3D Object Representations 2005, Fall

Course Syllabus Image Processing Modeling Rendering Animation

Modeling How do we..  Represent 3D objects in a computer?  Construct representations quickly and/or automatically with a computer?  Manipulate 3D objects with a computer? Different methods for different object representations

Representations of Geometry 3D Representations provide the foundations for  Computer Graphics, Computer-Aided Geometric Design, Visualization, Robotics They are languages for describing geometry SemanticsSyntax valuesdata structures operations algorithms Data structures determine algorithms!

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?

Polygon Soup Evaluation What are the advantages?  It ’ s very simple to read, write, transmit, etc.  A common output format from CAD modelers  The format required for OpenGL BIG disadvantage: No higher order information  No information about neighbors  No open/closed information  No guarantees on degeneracies

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

Surfaces What makes good surface representation?  Accurate  Concise  Intuitive specification  Local support  Affine invariant  Arbitrary topology  Guaranteed continuity  Natural parameterization  Efficient display  Efficient intersections

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

Key Questions  How refine mesh? Aim for propertied like smoothness Loop Subdivision Scheme  How store mesh? Aim for efficiency for implementing subdivision rules

Subdivision Surface Advantages  Simple method for describing complex surfaces  Relatively easy to implement  Arbitrary topology  Local support  Guaranteed continuity  Multiresolution Difficulties  Intuitive specification  Parameterization  Intersections

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

Parametric Surface Advantages  Easy to enumerate points on surface  Possible to describe complex shapes Disadvantages  Control mesh must be quadrilaterals  Continuity constrains difficult to maintain  Hard to find intersections

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

Implicit Surface Examples

Implicit Surfaces Advantages  Easy to test if point is on surface  Easy to intersect two surfaces  Easy to compute z given x and y Disadvantages  Hard to describe specific complex shapes  Hard to enumerate points on surface

Comparison Feature Polygon Mesh Implicit Surface Parametric Surface Subdivision Surface Accurate Concise Intuitive specification Local support Affine invariant Arbitrary topology Guaranteed continuity Natural parameterization Efficient display Efficient intersections No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No

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

Solid Modeling Motivation  Some acquisition methods generate solids (Ex: CAT scan)  Some applications requires solids (Ex: CAD/CAM)  Some algorithms require solids (Ex: RT with refraction)

Solid Modeling Representations What makes a good solid representation?  Accurate  Concise  Affine invariant  Easy acquisition  Guaranteed validity  Efficient boolean operation  Efficient display

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.

Voxel Acquisition Scanning devices  MRI (Magnetic Resonance Imaging)  CAT (Computed Axial Tomography) Simulation  FEM (Finite Element Method )

Voxels

Advantage  Simple, intuitive, unambiguous  Same complexity for all objects  Natural acquisition for some applications  Trivial boolean operations Disadvantages  Approximates  Not affine invariant  Large scale requirement  Expensive display

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

Comparison FeatureVoxelsOctreeBSPCSG Accurate Concise Affine invariant Easy acquisition Guaranteed validity Efficient boolean operations Efficient display No Some Yes No Some Yes No Some No Yes No Yes Some Yes Some No Yes No

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

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

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