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Princeton COS 426 - Overview of 3D Object Representations

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11Overview of3D Object RepresentationsEmil Praun(covering for Finkelstein)Princeton UniversityC0S 426, Fall 20002Course SyllabusI. Image processingII. RenderingIII. ModelingIV. AnimationImage Processing(Rusty Coleman, CS426, Fall99)Modeling(Dennis Zorin, CalTech)Animation(Angel, Plate 1)Rendering(Michael Bostock, CS426, Fall99)3Course SyllabusI. Image processingII. RenderingIII. ModelingIV. AnimationImage Processing(Rusty Coleman, CS426, Fall99)Modeling(Dennis Zorin, CalTech)Animation(Angel, Plate 1)Rendering(Michael Bostock, CS426, Fall99)4Modeling• How do we ...o Represent 3D objects in a computer?o Construct such representationsquickly and/or automatically with a computer?o Manipulate 3D objects with a computer?Different methods for different object representations53D ObjectsHow can this object be represented in a computer?63D ObjectsThis one?H&B Figure 10.46273D ObjectsHow about this one?Stanford Graphics Laboratory83D ObjectsThis one?Lorensen93D ObjectsThis one?H&B Figure 9.9103D ObjectsThis one?11Representations of Geometry• 3D Representations provide the foundations foro Computer Graphics, Computer-Aided GeometricDesign, Visualization, Robotics• They are languages for describing geometrySemantics Syntaxvalues data structuresoperations algorithms• Data structures determine algorithms!123D Object Representations• Raw datao Point cloudo Range imageo Polygon soup•Surfaceso Mesho Subdivisiono Parametrico Implicit•Solidso Voxelso BSP treeo CSGo Sweep• High-level structureso Scene grapho Skeletono Application specific313Point Cloud• Unstructured set of 3D point sampleso Acquired from range finder, computer vision, etcHoppeHoppe14Range Image• Set of 3D points mapping to pixels of depth imageo Acquired from range scannerBrian CurlessSIGGRAPH 99Course #4 NotesRange Image Tesselation Range Surface15Polygon Soup• Unstructured set of polygonso Created with interactive modeling systems?Larson163D Object Representations• Raw datao Point cloudo Range imageo Polygon soup•Surfaceso Mesho Subdivisiono Parametrico Implicit•Solidso Voxelso BSP treeo CSGo Sweep• High-level structureso Scene grapho Skeletono Application specific17Mesh• Connected set of polygons (usually triangles)o May not be closedStanford Graphics Laboratory18Subdivision Surface• Coarse mesh & subdivision ruleo Define smooth surface as limit ofsequence of refinementsZorin & SchroederSIGGRAPH 99Course Notes419Parametric Surface• Tensor product spline patchso Careful constraints to maintain continuityFvDFH Figure 11.4420Implicit Surface• Points satisfying: F(x,y,z) = 0Polygonal ModelImplicit ModelBill LorensenSIGGRAPH 99Course #4 Notes213D Object Representations• Raw datao Point cloudo Range imageo Polygon soup•Surfaceso Mesho Subdivisiono Parametrico Implicit•Solidso Voxelso BSP treeo CSGo Sweep• High-level structureso Scene grapho Skeletono Application specific22Voxels• Uniform grid of volumetric sampleso Acquired from CAT, MRI, etc.FvDFH Figure 12.20Stanford Graphics Laboratory23BSP Tree• Binary space partition with solid cells labeledo Constructed from polygonal representationsabcdef1237456abcdefgObjectabcdef1234567Binary Spatial PartitionBinary TreeNaylor24CSG• Hierarchy of boolean set operations (union,difference, intersect) applied to simple shapesFvDFH Figure 12.27H&B Figure 9.9525Sweep• Solid swept by curve along trajectoryRemoval Path Sweep ModelBill LorensenSIGGRAPH 99Course #4 Notes263D Object Representations• Raw datao Point cloudo Range imageo Polygon soup•Surfaceso Mesho Subdivisiono Parametrico Implicit•Solidso Voxelso BSP treeo CSGo Sweep• High-level structureso Scene grapho Skeletono Application specific27Scene Graph• Union of objects at leaf nodesBell Laboratoriesavalon.viewpoint.com28Skeleton• Graph of curves with radiiStanford Graphics LaboratorySGI29Application SpecificApo A-1(Theoretical Biophysics Group,University of Illinois at Urbana-Champaign)Architectural Floorplan(CS Buildi ng, Princet on Uni versi ty)30Taxonomy of 3D RepresentationsDiscrete ContinuousCombinatorialFunctionalParametric ImplicitTopological Set MembershipVoxelsMeshSubdivisionBSP TreeCell ComplexBezierB-SplineAlgebraicNaylor631Equivalence of Representations• Thesis:o Each fundamental representation has enoughexpressive power to model the shapeof any geometric objecto It is possible to perform all geometric operations withany fundamental representation!• Analogous to Turing-Equivalence:o All computers today are turing-equivalent,but we still have many different processors32Computational Differences• Efficiencyo Combinatorial complexity (e.g. O( n log n ) )o Space/time trade-offs (e.g. z-buffer)o Numerical accuracy/stability (degree of polynomial)• Simplicityo Ease of acquisitiono Hardware accelerationo Software creation and maintenance•Usabilityo Designer interface vs. computational engine33Complexity vs. Verbosity TradeoffVerbosity / InaccuracyComplexity / Accuracypixels/ voxelspiecewise linear polyhedralow degree piecewise non-linearsingle general functions34Summary• Raw datao Point cloudo Range imageo Polygon soup•Surfaceso Mesho Subdivisiono Parametrico Implicit•Solidso Voxelso BSP treeo CSGo Sweep• High-level structureso Scene grapho Skeletono Application


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Princeton COS 426 - Overview of 3D Object Representations

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