisition
Processing
Representation
Polygonal Mesh Processing
Storageo Compressiono Transmission
Analysiso Parameterizationo Differential geometryo Feature detectiono Segmentation
Editingo Smoothing, sharpening, etc.o Deformationo Completion
Polygonal Mesh Processing
StorageØ Compressiono Transmission
Analysiso Parameterizationo Differential geometryo Feature detectiono Segmentation
Editingo Smoothing, sharpening, etc.o Deformationo Completion
Lossy Compression(Simplification)
Garland
Polygonal Mesh Processing
Storageo CompressionØ Transmission
Analysiso Parameterizationo Differential geometryo Feature detectiono Segmentation
Editingo Smoothing, sharpening, etc.o Deformationo Completion
4
Polygonal Mesh Processing
Storageo Compressiono Transmission
AnalysisØ Parameterizationo Differential geometryo Feature detectiono Segmentation
Editingo Smoothing, sharpening, etc.o Deformationo Completion Sheffer
Polygonal Mesh Processing
Storageo Compressiono Transmission
Analysiso ParameterizationØ Differential geometryo Feature detectiono Segmentation
Editingo Smoothing, sharpening, etc.o Deformationo Completion Sheffer
Polygonal Mesh Processing
Storageo Compressiono Transmission
Analysiso Parameterizationo Differential geometryØ Feature detectiono Segmentation
Editingo Smoothing, sharpening, etc.o Deformationo Completion Novatnek et al.
Polygonal Mesh Processing
Storageo Compressiono Transmission
Analysiso Parameterizationo Differential geometryo Feature detectionØ Segmentation
Editingo Smoothing, sharpening, etc.o Deformationo Completion Katz & Tal
Polygonal Mesh Processing
Storageo Compressiono Transmission
Analysiso Parameterizationo Differential geometryo Feature detectiono Segmentation
EditingØ Smoothing, sharpening, etc.o Deformationo Completion Desbrun
Smoothing
Sharpening
Polygonal Mesh Processing
Storageo Compressiono Transmission
Analysiso Parameterizationo Differential geometryo Feature detectiono Segmentation
Editingo Smoothing, sharpening, etc.Ø Deformationo Completion Sheffer
5
Polygonal Mesh Processing
Storageo Compressiono Transmission
Analysiso Parameterizationo Differential geometryo Feature detectiono Segmentation
Editingo Smoothing, sharpening, etc.o DeformationØ Completion Podolak
Outline
Acquisition
Processing
Representation
Polygon Mesh Representation
Data structures determine algorithmso Data structure must support key operations of
algorithm efficiently
Examples:o Drawing a mesho Removing a vertexo Computing per-vertex normals
Different data structures for different algorithms
Polygon Mesh Representation
Important properties of mesh representation?
Polygon Mesh Representation
Important properties of mesh representation?o Efficient traversal of topologyo Efficient use of memory
Polygon Mesh Representation
Possible data structureso List of independent faceso Vertex and face tableso Adjacency listso Winged edgeo Half edgeo etc.
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Independent Faces
Each face lists vertex coordinateso Redundant verticeso No topology information
Vertex and Face Tables
Each face lists vertex referenceso Shared verticeso Still no topology information
Adjacency Lists
Store all vertex, edge, and face adjacencies o Efficient topology traversalo Extra storage
Partial Adjacency Lists
Can we store only some adjacency relationshipsand derive others?
Winged Edge
Adjacency encoded in edgeso All adjacencies in O(1) timeo Little extra storage (fixed records)o Arbitrary polygons
Winged Edge
Example:
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Half Edge
Adjacency encoded in edgeso All adjacencies in O(1) timeo Little extra storage (fixed records)o Arbitrary polygons
Similar to winged-edge,except adjacency encoded in half-edges
e heinv
vbegin
Fleft
henext
Summary
Polygonal mesh overviewo Acquisitiono Processingo Representation
Summary
Do polygonal mesh reps have these properties?o Easy to acquireo Accurateo Conciseo Efficient displayo Efficient intersectionso Efficient deformationso Efficient topology changeso Guaranteed validityo Guaranteed smoothnesso Intuitive editing controls
Summary
Next time: Laplacian Surface Editing