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Implicit Representations of Surfaces and Polygonalization Algorithms
Dr. Scott Schaefer
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Polygon Models
AdvantagesExplicit connectivity informationEasy to render(Relatively) small storage
DisadvantagesTopology changes difficultInside/Outside test hard
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Implicit Representations of Shape
Shape described by solution to f(x)=c
9),( 22 yxyxf
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Implicit Representations of Shape
Shape described by solution to f(x)=c
9),( 22 yxyxf
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Implicit Representations of Shape
Shape described by solution to f(x)=c
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9),( 22 yxyxf
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Implicit Representations of Shape
Shape described by solution to f(x)=c
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9),( 22 yxyxf
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
Union
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
Union
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
Union
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
Union
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
Union
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
Union
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+
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
UnionIntersection
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
UnionIntersection - -
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
UnionIntersectionSubtraction
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
UnionIntersectionSubtraction
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++
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+--
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+++
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
UnionIntersectionSubtraction
-- -
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Advantages
No topology to maintain Always defines a closed surface! Inside/Outside test CSG operations
UnionIntersectionSubtraction
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Disadvantages
Hard to render - no polygons Creating polygons amounts to root finding Arbitrary shapes hard to represent as an
analytic function Certain operations (like simplification) can be
difficult
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Non-Analytic Implicit Functions
Sample functions over grids
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Non-Analytic Implicit Functions
Sample functions over grids
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Data Sources
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Data Sources
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Data Sources
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Data Sources
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Data Sources
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Data Sources
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2D Surface Reconstruction
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2D Surface Reconstruction
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2D Surface Reconstruction
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2D Surface Reconstruction
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2D Surface Reconstruction
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Marching Cubes
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Marching Cubes
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Dual Contouring
Place vertices inside of square Generate segments across edges with zero Dual to polygons produced by MC
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Comparison of Primal/Dual
Produces well-shaped quads Allows more freedom in positioning vertices
Marching Cubes (Primal) Dual Contouring (Dual)
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Dual Contouring With Hermite Data
Place vertices at minimizer of QEFs Generate segments across edges with zeros
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Comparison
Marching Cubes Dual Contouring
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Contouring Signed Octrees
For each minimal edge with zero,Connect vertices of cubes containing edge
Constructs closed surface mesh for any octree
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Fast Polygon Generation
Recursive octree traversal Linear time in size of octree
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Extensions
Multiple materials CSG operations Simplification via QEFs Topological safety
Dual Marching Cubes
Generate cells for contouring using the dual of the octree
Creates adaptive, crack-free partitioning of space
Use Marching Cubes on dual
cells to construct polygons
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Dual Marching Cubes
Enumerate dual grid using
recursive walkThree types of recursive calls
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Dual Marching Cubes
AdvantagesAlways creates a manifold surfaceSame as Marching Cubes over uniform gridsWorks well for data centered in cells
DisadvantagesOctrees with data at vertices instead
of cells?…
