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## Implicit Representations of Surfaces and Polygonalization Algorithms

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### Title: Marching Cubes: A High Resolution Surface Construction Algorithm Author: Scott Last modified by: schaefer Created Date: 11/14/2007 11:22:38 PM – PowerPoint PPT presentation

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Title: Implicit Representations of Surfaces and Polygonalization Algorithms

1
Implicit Representations of Surfaces and
Polygonalization Algorithms
Dr. Scott Schaefer
2
Polygon Models
• Explicit connectivity information
• Easy to render
• (Relatively) small storage
• Topology changes difficult
• Inside/Outside test hard

3
Implicit Representations of Shape
• Shape described by solution to f(x)c

4
Implicit Representations of Shape
• Shape described by solution to f(x)c

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Implicit Representations of Shape
• Shape described by solution to f(x)c

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Implicit Representations of Shape
• Shape described by solution to f(x)c

7
• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations

8
• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations

9
• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations

10
• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union

11
• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union

12
• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union
• Intersection

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union
• Intersection

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union
• Intersection
• Subtraction

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union
• Intersection
• Subtraction

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union
• Intersection
• Subtraction

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• No topology to maintain
• Always defines a closed surface!
• Inside/Outside test
• CSG operations
• Union
• Intersection
• Subtraction

22
• 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

23
Non-Analytic Implicit Functions
• Sample functions over grids

24
Non-Analytic Implicit Functions
• Sample functions over grids

25
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
31
2D Surface Reconstruction
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2D Surface Reconstruction
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2D Surface Reconstruction
34
2D Surface Reconstruction
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2D Surface Reconstruction
36
Marching Cubes
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Marching Cubes
38
Dual Contouring
• Place vertices inside of square
• Generate segments across edges with zero
• Dual to polygons produced by MC

39
Comparison of Primal/Dual
• Allows more freedom in positioning vertices

Marching Cubes (Primal)
Dual Contouring (Dual)
40
Dual Contouring With Hermite Data
• Place vertices at minimizer of QEFs
• Generate segments across edges with zeros

41
Comparison
Marching Cubes
Dual Contouring
42
Contouring Signed Octrees
• For each minimal edge with zero,
• Connect vertices of cubes containing edge
• Constructs closed surface mesh for any octree

43
Fast Polygon Generation
• Recursive octree traversal
• Linear time in size of octree

44
Extensions
• Multiple materials
• CSG operations
• Simplification via QEFs
• Topological safety

45
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

46
Dual Marching Cubes
• Enumerate dual grid using
• recursive walk
• Three types of recursive calls

47
Dual Marching Cubes