[PPT] - An Efficient Algorithm for Determining an Aesthetic Shape PowerPoint Presentation

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An Efficient Algorithm for Determining an Aesthetic Shape Connecting Unorganised 2D Points

Stefan Ohrhallinger1,2 and Sudhir Mudur2

1Vienna University of Technology, 2Concordia University, Montréal

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Stefan Ohrhallinger and Sudhir Mudur 2

Connect The Dots

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Stefan Ohrhallinger and Sudhir Mudur 3

Now Try Without The Numbers

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Stefan Ohrhallinger and Sudhir Mudur 4

Now Try Without The Numbers

Boundary reconstruction = recovering connectivity

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Stefan Ohrhallinger and Sudhir Mudur 5

Now Try Without The Numbers

Boundary reconstruction = recovering connectivity

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Stefan Ohrhallinger and Sudhir Mudur 6

30 Years of Research Effort

α-shapes [EKS83], β-skeleton [KR85], γ-n'hood [Vel93], r-regular [Att97]

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

1 please see paper for references.

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Stefan Ohrhallinger and Sudhir Mudur 7

30 Years of Research Effort

α-shapes [EKS83], β-skeleton [KR85], γ-n'hood [Vel93], r-regular [Att97]

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

1 please see paper for references.

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Stefan Ohrhallinger and Sudhir Mudur 8

30 Years of Research Effort

α-shapes [EKS83], β-skeleton [KR85], γ-n'hood [Vel93], r-regular [Att97]

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

Crust [ABE98], [DK99], [DMR99]

1 please see paper for references.

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Stefan Ohrhallinger and Sudhir Mudur 9

30 Years of Research Effort

α-shapes [EKS83], β-skeleton [KR85], γ-n'hood [Vel93], r-regular [Att97]

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

Crust [ABE98], [DK99], [DMR99] Gathan [DW01], GathanG [DW02]

1 please see paper for references.

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Stefan Ohrhallinger and Sudhir Mudur 10

30 Years of Research Effort

α-shapes [EKS83], β-skeleton [KR85], γ-n'hood [Vel93], r-regular [Att97]

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

Crust [ABE98], [DK99], [DMR99] Gathan [DW01], GathanG [DW02] DISCUR [ZNYL08], VICUR [NZ08]

1 please see paper for references.

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Stefan Ohrhallinger and Sudhir Mudur 11

Global Approach for Closed Curves

Related: Travelling Salesman Problem

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

MST-based [OM11]

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Stefan Ohrhallinger and Sudhir Mudur 12

Global Approach for Closed Curves

Related: Travelling Salesman Problem

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

Heuristics [AMNS00] MST-based [OM11]

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Stefan Ohrhallinger and Sudhir Mudur 13

Global Approach for Closed Curves

Related: Travelling Salesman Problem

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

Heuristics [AMNS00] Exact solvers [ABCC11]

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Stefan Ohrhallinger and Sudhir Mudur 14

Global Approach for Closed Curves

Related: Travelling Salesman Problem

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

Heuristics [AMNS00] Exact solvers [ABCC11] MST-based [OM11]

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Stefan Ohrhallinger and Sudhir Mudur 15

Global Approach for Closed Curves

Related: Travelling Salesman Problem

non-uniform sharp corners sparse sampling parameter-free fast O(n log n) α,β,γ,...

Heuristics [AMNS00] Exact solvers [ABCC11] MST-based [OM11] Our method [OM13]

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Stefan Ohrhallinger and Sudhir Mudur 16

The Fundamental Problem

B reconstructs C C

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Stefan Ohrhallinger and Sudhir Mudur 17

The Fundamental Problem

B reconstructs C C noisy samples

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Stefan Ohrhallinger and Sudhir Mudur 18

The Fundamental Problem

B reconstructs C C noisy samples very noisy + outliers

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Stefan Ohrhallinger and Sudhir Mudur 19

The Fundamental Problem

B reconstructs C C benefit cost entropy noisy samples very noisy + outliers

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Stefan Ohrhallinger and Sudhir Mudur 20

The Fundamental Problem

B reconstructs C C benefit cost entropy noisy samples very noisy + outliers

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Stefan Ohrhallinger and Sudhir Mudur 21

The Fundamental Problem

B reconstructs C C benefit cost entropy noisy samples very noisy + outliers aesthetic

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Stefan Ohrhallinger and Sudhir Mudur 22

Our Approach

What do we know? Sampling process ?

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Stefan Ohrhallinger and Sudhir Mudur 23

Our Approach

What do we know? Sampling process ? Sampled object is a solid

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Stefan Ohrhallinger and Sudhir Mudur 24

Our Approach

What do we know? Proximity Sampling process ? Sampled object is a solid Correlates to Gestalt principles

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Stefan Ohrhallinger and Sudhir Mudur 25

Our Approach

What do we know? Proximity Closure Sampling process ? Sampled object is a solid Correlates to Gestalt principles

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Stefan Ohrhallinger and Sudhir Mudur 26

Our Approach

What do we know? Proximity Closure Sampling process ? Sampled object is a solid Formalized as Bmin Correlates to Gestalt principles

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Stefan Ohrhallinger and Sudhir Mudur 27

Our Approach

What do we know? Proximity Closure Sampling process ? Sampled object is a solid vertex degree c=2 Formalized as Bmin Correlates to Gestalt principles

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Stefan Ohrhallinger and Sudhir Mudur 28

Our Approach

What do we know? Proximity Closure Sampling process ? Sampled object is a solid vertex degree c=2 Goal: O(n log n) for aesthetic point sets, heuristic for remaining class Formalized as Bmin Correlates to Gestalt principles

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Stefan Ohrhallinger and Sudhir Mudur 29

A Naïve Attempt

P

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Stefan Ohrhallinger and Sudhir Mudur 30

A Naïve Attempt

Convex hull P

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Stefan Ohrhallinger and Sudhir Mudur 31

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1 P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 32

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 33

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 34

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 35

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 36

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 37

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 38

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 39

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 40

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 41

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 42

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

local minimum P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 43

A Naïve Attempt

Convex hull: Sculpturing [Boi84]1

find ti candidates select ti: min Δ|B| remove ti from B

local minimum P

1 Boissonnat. Geometric structures for three-dimensional shape representation. TOG 1984.

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Stefan Ohrhallinger and Sudhir Mudur 44

Start With A Minimum Set

P

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Stefan Ohrhallinger and Sudhir Mudur 45

Start With A Minimum Set

P NP-hard

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Stefan Ohrhallinger and Sudhir Mudur 46

Start With A Minimum Set

P MST NP-hard

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Stefan Ohrhallinger and Sudhir Mudur 47

Start With A Minimum Set

Vertex degree P MST NP-hard

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Stefan Ohrhallinger and Sudhir Mudur 48

Start With A Minimum Set

Vertex degree P MST NP-hard

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Stefan Ohrhallinger and Sudhir Mudur 49

Start With A Minimum Set

Vertex degree P MST NP-hard

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Stefan Ohrhallinger and Sudhir Mudur 50

Start With A Minimum Set

Vertex degree P MST NP-hard

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Stefan Ohrhallinger and Sudhir Mudur 51

Start With A Minimum Set

Vertex degree P MST NP-hard NP-hard?

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Stefan Ohrhallinger and Sudhir Mudur 52

Start With A Minimum Set

Vertex degree P MST NP-hard

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Stefan Ohrhallinger and Sudhir Mudur 53

Start With A Minimum Set

Vertex degree P MST ? NP-hard

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Stefan Ohrhallinger and Sudhir Mudur 54

Overview Of Our 'Connect2D' Algorithm

Input

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Stefan Ohrhallinger and Sudhir Mudur 55

Overview Of Our 'Connect2D' Algorithm

Input

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Stefan Ohrhallinger and Sudhir Mudur 56

Overview Of Our 'Connect2D' Algorithm

Input

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Stefan Ohrhallinger and Sudhir Mudur 57

Overview Of Our 'Connect2D' Algorithm

Input not manifold

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Stefan Ohrhallinger and Sudhir Mudur 58

Overview Of Our 'Connect2D' Algorithm

Input Inflate min Δ|B|

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Stefan Ohrhallinger and Sudhir Mudur 59

Overview Of Our 'Connect2D' Algorithm

Input Inflate min Δ|B|

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Stefan Ohrhallinger and Sudhir Mudur 60

Overview Of Our 'Connect2D' Algorithm

Input Inflate Sculpture min Δ|B| min Δ|B|

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Stefan Ohrhallinger and Sudhir Mudur 61

Overview Of Our 'Connect2D' Algorithm

Input Inflate Sculpture Dual min Δ|B| min Δ|B|

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Stefan Ohrhallinger and Sudhir Mudur 62

Overview Of Our 'Connect2D' Algorithm

Input Inflate Sculpture Theorem 1: Our algorithm constructs Bout in O(n log n) time. Dual min Δ|B| min Δ|B|

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Stefan Ohrhallinger and Sudhir Mudur 63

Improved For Sparse Sampling

Points

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Stefan Ohrhallinger and Sudhir Mudur 64

Improved For Sparse Sampling

Points Gathan [DW01]

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Stefan Ohrhallinger and Sudhir Mudur 65

Improved For Sparse Sampling

Points Gathan [DW01] Ours [OM13]

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Stefan Ohrhallinger and Sudhir Mudur 66

Large Sets: Easy

Gathan [DW01]

10k points

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Stefan Ohrhallinger and Sudhir Mudur 67

Large Sets: Easy

Gathan [DW01] Ours [OM13]: manifold

10k points

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Stefan Ohrhallinger and Sudhir Mudur 68

Challenge: Extremely Sparse Sampling

Points

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Stefan Ohrhallinger and Sudhir Mudur 69

Challenge: Extremely Sparse Sampling

Points Gathan [DW01]

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Stefan Ohrhallinger and Sudhir Mudur 70

Challenge: Extremely Sparse Sampling

Points Gathan [DW01] Ours

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Stefan Ohrhallinger and Sudhir Mudur 71

Failure Cases

[OM11]

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Stefan Ohrhallinger and Sudhir Mudur 72

Failure Cases

[OM11] Ours [OM13] local minimum

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Stefan Ohrhallinger and Sudhir Mudur 73

Not Bmin? Insert Points, Manually

Not a solid

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Stefan Ohrhallinger and Sudhir Mudur 74

Not Bmin? Insert Points, Manually

Not a solid local minimum

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Stefan Ohrhallinger and Sudhir Mudur 75

Not Bmin? Insert Points, Manually

Not a solid local minimum undersampled

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Stefan Ohrhallinger and Sudhir Mudur 76

Not Bmin? Insert Points, Manually

+

Not a solid local minimum undersampled

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Stefan Ohrhallinger and Sudhir Mudur 77

Not Bmin? Insert Points, Manually

+ +

Not a solid local minimum undersampled

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Stefan Ohrhallinger and Sudhir Mudur 78

Robust To Noise

[MTSM10]

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Stefan Ohrhallinger and Sudhir Mudur 79

Robust To Noise

[MTSM10] Ours [OM13]: manifold+interpolating

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Stefan Ohrhallinger and Sudhir Mudur 80

Class Of Point Sets

C

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Stefan Ohrhallinger and Sudhir Mudur 81

Class Of Point Sets

C M Medial axis M

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Stefan Ohrhallinger and Sudhir Mudur 82

Class Of Point Sets

C M Medial axis M Local feature size f(x)=||x, M||

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Stefan Ohrhallinger and Sudhir Mudur 83

Class Of Point Sets

C M Medial axis M Local feature size f(x)=||x, M||

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Stefan Ohrhallinger and Sudhir Mudur 84

Class Of Point Sets

C M Medial axis M Local feature size f(x)=||x, M|| ε-sampling:

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Stefan Ohrhallinger and Sudhir Mudur 85

Class Of Point Sets

C M Medial axis M Local feature size f(x)=||x, M|| ε-sampling:

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Stefan Ohrhallinger and Sudhir Mudur 86

Class Of Point Sets

C M Medial axis M Local feature size f(x)=||x, M|| ε-sampling:

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Stefan Ohrhallinger and Sudhir Mudur 87

Class Of Point Sets

Theorem 2 BC0 reconstructs ε-sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609. C M

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Stefan Ohrhallinger and Sudhir Mudur 88

Class Of Point Sets

Theorem 2 BC0 reconstructs ε-sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609. C M

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Stefan Ohrhallinger and Sudhir Mudur 89

Class Of Point Sets

Theorem 2 BC0 reconstructs ε-sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609.

1 please see paper for references.

C M [ABE98]1

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Stefan Ohrhallinger and Sudhir Mudur 90

Class Of Point Sets

Theorem 2 BC0 reconstructs ε-sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609.

1 please see paper for references.

C M [ABE98]1 [DK99]1

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Stefan Ohrhallinger and Sudhir Mudur 91

Class Of Point Sets

Theorem 2 BC0 reconstructs ε-sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609.

1 please see paper for references.

C M BC0=Bmin [ABE98]1 [DK99]1

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Stefan Ohrhallinger and Sudhir Mudur 92

Class Of Point Sets

Theorem 2 BC0 reconstructs ε-sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609.

1 please see paper for references.

C M B=Bmin (conj) BC0=Bmin [ABE98]1 [DK99]1

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Stefan Ohrhallinger and Sudhir Mudur 93

Why Limited Non-Uniformity?

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Stefan Ohrhallinger and Sudhir Mudur 94

Why Limited Non-Uniformity?

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Stefan Ohrhallinger and Sudhir Mudur 95

Future Work (2D)

Local minimum → fill hole

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Stefan Ohrhallinger and Sudhir Mudur 96

Future Work (2D)

Local minimum → fill hole Multiply connected components

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Stefan Ohrhallinger and Sudhir Mudur 97

Future Work (2D)

Local minimum → fill hole Prove Bmin for ε<1 Multiply connected components

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Stefan Ohrhallinger and Sudhir Mudur 98

Future Work (2D)

Local minimum → fill hole Sampling: tighter bound Prove Bmin for ε<1 Multiply connected components ε<0.5

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Stefan Ohrhallinger and Sudhir Mudur 99

Future Work (2D)

Local minimum → fill hole Sampling: tighter bound Prove Bmin for ε<1 Multiply connected components Open curves (vs. sparse) ε<0.5

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Stefan Ohrhallinger and Sudhir Mudur 100

Future Work (3D)

See my talk at SMI'13 (July 11th), Bournemouth, UK

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Stefan Ohrhallinger and Sudhir Mudur 101

Contributions

C Source: http://sf.net/p/connect2dlib/

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Stefan Ohrhallinger and Sudhir Mudur 102

Contributions

C Source: http://sf.net/p/connect2dlib/

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Stefan Ohrhallinger and Sudhir Mudur 103

Contributions

MST C Source: http://sf.net/p/connect2dlib/

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Stefan Ohrhallinger and Sudhir Mudur 104

Contributions

MST Inflate as Sculpture Dual C Source: http://sf.net/p/connect2dlib/

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Stefan Ohrhallinger and Sudhir Mudur 105

Contributions

MST Inflate as Sculpture Dual Theorem 2 ε < 0.5, u < 1.609 Experimental evidence + Proof C Source: http://sf.net/p/connect2dlib/

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Stefan Ohrhallinger and Sudhir Mudur 106

Contributions

MST Inflate as Sculpture Dual Theorem 2 ε < 0.5, u < 1.609 Experimental evidence + Proof Noise robust C Source: http://sf.net/p/connect2dlib/

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Stefan Ohrhallinger and Sudhir Mudur 107

Contributions

MST Inflate as Sculpture Dual Theorem 2 ε < 0.5, u < 1.609 Experimental evidence + Proof Noise robust See SMI'13 C Source: http://sf.net/p/connect2dlib/