3D Object Representations 2011, 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

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!

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.

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

3D point Specifies a location  Represented by three coordinates  Infinitely small

3D Vector Specifies a direction and a magnitude  Represented by three coordinates  Magnitude  Has no location

3D Vector Dot(=Scalar) product of two 3D vectors

3D Vector Cross(=Vector) product of two 3D vectors  V 1 X V 2 = vector perpendicular V 1 and V 2 (dx2, dy2, dz2) (dx1, dy1, dz1)

3D Line Segment Linear path between two points Use a linear combination of two points  Parametric representation

3D Ray Line segment with one endpoint at infinity  Parametric representation

3D Line Line segment with both endpoints at infinity  Parametric representation

3D Plane A linear combination of three points  Implicit representation, or  N is the plane “normal” Unit-length vector Perpendicular to plane

3D Polygon Area “inside” a sequence of coplanar points  Triangle  Quadrilateral  Convex  Star-shaped  Concave  Self-intersection  Holes (use > 1 polygon struct) Points are in counter-clockwise order

3D Sphere All points at distance “r” from point “(c x, c y, c z )”  Implicit representation  Parametric representation

3D Scenes Comprise set of geometric primitives

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

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

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