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