three-dimensional object representation jehee lee seoul national university
TRANSCRIPT
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Three-DimensionalObject Representation
Jehee Lee
Seoul National University
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Object Representation
• Graphics scenes can contain many different kinds of objects and material surfaces– Trees, flowers, clouds, rocks, water, bricks, wood paneling,
rubber, paper, marble, steel, glass, plastic and cloth
• So it may not be surprising that there is no single method that we can use to describe all the characteristics of these different shapes/materials
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Object Representation
• Scene = an assembly of one or more models
• A model contains– Structural description: Geometry of the shape– Surface description: Appearance and light
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3D Object Representations
• Boundary representation– A set of surfaces that separate the object interior from the
environment– Eg) Polyhedra, curved boundary surfaces
• Space-partitioning– Partitioning the spatial region into a set of small, nonoverlapping,
contiguous solids (usually cubes)– Eg) Volumetric data, octrees, BSP trees
• Procedural methods– Fractals, shape grammars
• Constructive solid geometry• Physically-based modeling
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Issues in Model Selection
• Computational cost– Storage space– Model construction time– Display time
• Effectiveness in modeling the desired phenomena– Geometry
• Looks good for image synthesis• Accuracy for simulation
– Appearance• Looks / Accuracy
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Issues in Model Selection
• Implementation complexity– The number of primitives– The complexity of each primitives
• Ease of acquiring (or creating) data• Ease of manipulation
– Operations on models
• Ease of animation– Match to simulator– Cost of conversion– Physics of motion
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Polyhedra
• A polyhedron is a 3D solid which consists of a collection of polygons, usually joined at their edges
• The inside of the solid is divided by the polygons from the outside of the solid
• Regular 2-manifold boundary surfaces
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Polyhedra
• A polyhedron with zero-genus can be considered as a planar graph
• Euler formula– Determines the validity of planar graphs
– V: The number of vertices– E: The number of edges– F: The number of faces
2 FEV
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Polyhedra
• Euler-Poincare formula– Generalizes the Euler formula for arbitrary dimension
and genus– Polyhedra in three-dimension space follows
– L: The number of Inner loops– S: The number of shells– G: The number of holds
)(2 GSLFEV
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Functional Representation
• Explicit surfaces
• Implicit surfaces
• Parametric surfaces
• Issues– Representation power– Easy to render– Easy to manipulate (translate, rotate, and deform)
),( yxfz
0),,( zyxf
],[for )(),(),( battztytx
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Implicit Surfaces
• Quadric surfaces– Implicit second-order polynomial equations
• Superquadric surfaces– A generalization of quadric surfaces
• Blobby objects (metaballs)– A collection of spherical density functions
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Quadric surfaces
• Double cones
• Ellipsoids
• Hyperboloids of one sheet
• Hyperboloids of two sheets
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Quadric surfaces
• Elliptic paraboloids
• Hyperbolic paraboloids
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Superquadrics
• A generalization of quadric surfaces, formed by incorporating additional parameters into quadric equations– Increased flexibility for adjusting object shapes
• Superellipses 1/2/2
SS
b
y
a
x
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Superquadrics
• Superellipsoid
11
1221 /2//2/2
SSSSS
c
z
b
y
a
x
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Blobby Objects
• A collection of density functions
• Equi-density surfaces
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Spatial Partitioning
• Volume data– Use identical cells (voxels)– Space-filling tesselation with cubes or parallelopipeds– Expensive storage but simple data structure– Useful for medical imaging: volume visualization
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Spatial Partitioning
• Octrees– Partition space into 8 cubes, recursively– Increase space efficiency of solid tesselations
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Spatial Partitioning
• Binary Space Partitioning (BSP) trees– Subdivide a scene into two sections at each step with
a plane that can be at any position and orientation
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Constructive Solid Geometry
• Recursively combine simple primitives by boolean operations– Simple primitives
– Transform or deform– Tree structure
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Constructive Solid Geometry
Primitives Transformed Combined
Tree structure
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Constructive Solid Geometry
• Ray-casting methods for rendering CSG objects– Cast a ray through each pixel– Perform boolean operations along each ray
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Procedural Modeling
• Self-similar fractals– Substitution ruls
– Example: Koch curve
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Procedural Modeling
• Substitution rules
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Procedural Modeling
• Terrain by random perturbation
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Procedural Modeling
• Natural scenes with trees, flowers, and grass
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Physically Based Modeling
• Particle systems– Shape description is combined with physical simulation
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Physically Based Modeling
• Procedural modeling + physically based simulation