3D Object Representation. Michael Kazhdan ( /657)

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Transcription:

3D Object Representation Michael Kazhdan (601.457/657)

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

3D Objects This one? H&B Figure 10.46

3D Objects This one? H&B Figure 9.9

3D Objects This one?

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

Point Clouds Unstructured set of 3D point samples Acquired from random sampling, particle system implementations, etc. Hoppe Hoppe

Range Images An image storing depth instead of / as well as color Acquired from 3D scanners Range Image Tesselation Range Surface Brian Curless SIGGRAPH 99 Course #4 Notes

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

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

(Manifold) Meshes Connected set of polygons (usually triangles) Stanford Graphics Laboratory

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

Parametric Surfaces Tensor product spline patches Careful use of constraints to maintain continuity FvDFH Figure 11.44

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

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

Voxels Uniform grid of volumetric samples Acquired from CT, MRI, etc. FvDFH Figure 12.20 Stanford Graphics Laboratory

BSP Trees Binary space partition with solid cells labeled Constructed from polygonal representations a b c d e f 1 2 3 7 4 5 6 a b c d e f g Object a b c d e f 1 2 3 4 5 6 7 Binary Spatial Partition Binary Tree Naylor

Constructive Solid Geometry (CSG) Hierarchy of boolean set operations (union, difference, intersect) applied to simple shapes FvDFH Figure 12.27 H&B Figure 9.9

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

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

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

Skeletons Graph of curves with radii Stanford Graphics Laboratory

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

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

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

Surfaces What makes a good surface representation? Concise Local support Affine invariant Arbitrary topology Guaranteed smoothness Natural parameterization Efficient display Efficient intersections Applying an affine transformation to the surface does not fundamentally change its representation.

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

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

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

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

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