1. 3D Object Representations
2017, Fall

2. Introduction What is CG? Imaging : Representing 2D images
Modeling : Representing 3D objects
Rendering : Constructing 2D images from 3D objects
Animation : Simulating changes over time

3. Course Syllabus
Image Processing
Modeling
Rendering
Animation

4. Modeling
How do we..
Represent 3D objects in a computer?
Acquire computer representations of 3D objects?
Manipulate computer representations of 3D objects?
Analyze computer representations of 3D objects?
Different methods for different object representations

5. 3D Object

6. Why Different Representations?
Efficiency for different tasks
Acquisition
Rendering
Manipulation
Animation
Analysis
The choice of 3D object representation can have great impact on algorithms
Data structures determine algorithms!

7. 3D Object Representations
Desirable properties depend on intended use
Easy acquisition
Accurate
Concise
Intuitive editing
Efficient editing
Efficient display
Efficient intersections
Guaranteed validity
Guaranteed smoothness
Etc.

8. 3D Scene Representation
Scene is usually approximated by 3D primitives
Point
Line segment
Polygon
Polyhedron
Curved surface
Solid object
etc.

9. Other Geometric Primitives
More detail on 3D modeling in course
Point
Line segment
Polygon
Polyhedron
Curved surface
Solid object
etc.

10. 3D Object Representations
Raw data
Point cloud
Range Image
Polygon soup
Surface
Mesh
Subdivision
Parametric
Implicit
Solids
Voxels
BSP tree
CSG
Sweep
High-level structures
Scene graph
Skeleton
Application specific

11. Point Cloud Unstructured set of 3D point samples
Acquired from range finer, computer vision, etc

12. Range Image Set of 3D points mapping to pixels of depth Image
Acquired from range scanner

13. Point Sample Rendering
an object representation consisting of a dense set of surface point samples, which contain color, depth and normal information
Point Sample Rendering (Surfel)

14. Polygon Soup Unstructured set of polygons
Many polygon models are just lists of polygons
Created with interactive modeling systems?

15. 3D Object Representations
Raw data
Point cloud
Range Image
Polygon soup
Surface
Mesh
Subdivision
Parametric
Implicit
Solids
Voxels
BSP tree
CSG
Sweep
High-level structures
Scene graph
Skeleton
Application specific

16. Curved Surfaces Motivation
Exact boundary representation for some objects
More concise representation than polygonal mesh

17. Mesh
Connected set of polygons (usually triangles)
May not be closed

18. Subdivision Surface Coarse mesh & subdivision rule
Define smooth surfaces as limit of sequence of refinements
Subdivision Surface
Subdivision (Smooth Curve)

19. Parametric Surface Boundary defined by parametric functions
x = fx (u, v)
y = fy (u, v)
z = fz (u, v)
Example: ellipsoid

20. Parametric Surface Tensor product spline patchs
Each patch is defined by blending control points
Careful constrains to maintain continuity

21. Implicit Surfaces Boundary defined by implicit function Example
f(x, y, z) = 0
Example
linear (plane)
ax + by + cz + d = 0
Ellipsoid

22. Implicit Surface Examples

23. 3D Object Representations
Raw data
Point cloud
Range Image
Polygon soup
Surface
Mesh
Subdivision
Parametric
Implicit
Solids
Voxels
BSP tree
CSG
Sweep
High-level structures
Scene graph
Skeleton
Application specific

24. Solid Modeling Represent solid interiors of objects
Surface may not be described explicitly

25. Voxels Partition space into uniform grid
Grid cells are called a voxels (like pixels)
Store properties of solid object with each voxel
Occupancy
Color
Density
Temperature
Etc.

26. Quadtrees & Octrees Refine resolution of voxels hierarchically
More concise and efficient for non-uniform objects
<Enumeration vs Quadtree >

27. Quadtree Display

28. Binary Space Partitions (BSPs)
Recursive partition of space by planes
Mark leaf cells as inside or outside object

29. Binary Space Partitions (BSPs)
recursively divide space into pairs of subspaces
each separated by a plane of arbitrary orientation and position

30. Constructive Solid Geometry (CSG)
Represent solid object as hierarchy of boolean operations
Union
Intersection
Difference

31. Constructive Solid Geometry

32. Constructive Solid Geometry (CSG)
CSG Acquisition
Interactive modeling programs
CAD/CAM

33. Surface of Revolution (SOR)
To generate a 3-D surface, revolve a two dimensional entity, e.g., a line or plane about the axis in space.
called surfaces of revolution

35. Sweep surfaces (1/2) Generates curve
A 3-D surface is obtained by traversing an entity such as a line, polygon or curve, along a path in space
the resulting surfaces are called sweep surfaces
Frequently used in Geometric modeling
ex) entity : point
path : curve
Generates curve

37. Sweep surfaces (2/2)
Closed polygons and curves generates finite volume by sweeping transformation
ex) square or rectangle swept along a
- straight path yields a parallel piped
- circle on straight path cylinder
- Rotation is also possible

38. Sweep
Solid swept by curve along trajectory

39. 3D Object Representations
Raw data
Point cloud
Range Image
Polygon soup
Surface
Mesh
Subdivision
Parametric
Implicit
Solids
Voxels
Octree
BSP tree
CSG
Sweep
High-level structures
Scene graph
Skeleton
Application specific

40. Scene Graph
Union of objects at leaf nodes

41. Skeleton
Graph of curves with radii

42. Application Specific

43. Taxonomy of 3D Representations

44. Computational Differences
Efficiency
Combinatorial complexity (Ex: O( n log n))
Space/time trade-offs (Ex: Z-buffer)
Numerical accuracy/stability (Degree of polynomial)
Simplicity
Ease of acquisition
Hardware Acceleration
Software creation and maintenance
Usability
Designer interface vs. computational engine

45. Advanced Modeling Advanced Modeling Procedural Modeling
Fractal Modeling
Grammar-based Modeling
Particle System
Physically Based Modeling