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
5 3D Objects How about this one? Stanford Graphics Laboratory
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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