In this talk, I argue that mesh generation is good tool to preprocess point clouds for geometric inference and discuss how to compute such meshes.

1. Learning with Nets and
Meshes
Don Sheehy

2. 2
Thanks to my collaborators:
Benoit Hudson
Gary Miller
Todd Phillips
Steve Oudot

3. Point Clouds in low to medium dimensional 3
ambient space.
d2
Rule of thumb: d! or 2 is okay but nd is not.

4. There are many geometric inference problems 4
that could benefit from meshing.
Discretize Space
Approximate Functions
Adapt to density
Describe the space around the input.

5. Meshing 5
Input: P ⊂ Rd n = |P |
Output: M ⊃ P with a “nice” Voronoi diagram

6. Meshing 5
Input: P ⊂ Rd n = |P |
Output: M ⊃ P with a “nice” Voronoi diagram

7. Points, offsets, homology, and persistence.

8. Points, offsets, homology, and persistence.
Input: P ⊂ Rd

9. Points, offsets, homology, and persistence.
Input: P ⊂ Rd
α
P = ball(p, α)
p∈P

10. Points, offsets, homology, and persistence.
Input: P ⊂ Rd
α
P = ball(p, α)
p∈P

11. Points, offsets, homology, and persistence.
Input: P ⊂ Rd
α
P = ball(p, α)
p∈P

12. Points, offsets, homology, and persistence.
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P

13. Points, offsets, homology, and persistence.
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

14. Points, offsets, homology, and persistence.
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

15. Points, offsets, homology, and persistence.
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

16. Points, offsets, homology, and persistence.
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

17. Points, offsets, homology, and persistence.
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

18. Points, offsets, homology, and persistence.
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Compute the Homology

19. Points, offsets, homology, and persistence.
Offsets Input: P ⊂ Rd
α
P = ball(p, α)
p∈P
Persistent
Compute the Homology

20. Geometric Topologically
Approximation Equivalent

21. Geometric Topologically
Approximation Equivalent

22. Complexity: How big is the mesh? 8

23. Complexity: How big is the mesh? 8
How many Steiner Points?

24. Complexity: How big is the mesh? 8
How many Steiner Points?
1
|M | = dx
Ω lfsP (x)d

25. Complexity: How big is the mesh? 8
How many Steiner Points?
1
|M | = dx
Ω lfsP (x)d
|M| = O(n) as long as there are no big empty annuli
with 2 or more points inside [MPS08].

26. Complexity: How big is the mesh? 8
How many Steiner Points?
1
|M | = dx
Ω lfsP (x)d
|M| = O(n) as long as there are no big empty annuli
with 2 or more points inside [MPS08].
How many simplices?

27. Complexity: How big is the mesh? 8
How many Steiner Points?
1
|M | = dx
Ω lfsP (x)d
|M| = O(n) as long as there are no big empty annuli
with 2 or more points inside [MPS08].
How many simplices?
Only O(|M|) simplices.
d/2
Compare to |M | for general Delaunay triangulations.
Constants depend on aspect ratio.

28. Complexity: How hard is it to compute a mesh? 9

29. Complexity: How hard is it to compute a mesh? 9
O(n log n + |M |)

30. Complexity: How hard is it to compute a mesh? 9
O(n log n + |M |)
Point Location

31. Complexity: How hard is it to compute a mesh? 9
O(n log n + |M |)
Point Location Output Sensitive

32. Complexity: How hard is it to compute a mesh? 9
O(n log n + |M |)
Point Location Output Sensitive
This is optimal in the comparison model.

33. Complexity: How hard is it to compute a mesh? 9
O(n log n + |M |)
Point Location Output Sensitive
This is optimal in the comparison model.
O(n log n)

34. Complexity: How hard is it to compute a mesh? 9
O(n log n + |M |)
Point Location Output Sensitive
This is optimal in the comparison model.
O(n log n)
Compute a hierarchical quality mesh.

35. Complexity: How hard is it to compute a mesh? 9
O(n log n + |M |)
Point Location Output Sensitive
This is optimal in the comparison model.
O(n log n) + O(|M |)
Compute a hierarchical quality mesh. Finishing post-process.

36. The Delaunay Triangulation is the dual 10
of the Voronoi Diagram.

37. The Delaunay Triangulation is the dual 10
of the Voronoi Diagram.

38. The Delaunay Triangulation is the dual 10
of the Voronoi Diagram.

39. The Delaunay Triangulation is the dual 10
of the Voronoi Diagram.
D-ball

40. Quality Meshes 11

41. Quality Meshes 11
D-Balls are Constant Ply

42. Quality Meshes 11
D-Balls are Constant Ply
Total Complexity is linear in |M|

43. Quality Meshes 11
D-Balls are Constant Ply
Total Complexity is linear in |M|
No D-ball intersects more than O(1) others.

44. Quality Meshes 11
D-Balls are Constant Ply
Total Complexity is linear in |M|
No D-ball intersects more than O(1) others.
Refine poor-quality cells by adding a Steiner point
at its farthest vertex.

45. We replace quality with hierarchical quality. 12

46. We replace quality with hierarchical quality. 12

47. We replace quality with hierarchical quality. 12

48. We replace quality with hierarchical quality. 12

49. We replace quality with hierarchical quality. 12
Inside the cage: Old definition of quality.

50. We replace quality with hierarchical quality. 12
Inside the cage: Old definition of quality.
Outside: Treat the whole cage as a single object.

51. We replace quality with hierarchical quality. 12
Inside the cage: Old definition of quality.
Outside: Treat the whole cage as a single object.
All the same properties as quality meshes: ply, degree, etc.

52. The sparse meshing model: 13

53. The sparse meshing model: 13
1 Build a Voronoi diagram incrementally.

54. The sparse meshing model: 13
1 Build a Voronoi diagram incrementally.
2 Interleave input and Steiner point insertions.

55. The sparse meshing model: 13
1 Build a Voronoi diagram incrementally.
2 Interleave input and Steiner point insertions.
3 Recover quality after each input point.

56. The sparse meshing model: 13
1 Build a Voronoi diagram incrementally.
2 Interleave input and Steiner point insertions.
3 Recover quality after each input point.
This is how we avoid the worst-case Voronoi bounds.

57. Point Location and Geometric D&C 14
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.

58. Point Location and Geometric D&C 14
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.

59. Point Location and Geometric D&C 14
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.

60. Point Location and Geometric D&C 14
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.

61. Point Location and Geometric D&C 14
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.

62. Point Location and Geometric D&C 14
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.
Lemma (HMP06). A point is touched at most
a constant number of times before the radius of
the largest D-ball containing it goes down by a
factor of 2.

63. Point Location and Geometric D&C 14
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.
Lemma (HMP06). A point is touched at most
a constant number of times before the radius of
the largest D-ball containing it goes down by a
factor of 2.
Geometric Divide and Conquer: O(n log ∆)

64. With hierarchical meshes, we can add inputs 15
in any order!

65. With hierarchical meshes, we can add inputs 15
in any order!

66. With hierarchical meshes, we can add inputs 15
in any order!

67. With hierarchical meshes, we can add inputs 15
in any order!
Goal: Combinatorial Divide and Conquer

68. With hierarchical meshes, we can add inputs 15
in any order!
Goal: Combinatorial Divide and Conquer
Old: Progress was decreasing the radius by a factor of 2.

69. With hierarchical meshes, we can add inputs 15
in any order!
Goal: Combinatorial Divide and Conquer
Old: Progress was decreasing the radius by a factor of 2.
New: Decrease number of points in a ball by a factor of 2.

70. Range Nets 16
“Nets catch everything that’s big.”

71. Range Nets 16
“Nets catch everything that’s big.”
Definition. A range space is a pair (X, R),
where X is a set (the vertices) and R is a
collection of subsets (the ranges).

72. Range Nets 16
“Nets catch everything that’s big.”
Definition. A range space is a pair (X, R),
where X is a set (the vertices) and R is a
collection of subsets (the ranges).
For us X = P and R is the set of open balls.

73. Range Nets 16
“Nets catch everything that’s big.”
Definition. A range space is a pair (X, R),
where X is a set (the vertices) and R is a
collection of subsets (the ranges).
For us X = P and R is the set of open balls.
Definition. Given a range space (X, R),
a set N ⊂ X is a range space ε-net if
for all ranges r ∈ R that contain at least
ε|X| vertices, r contains a vertex from N .

74. Range Nets 16
“Nets catch everything that’s big.”
Definition. A range space is a pair (X, R),
where X is a set (the vertices) and R is a
collection of subsets (the ranges).
For us X = P and R is the set of open balls.
Definition. Given a range space (X, R),
a set N ⊂ X is a range space ε-net if
for all ranges r ∈ R that contain at least
ε|X| vertices, r contains a vertex from N .
Theorem: [Chazelle & Matousek 96] For ε, d fixed constants,
ε-nets of size O(1) can be computed in O(n) deterministic time.

75. Ordering the inputs 17
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

76. Ordering the inputs 17
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

77. Ordering the inputs 17
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

78. Ordering the inputs 17
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

79. Ordering the inputs 17
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

80. Ordering the inputs 17
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.
Lemma. Let M be a set of vertices. If an open ball B
contains no points of M , then B is contained in the
union of d D-balls of M .

81. Ordering the inputs 17
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.
log(n) Rounds
Lemma. Let M be a set of vertices. If an open ball B
contains no points of M , then B is contained in the
union of d D-balls of M .

82. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.

83. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
x

84. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
x

85. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y
x

86. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y
x

87. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
x

88. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
x

89. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x

90. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
x

91. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
x

92. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
x

93. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
Only O(1) points are added x
to any D-ball in a round.

94. Amortized Cost of a Round is O(n) 18
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
Only O(1) points are added x
to any D-ball in a round.
O(n) total work per round.

95. Some Takeaways 19

96. Some Takeaways 19
Voronoi refinement meshes give structure to points,
especially with respect to the ambient space.

97. Some Takeaways 19
Voronoi refinement meshes give structure to points,
especially with respect to the ambient space.
For d not too big, meshing is fast (O(n log n)).

98. Some Takeaways 19
Voronoi refinement meshes give structure to points,
especially with respect to the ambient space.
For d not too big, meshing is fast (O(n log n)).
There are many possible new applications, waiting to be
discovered (i.e. any time you would have used a Voronoi
diagram).

99. Thanks.