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3D Object Representations 2009, Fall
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Introduction What is CG? Imaging : Representing 2D images Modeling : Representing 3D objects Rendering : Constructing 2D images from 3D objects Animation : Simulating changes over time
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Course Syllabus Image Processing Modeling Rendering Animation
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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
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3D Object
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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!
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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
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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
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Point Cloud Unstructured set of 3D point samples Acquired from range finer, computer vision, etc
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Range Image Set of 3D points mapping to pixels of depth Image Acquired from range scanner
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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)
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Polygon Soup Unstructured set of polygons Many polygon models are just lists of polygons Created with interactive modeling systems?
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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
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Curved Surfaces Motivation Exact boundary representation for some objects More concise representation than polygonal mesh
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Mesh Connected set of polygons (usually triangles) May not be closed
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Subdivision Surface Coarse mesh & subdivision rule Define smooth surfaces as limit of sequence of refinements Subdivision (Smooth Curve) Subdivision Surface
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Parametric Surface Boundary defined by parametric functions x = f x (u, v) y = f y (u, v) z = f z (u, v) Example: ellipsoid
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Parametric Surface Tensor product spline patchs Each patch is defined by blending control points Careful constrains to maintain continuity
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Implicit Surfaces Boundary defined by implicit function f(x, y, z) = 0 Example linear (plane) ax + by + cz + d = 0 Ellipsoid
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Implicit Surface Examples
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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
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Solid Modeling Represent solid interiors of objects Surface may not be described explicitly
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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.
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Quadtrees & Octrees Refine resolution of voxels hierarchically More concise and efficient for non-uniform objects
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Quadtree Display
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Binary Space Partitions (BSPs) Recursive partition of space by planes Mark leaf cells as inside or outside object
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Binary Space Partitions (BSPs) recursively divide space into pairs of subspaces each separated by a plane of arbitrary orientation and position
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Constructive Solid Geometry (CSG) Represent solid object as hierarchy of boolean operations Union Intersection Difference
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Constructive Solid Geometry
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Constructive Solid Geometry (CSG) CSG Acquisition Interactive modeling programs CAD/CAM
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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)
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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
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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)
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Sweep Solid swept by curve along trajectory
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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
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Scene Graph Union of objects at leaf nodes
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Skeleton Graph of curves with radii
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Application Specific
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Taxonomy of 3D Representations
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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
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Complexity vs. Verbosity Tradeoff
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Advanced Modeling Procedural Modeling Fractal Modeling Grammar-based Modeling Particle System Physically Based Modeling
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