Gokul Varadhan

1
Accurate Minkowski Sum Approximation of
Polyhedral Models

• Gokul Varadhan
• Dinesh Manocha

University of North Carolina at Chapel Hill
http//gamma.cs.unc.edu/recons
2
Minkowski Sum
3
Minkowski Sum
4
Minkowski Sum
5
Minkowski Sum
6
Applications

• Motion planning
• Penetration depth computation dynamic
  simulation
• Packing
• CAD offsets
• Animation

7
Motion Planning
Workspace
Config Space
O
R
8
Dynamic Simulation

• Collision Detection
• Penetration Depth
• Computation

Kim et al. 2002
9
Packing
10
Offsetting
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Morphing
A
B
Morph
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Minkowski Sum Computation

• Difficult to compute the Minkowski sum exactly
• High complexity
• Minkowski sum of two nonconvex objects of size n
  can have O(n6) complexity

2D example Agarwal et al. 02
13
Minkowski Sum Computation

• Minkowski sum of two convex objects much easier
  to compute
• O(n2) complexity
• Common approach
• Decompose the two objects into convex pieces
• Compute pairwise Minkowski sums between
  all pairs of convex pieces
• Compute the union of pairwise Minkowski sums

14
Minkowski Sum Computation

• Main bottleneck
• Union computation
• Large number of primitives
• Given two objects with m and n convex pieces,
  perform a union of mn primitives
• Typically, mn very large (in the order of
  thousands)
• Exact union computation is prone to robustness
  problems and degeneracies

15
Previous Work

• 3D Minkowski sum
• Lozano-Perez 83
• Evans et al. 92
• Ghosh 93
• Special cases
• 2D Lee et al. 98 Flato Halperin 00
• Convex Guibas Seidel 87
• Slope monotone surfaces Seong et al. 02
• Superset of the Minkowski sum
• Kaul Rossignac 91, Basch et al. 96

16
Our Approach

• Do not compute exactly
• Approximate it using distance field based
  volumetric techniques

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Volumetric Approach
A
M
Sampling
Reconstruction
Operation
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Volumetric Approach

• Sampling
• Generate a volumetric grid
• At each grid point, compute
• Distance to the surface
• Sign i.e. inside ( ) /outside ( ) status

D Distance Sign
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Volumetric Approach

• 2. Operation
• For each geometric operation (e.g.
  union/intersection), perform a min/max operation
  on the distance fields

D min (D1, D2)
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Volumetric Approach

• 3. Reconstruction
• Perform isosurface extraction using Marching
  Cubes (MC) or its variant

A
21
Challenges

• 1. Accuracy
• Depends on rate of sampling, i.e., resolution of
  the underlying volumetric grid
• Undersampling can result in poor approximation

A
M
MC
22
Challenges

• 2. Efficiency
• Large number of primitives
• Inefficient to define sign and distance queries
  in terms of all the primitives
• Need culling techniques

23
Main Result

• An accurate and efficient Minkowski sum
  approximation algorithm

24
Main Result

• An accurate and efficient Minkowski sum
  approximation algorithm
• Geometric Error Bound
• Bound the two-sided Hausdorff distance between M
  A
• Topological Guarantee
• A has the same topology as M

Together these two guarantees ensure a good
quality of approximation
25
Main Result

• An accurate and efficient Minkowski sum
  approximation algorithm
• Culling techniques that improve performance

26
Outline

• Approximate Algorithm
• Culling Techniques
• Results
• Discussion

27
Approximate Algorithm

• Sampling
• Generate an adaptive volumetric grid satisfying a
  sampling condition
• Operation
• Compute a distance field on the grid by
  performing min/max operations
• Reconstruction
• Obtain an approximation by extracting an
  isosurface from the adaptive distance field

28
Approximate Algorithm

• Sampling
• Generate an adaptive volumetric grid satisfying a
  sampling condition
• Operation
• Compute a distance field on the grid by
  performing min/max operations
• Reconstruction
• Obtain an approximation by extracting an
  isosurface from the adaptive distance field

29
Main Idea

• Ensure that within each grid cell, M should be
  simple

• A cell that is intersected by the boundary of M
  should exhibit a sign change
• The boundary of M restricted to the cell should
  be a topological disk

30
Complex Cell Test

• M intersects the cell (volume) and the grid
  vertices belonging to the cell do not exhibit a
  sign change

No cell in the grid can be complex
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Main Idea

• Ensure that within each grid cell, M is simple

• A cell that is intersected by the boundary of M
  should exhibit a sign change
• The boundary of M restricted to the cell should
  be a topological disk

32
Star-shapedness

• A primitive is star-shaped if there exists a
  point inside the primitive that can see every
  point in the primitive

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Star-shaped Test

• M restricted to the cell should be star-shaped

Not star-shaped
Star-shaped
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Topological Guarantee

• Theorem If every grid cell C satisfies the
  following criteria
• then the approximate Minkowski sum A has
  the same topology as the exact Minkowski sum M

• Complex cell test
• M must not be complex w.r.t C
• Star-shaped test
• M must be star-shaped w.r.t C

35
Adaptive Grid Generation

• Start with a grid cell that bounds M
• Perform adaptive subdivision
• Each grid cell must satisfy two criteria
• Complex cell test
• Star-shaped test
• Otherwise subdivide the grid cell

36
Adaptive Grid Generation
A
D
C
B
37
Adaptive Grid Generation

• Easy to implement these tests
• Complex cell test ? Max-norm distance computation
  Varadhan et al. 2003
• Star-shaped test ? Linear programming

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Adaptive Grid Generation

• These tests do not require an explicit
  representation of the surface
• Applicable when the surface is defined as a
  Boolean expression
• Well suited to our problem as M is defined as a
  Boolean expression

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Star-shaped Test

• For polyhedral primitives, it reduces to linear
  programming

Linear constraint
n (c - p) gt 0
n
c
p
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Star-shaped Test

• Check if the linear program has a feasible
  solution

p
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Star-shaped Test

• Booleans on primitives
• If both A and B are star-shaped w.r.t a common
  point p, then so is

• Combine the linear
• constraints
• Conservative

p
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Geometric Error Bound

• We can bound the two-sided Hausdorff error to be
  less than ?
• Add an additional criterion
• Subdivide a cell if
• Hausdorff distance between the approximation and
  some primitive is greater than ?

43
Outline

• Approximate Algorithm
• Culling Techniques
• Results
• Discussion

44
Culling Techniques

• Cell culling
• Eliminate cells that do not contain a part of M
• Primitive culling
• When performing a query, disregard primitives
  that do not contribute to the answer

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Cell Culling

• Eliminate cells do not contain a part of M

P2
A
B
P1
C
D
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Primitive Culling

• The Minkowski sum is defined in terms of a large
  number of primitives
• Need to define the sign and distance queries in
  terms of fewer primitives

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Primitive Culling

• A primitive influences the query only if its
  boundary intersects the cell (volume)

P
P
P
C
C
C
p
p
p
Disregard cell C (cell culling)
Disregard P
Consider P
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Outline

• Approximate Algorithm
• Culling Techniques
• Results
• Discussion

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Implementation

• 2 GHz Pentium IV PC, 1GB RAM
• Choice of isosurface extraction
• Dual Contouring Ju et al. 2002

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Spoon (336 tris)
Anvil (144 tris)
Union of 4,446 primitives
1 min
15K tris
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Rod (24 tris)
Brake Hub (4,736 tris)
Union of 1,777 primitives
2.3 mins
45K tris
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Spiral (500 tris)
Wrench (772 tris)
Union of 38,703 primitives
5 mins
25K tris
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Scissors (636 tris)
Knife (516 tris)
Union of 63,790 primitives
13 mins
26K tris
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444 tris
1,134 tris
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Union of 66,667 primitives
52 mins
358K tris
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Offsetting
Cup Offset 33 secs 14K tris
Cup (1,000 tris)
Gear Offset 84 secs 22K tris
Gear 2,382 tris)
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Motion Planning
A Simple Algorithm for Complete Motion Planning
of Translating Polyhedral Robots, WAFR 2004
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Assembly Planning
A Simple Algorithm for Complete Motion Planning
of Translating Polyhedral Robots, WAFR 2004
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Outline

• Approximate Algorithm
• Culling Techniques
• Results
• Discussion

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Benefit of Culling

• Cell and primitive culling techniques improve
  performance by over two orders of magnitude
• More than 90 of the speedup comes from primitive
  culling
• Anvil Spoon example
• 7 hours without culling
• 1 minute with culling

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Limitations

• Axis-aligned subdivision cannot handle degenerate
  cases where two primitives touch each other
• Use other subdivision strategies Sriram et al.
  2004
• Cannot handle all inputs
• Self intersections, non-manifolds
• Sampling condition is conservative

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Main Result

• An accurate and efficient Minkowski sum
  approximation algorithm
• Good quality of approximation
• Guaranteed topology and a tight two-sided
  Hausdorff error bound

63
Main Result

• An accurate and efficient Minkowski sum
  approximation algorithm
• Good quality of Minkowski sum approximation
• Use of culling techniques significantly improves
  performance

64
Main Result

• An accurate and efficient Minkowski sum
  approximation algorithm
• Good quality of Minkowski sum approximation
• Use of culling techniques significantly improves
  performance
• Simple to implement and requires only two tests
• Complex cell test ? max-norm distance computation
• Star-shaped test ? linear programming

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Conclusion

• Practical algorithm
• Applied it to complex polyhedral models
• Offset computation and motion planning

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Ongoing Future Work

• Swept volume computation
• Approximate Voronoi diagram computation
• Less conservative scheme

67
Acknowledgement

• Danny Halperin for useful discussions on
  Minkowski sum
• Young Kim, Shankar Krishnan and TVN Sriram
• David Applegate for a modified dual LP code
• Joe Warren and Scot Schaefer for dual contouring
  code
• Members of UNC Gamma group

68
Acknowledgement

• ARO Contracts DAAD19-02-1-0390 and
  W911NF-04-1-0088
• NSF awards ACI 9876914 and ACR-0118743
• ONR Contracts N00014-01-1-0067 and
  N00014-01-1-0496
• DARPA Contract N61339-04-C-0043
• Intel

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Thank you!
URL http//gamma.cs.unc.edu/recons
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Performance

• 2 GHz Pentium IV PC, 1GB RAM

71
Star-shaped Test

• Need to enforce the topological disk property
• Obvious approach
• Test if the Euler characteristic is equal to 1
• Issue We do not have a polygonization of the
  boundary of M
• We use the star-shaped test to enforce this
  property

72
Performance

• Overall performance dominated by the sampling
  step
• More subdivision takes place in the vicinity of
  complex features such as thin sheets, needle like
  features, small holes

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Level of Subdivision
Number of Voxels
Level of Subdivision
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Intersection Test

• Calculate distance under max-norm
• Test if

d Max-norm distance at O L length of the cell
P
A
B
O
d
C
D
L
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Cell Culling

• If the intersection test fails, do not consider
  the cell any further

Max-norm distance at O
P2
A
B
P1
O
C
D
76
Sign Query

• Strictly speaking, the sign query is a global
  query
• Sign of a point depends on all the primitives

• This definition results in a considerable
  overhead because the sign query is used
  frequently

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Primitive Culling

• Transform it to a local query
• Suppose we wish to perform a sign query at point
  p contained in a grid cell C
• Consider only those primitives whose boundary
  intersects C
• This preserves correctness of the query

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Primitive Culling

• Distance query can be handled similarly
• We care only for a local distance