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

Start With A Minimum Set NP-hard P MST Vertex degree Stefan Ohrhallinger and Sudhir Mudur 49

Start With A Minimum Set NP-hard P MST Vertex degree Stefan Ohrhallinger and Sudhir Mudur 50

Start With A Minimum Set NP-hard? NP-hard P MST Vertex degree Stefan Ohrhallinger and Sudhir Mudur 51

Start With A Minimum Set NP-hard P MST Vertex degree Stefan Ohrhallinger and Sudhir Mudur 52

Start With A Minimum Set NP-hard ? P MST Vertex degree Stefan Ohrhallinger and Sudhir Mudur 53

Overview Of Our 'Connect2D' Algorithm Input Stefan Ohrhallinger and Sudhir Mudur 54

Overview Of Our 'Connect2D' Algorithm Input Stefan Ohrhallinger and Sudhir Mudur 55

Overview Of Our 'Connect2D' Algorithm Input Stefan Ohrhallinger and Sudhir Mudur 56

Overview Of Our 'Connect2D' Algorithm not manifold Input Stefan Ohrhallinger and Sudhir Mudur 57

Overview Of Our 'Connect2D' Algorithm Inflate min Δ|B| Input Stefan Ohrhallinger and Sudhir Mudur 58

Overview Of Our 'Connect2D' Algorithm Inflate min Δ|B| Input Stefan Ohrhallinger and Sudhir Mudur 59

Overview Of Our 'Connect2D' Algorithm Inflate Sculpture min Δ|B| min Δ|B| Input Stefan Ohrhallinger and Sudhir Mudur 60

Overview Of Our 'Connect2D' Algorithm Inflate Dual Sculpture min Δ|B| min Δ|B| Input Stefan Ohrhallinger and Sudhir Mudur 61

Overview Of Our 'Connect2D' Algorithm Inflate Dual Sculpture min Δ|B| min Δ|B| Input Theorem 1 : Our algorithm constructs B out in O(n log n) time. Stefan Ohrhallinger and Sudhir Mudur 62

Improved For Sparse Sampling Points Stefan Ohrhallinger and Sudhir Mudur 63

Improved For Sparse Sampling Points Gathan [DW01] Stefan Ohrhallinger and Sudhir Mudur 64

Improved For Sparse Sampling Points Gathan [DW01] Ours [OM13] Stefan Ohrhallinger and Sudhir Mudur 65

Large Sets: Easy 10k points Gathan [DW01] Stefan Ohrhallinger and Sudhir Mudur 66

Large Sets: Easy 10k points Gathan [DW01] Ours [OM13]: manifold Stefan Ohrhallinger and Sudhir Mudur 67

Challenge: Extremely Sparse Sampling Points Stefan Ohrhallinger and Sudhir Mudur 68

Challenge: Extremely Sparse Sampling Points Gathan [DW01] Stefan Ohrhallinger and Sudhir Mudur 69

Challenge: Extremely Sparse Sampling Points Gathan [DW01] Ours Stefan Ohrhallinger and Sudhir Mudur 70

Failure Cases [OM11] Stefan Ohrhallinger and Sudhir Mudur 71

Failure Cases local minimum [OM11] Ours [OM13] Stefan Ohrhallinger and Sudhir Mudur 72

Not Bmin? Insert Points, Manually Not a solid Stefan Ohrhallinger and Sudhir Mudur 73

Not Bmin? Insert Points, Manually local minimum Not a solid Stefan Ohrhallinger and Sudhir Mudur 74

Not Bmin? Insert Points, Manually local minimum undersampled Not a solid Stefan Ohrhallinger and Sudhir Mudur 75

Not Bmin? Insert Points, Manually local minimum + undersampled Not a solid Stefan Ohrhallinger and Sudhir Mudur 76

Not Bmin? Insert Points, Manually local minimum + + undersampled Not a solid Stefan Ohrhallinger and Sudhir Mudur 77

Robust To Noise [MTSM10] Stefan Ohrhallinger and Sudhir Mudur 78

Robust To Noise [MTSM10] Ours [OM13]: manifold+interpolating Stefan Ohrhallinger and Sudhir Mudur 79

Class Of Point Sets C Stefan Ohrhallinger and Sudhir Mudur 80

Class Of Point Sets C M Medial axis M Stefan Ohrhallinger and Sudhir Mudur 81

Class Of Point Sets C M Medial axis M Local feature size f(x)=||x, M|| Stefan Ohrhallinger and Sudhir Mudur 82

Class Of Point Sets C M Medial axis M Local feature size f(x)=||x, M|| Stefan Ohrhallinger and Sudhir Mudur 83

Class Of Point Sets C M Medial axis M Local feature size f(x)=||x, M|| ε -sampling: Stefan Ohrhallinger and Sudhir Mudur 84

Class Of Point Sets C M Medial axis M Local feature size f(x)=||x, M|| ε -sampling: Stefan Ohrhallinger and Sudhir Mudur 85

Class Of Point Sets C M Medial axis M Local feature size f(x)=||x, M|| ε -sampling: Stefan Ohrhallinger and Sudhir Mudur 86

Class Of Point Sets C M Theorem 2 BC 0 reconstructs ε -sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609. Stefan Ohrhallinger and Sudhir Mudur 87

Class Of Point Sets C M Theorem 2 BC 0 reconstructs ε -sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609. Stefan Ohrhallinger and Sudhir Mudur 88

Class Of Point Sets C M Theorem 2 [ABE98] 1 BC 0 reconstructs ε -sampled C from P with ε < 0.5 and a local non-uniformity u < 1.609. 1 please see paper for references. Stefan Ohrhallinger and Sudhir Mudur 89

Class Of Point Sets C M Theorem 2 [ABE98] 1 BC 0 reconstructs ε -sampled C from P with [DK99] 1 ε < 0.5 and a local non-uniformity u < 1.609. 1 please see paper for references. Stefan Ohrhallinger and Sudhir Mudur 90

Class Of Point Sets C M Theorem 2 [ABE98] 1 BC 0 reconstructs ε -sampled C from P with [DK99] 1 ε < 0.5 and a local non-uniformity u < 1.609. BC 0 =B min 1 please see paper for references. Stefan Ohrhallinger and Sudhir Mudur 91

Class Of Point Sets C M Theorem 2 [ABE98] 1 BC 0 reconstructs ε -sampled C from P with B=B min [DK99] 1 ε < 0.5 and a local non-uniformity u < 1.609. (conj) BC 0 =B min 1 please see paper for references. Stefan Ohrhallinger and Sudhir Mudur 92

Why Limited Non-Uniformity? Stefan Ohrhallinger and Sudhir Mudur 93

Why Limited Non-Uniformity? Stefan Ohrhallinger and Sudhir Mudur 94

Future Work (2D) Local minimum → fill hole Stefan Ohrhallinger and Sudhir Mudur 95

Future Work (2D) Local minimum → fill hole Multiply connected components Stefan Ohrhallinger and Sudhir Mudur 96

Future Work (2D) Prove B min for ε < 1 Local minimum → fill hole Multiply connected components Stefan Ohrhallinger and Sudhir Mudur 97

Future Work (2D) Prove B min for ε < 1 Local minimum → fill hole ε < 0.5 Sampling: tighter bound Multiply connected components Stefan Ohrhallinger and Sudhir Mudur 98

Future Work (2D) Prove B min for ε < 1 Local minimum → fill hole ε < 0.5 Sampling: tighter bound Multiply connected components Open curves (vs. sparse) Stefan Ohrhallinger and Sudhir Mudur 99

Future Work (3D) See my talk at SMI'13 (July 11th), Bournemouth, UK Stefan Ohrhallinger and Sudhir Mudur 100