3D Object Representations 2009, Fall

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

Course Syllabus Image Processing Modeling Rendering Animation

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

3D Object

3D Object Representations A computational representation of geometry can be viewed as a language or a data structure The choice of 3D object representation can have great impact on algorithms  Data structures determine algorithms!

3D Object Representations Desirable properties  Accurate  Concise  Easy acquisition  Intuitive editing  Local support  Affine invariant  Arbitrary topology  Guaranteed validity  Guaranteed continuity  Natural parameterization  Efficient display  Efficient intersections

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

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

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

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)

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

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

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

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

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

Parametric Surface Boundary defined by parametric functions  x = f x (u, v)  y = f y (u, v)  z = f z (u, v) Example: ellipsoid

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

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

Implicit Surface Examples

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

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

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.

Quadtrees & Octrees Refine resolution of voxels hierarchically  More concise and efficient for non-uniform objects

Quadtree Display

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

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

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

Constructive Solid Geometry

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

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 Surface of Revolution (SOR)

Sweep surfaces (1/2) 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

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 Sweep surfaces (2/2)

Sweep Solid swept by curve along trajectory

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

Scene Graph Union of objects at leaf nodes

Skeleton Graph of curves with radii

Application Specific

Taxonomy of 3D Representations

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

Complexity vs. Verbosity Tradeoff

Advanced Modeling  Procedural Modeling Fractal Modeling Grammar-based Modeling  Particle System  Physically Based Modeling