3D Object Representations 2017, 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.

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 Surface Subdivision (Smooth Curve)

Parametric Surface Boundary defined by parametric functions x = fx (u, v) y = fy (u, v) z = fz (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 Example 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 <Enumeration vs Quadtree >

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

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

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

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

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