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Overview of 3D Object Representations

Overview of 3D Object Representations. Thomas Funkhouser Princeton University C0S 597D, Fall 2003. 3D Object Representations. What makes a good 3D object representation?. Stanford and Hearn & Baker. 3D Object Representations. What makes a good 3D object representation?

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Overview of 3D Object Representations

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  1. Overview of3D Object Representations Thomas Funkhouser Princeton University C0S 597D, Fall 2003

  2. 3D Object Representations • What makes a good 3D object representation? Stanford andHearn & Baker

  3. 3D Object Representations • What makes a good 3D object representation? • Intuitive specification • Guaranteed continuity • Guaranteed validity • Efficient rendering • Efficient boolean operations • Accurate • Concise • Structure

  4. Raw data Point cloud Range image Voxels Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Octree BSP tree CSG High-level structures Scene graph 3D Reps for Computer Graphics

  5. Point Cloud • Unstructured set of 3D point samples • Acquired from range finder, computer vision, etc Hoppe Hoppe

  6. Range Image • Set of 3D points mapping to pixels of depth image • Acquired from range scanner Range Image Tesselation Range Surface Brian Curless SIGGRAPH 99 Course #4 Notes

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

  8. Polygon Soup • Unstructured set of polygons • Created with interactive modeling systems? Larson

  9. Raw data Point cloud Range image Voxels Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Octree BSP tree CSG High-level structures Scene graph 3D Reps for Computer Graphics

  10. Mesh • Connected set of polygons (usually triangles) • Efficient rendering Stanford Graphics Laboratory

  11. Subdivision Surface • Define surfaces as limit of refinement sequence • Guaranteed continuity, concise Zorin & Schroeder SIGGRAPH 99 Course Notes

  12. Parametric Surface • Tensor product spline patchs • Intuitive specification?, guaranteed continuity?, accurate?, concise FvDFH Figure 11.44

  13. Implicit Surface • Points satisfying: F(x,y,z) = 0 • Guaranteed continuity, guaranteed validity, efficient boolean operations, concise? Polygonal Model Implicit Model Bill Lorensen SIGGRAPH 99 Course #4 Notes

  14. Raw data Point cloud Range image Voxels Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Octree BSP tree CSG High-level structures Scene graph 3D Reps for Computer Graphics

  15. Octree • Binary space partition with solid cells labeled • Guaranteed validity, efficient boolean operations FvDFH Figure 12.25

  16. BSP Tree • Binary space partition with solid cells labeled • Guaranteed validity, efficient boolean operations a 1 b 1 a a g 6 2 c f f 3 5 e 7 e c d d d 3 4 e c 4 2 b b f 5 Object Binary Spatial Partition 6 7 Binary Tree Naylor

  17. CSG • Hierarchy of boolean set operations (union, difference, intersect) applied to simple shapes • Intuitive specification, guaranteed validity, efficient boolean operations FvDFH Figure 12.27 H&B Figure 9.9

  18. Raw data Point cloud Range image Voxels Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Octree BSP tree CSG High-level structures Scene graph 3D Reps for Computer Graphics

  19. Scene Graph • Union of objects at leaf nodes • Efficient rendering, high-level structure Bell Laboratories avalon.viewpoint.com

  20. Raw data Point cloud Range image Voxels Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Octree BSP tree CSG High-level structures Scene graph 3D Reps for Computer Graphics

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

  22. Computational Differences • Efficiency • Combinatorial complexity (e.g. O( n log n ) ) • Space/time trade-offs (e.g. z-buffer) • Numerical accuracy/stability (degree of polynomial) • Simplicity • Ease of acquisition • Hardware acceleration • Software creation and maintenance • Usability • Designer interface vs. computational engine

  23. 3D Reps for Computer Graphics • Different properties for different applications Display Editing Property • Intuitive specificationYesNo • Guaranteed continuityYesNo • Guaranteed validityYesNo • Efficient boolean operations YesNo • Efficient renderingYes Yes • AccurateYes Yes • Concise ? ? • Structure Yes Yes

  24. 3D Reps for Analysis & Retrieval • Different properties for different applications Retrieval Analysis Display Editing Property • Intuitive specificationYesNo No No • Guaranteed continuityYesNo No No • Guaranteed validityYesNo No No • Efficient boolean operations YesNoNo No • Efficient renderingYes YesNo No • AccurateYes Yes ? ? • Concise ? ? ? Yes • Structure Yes Yes Yes Yes

  25. Statistical examples Moments Wavelets Extended Gaussian Image Structural examples Medial axis Curve skeletons Deformable models 3D Reps for Analysis & Retrieval

  26. Moments • Define shape by moments of inertia: • Properties • Invertible • First-order moments give center of mass • Second-order moments give principal axes of rotation

  27. Wavelets • Define shape with wavelet coefficients • Properties • Invertible • Multiresolution 16,000 coefficients 400 coefficients 100 coefficients 20 coefficients Jacobs, Finkelstein, & Salesin 1995

  28. Extended Gaussian Image • Define shape with histogram of normal directions • Invertible for convex objects • Spherical function

  29. 3D Reps for Analysis & Retrieval • Statistical examples • Moments • Wavelets • Extended Gaussian Image • Structural examples • Medial axis • Curve skeletons • Deformable models

  30. Medial Axis • Define shape as union of centers of maximal balls Nina Amenta

  31. Curve Skeleton • Graph representing axis of local symmetry Stanford Graphics Laboratory

  32. Deformable Models • Represent model as union of “part” primitives Robert Osada

  33. Summary • Many possible 3D object representations • Most are “turing equivalent” • Different reps are more efficient for different tasks • Shape analysis & retrieval • Not same requirements as modeling and rendering • We will study several different reps in this course

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