# 3D Object Representation

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1 3D Object Representation Jason Lawrence CS445: Graphics Acknowledgment: slides by Misha Kazhdan, Allison Klein, Tom Funkhouser, Adam Finkelstein and David Dobkin

2 3D Object Representation How do we... o Represent 3D objects in a computer? o Construct such representations quickly and/or automatically with a computer? o Manipulate 3D objects with a computer? Different methods for different object representations

3 3D Objects How can this object be represented in a computer?

4 3D Objects This one? H&B Figure 10.46

6 3D Objects This one? H&B Figure 9.9

7 3D Objects This one?

8 Representations of Geometry 3D Representations provide the foundations for o Computer Graphics o Computer-Aided Geometric Design o Visualization o Robotics They are languages for describing geometry data structures algorithms Data structures determine algorithms!

9 3D Object Representations Raw data o Point cloud o Range image o Polygon soup Surfaces o Mesh o Subdivision o Parametric o Implicit Solids o Voxels o BSP tree o CSG o Sweep High-level structures o Scene graph o Skeleton o Application specific

10 Point Cloud Unstructured set of 3D point samples o Acquired from range finder, random sampling, particle system implementations, etc Hoppe Hoppe

11 Point Cloud Unstructured set of 3D point samples o Acquired from range finder, random sampling, particle system implementations, etc Can associate colors/normals/ Hoppe etc. to the points Hoppe

12 Range Image An image storing depth instead of color o Acquired from structured range scanners Range Image Tesselation Range Surface Brian Curless SIGGRAPH 99 Course #4 Notes

13 Polygon Soup Unstructured set of polygons o Created with interactive modeling systems, combining range images, etc. Larson

14 3D Object Representations Raw data o Point cloud o Range image o Polygon soup Surfaces o Mesh o Subdivision o Parametric o Implicit Solids o Voxels o BSP tree o CSG o Sweep High-level structures o Scene graph o Skeleton o Application specific

15 Mesh Connected set of polygons (usually triangles) o May not be closed Stanford Graphics Laboratory

16 Subdivision Surface Coarse mesh & subdivision rule o Define smooth surface as limit of sequence of refinements Zorin & Schroeder SIGGRAPH 99 Course Notes

17 Parametric Surface Tensor product spline patches o Careful use of constraints to maintain continuity FvDFH Figure 11.44

18 Implicit Surface Points satisfying: F(x,y,z) = 0 Polygonal Model Implicit Model Bill Lorensen SIGGRAPH 99 Course #4 Notes

19 3D Object Representations Raw data o Point cloud o Range image o Polygon soup Surfaces o Mesh o Subdivision o Parametric o Implicit Solids o Voxels o BSP tree o CSG o Sweep High-level structures o Scene graph o Skeleton o Application specific

20 Voxels Uniform grid of volumetric samples o Acquired from CT, MRI, etc. FvDFH Figure Stanford Graphics Laboratory

21 BSP Tree Binary space partition with solid cells labeled o Constructed from polygonal representations a b c d e f a b c d e f g Object a b c d e f Binary Spatial Partition Binary Tree Naylor

22 CSG Hierarchy of boolean set operations (union, difference, intersect) applied to simple shapes FvDFH Figure H&B Figure 9.9

23 Sweep Solid swept by curve along trajectory Stephen Chenney U Wisconsin

24 Sweep Solid swept by curve along trajectory Stephen Chenney U Wisconsin o Curve may be arbitrary o Sweep polygon may deform (scale, rotate) with respect to the path orientation

25 3D Object Representations Raw data o Point cloud o Range image o Polygon soup Surfaces o Mesh o Subdivision o Parametric o Implicit Solids o Voxels o BSP tree o CSG o Sweep High-level structures o Scene graph o Skeleton o Application specific

26 Scene Graph Union of objects at leaf nodes Bell Laboratories avalon.viewpoint.com

27 Skeleton Graph of curves with radii Stanford Graphics Laboratory SGI

28 Application Specific Apo A-1 (Theoretical Biophysics Group, University of Illinois at Urbana-Champaign) Architectural Floorplan

29 Equivalence of Representations Thesis: o Each fundamental representation has enough expressive power to model the shape of any geometric object o It is possible to perform all geometric operations with any fundamental representation! Analogous to Turing-Equivalence: o All computers today are Turing-equivalent, but we still have many different processors

30 Computational Differences Efficiency o Combinatorial complexity o Space/time trade-offs o Numerical accuracy/stability Simplicity o Ease of acquisition o Hardware acceleration Usability

31 Surfaces What makes a good surface representation? o Concise o Local support o Affine invariant o Arbitrary topology o Guaranteed continuity o Natural parameterization o Efficient display o Efficient intersections H&B Figure 10.46

32 Surfaces What makes a good surface representation? o Concise o Local support o Affine invariant o Arbitrary topology o Guaranteed continuity o Natural parameterization o Efficient display o Efficient intersections Not Local Support

33 Surfaces What makes a good surface representation? o Concise o Local support o Affine invariant o Arbitrary topology o Guaranteed continuity o Natural parameterization o Efficient display o Efficient intersections

34 Surfaces What makes a good surface representation? o Concise o Local support o Affine invariant o Arbitrary topology o Guaranteed continuity o Natural parameterization o Efficient display o Efficient intersections Topological Genus Equivalences

35 Surfaces What makes a good surface representation? o Concise o Local support o Affine invariant o Arbitrary topology o Guaranteed continuity o Natural parameterization o Efficient display o Efficient intersections

36 Surfaces What makes a good surface representation? o Concise o Local support o Affine invariant o Arbitrary topology o Guaranteed continuity o Natural parameterization o Efficient display o Efficient intersections A Parameterization (not necessarily natural)

37 Surfaces What makes a good surface representation? o Concise o Local support o Affine invariant o Arbitrary topology o Guaranteed continuity o Natural parameterization o Efficient display o Efficient intersections

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