MEDIAN TRAJECTORIES Kevin Buchin Maike Buchin Marc van Kreveld - - PowerPoint PPT Presentation

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MEDIAN TRAJECTORIES Kevin Buchin Maike Buchin Marc van Kreveld - - PowerPoint PPT Presentation

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MEDIAN TRAJECTORIES Kevin Buchin Maike Buchin Marc van Kreveld Maarten L offler Rodrigo Silveira Carola Wenk Lionov Wiratma OVERVIEW OVERVIEW III Introduction OVERVIEW III Introduction Motivation Data representatives

slide-1
SLIDE 1

Kevin Buchin Maike Buchin Marc van Kreveld Maarten L¨

  • ffler

Rodrigo Silveira Carola Wenk Lionov Wiratma MEDIAN TRAJECTORIES

slide-2
SLIDE 2

OVERVIEW

slide-3
SLIDE 3

OVERVIEW

  • III Introduction
slide-4
SLIDE 4

OVERVIEW

  • III Introduction
  • Motivation
  • Data representatives
slide-5
SLIDE 5

OVERVIEW

  • III Introduction
  • Motivation
  • Data representatives
  • III Defining the median
slide-6
SLIDE 6

OVERVIEW

  • III Introduction
  • Motivation
  • Data representatives
  • III Defining the median
  • Simple median
  • Homotopic median
slide-7
SLIDE 7

OVERVIEW

  • III Introduction
  • Motivation
  • Data representatives
  • III Defining the median
  • Simple median
  • Homotopic median
  • III Results
slide-8
SLIDE 8

OVERVIEW

  • III Introduction
  • Motivation
  • Data representatives
  • III Defining the median
  • Simple median
  • Homotopic median
  • III Results
  • Algorithms
  • Implementation
slide-9
SLIDE 9

OVERVIEW

  • III Introduction
  • Motivation
  • Data representatives
  • III Defining the median
  • Simple median
  • Homotopic median
  • III Results
  • Algorithms
  • Implementation
  • Conclusion
slide-10
SLIDE 10

I INTRODUCTION

slide-11
SLIDE 11

MOTIVATION

slide-12
SLIDE 12

MOTIVATION

  • Problem
slide-13
SLIDE 13

MOTIVATION

  • Planar domain
  • Problem
slide-14
SLIDE 14

MOTIVATION

  • Planar domain
  • Big collection of trajectories
  • Problem
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SLIDE 15

MOTIVATION

  • Planar domain
  • Big collection of trajectories
  • Problem
  • Build catalogue of ‘common’ trajectories
slide-16
SLIDE 16

MOTIVATION

  • Planar domain
  • Big collection of trajectories
  • Problem
  • Build catalogue of ‘common’ trajectories
  • Solution
slide-17
SLIDE 17

MOTIVATION

  • Planar domain
  • Big collection of trajectories
  • Problem
  • Build catalogue of ‘common’ trajectories
  • Solution
  • Cluster the trajectories
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SLIDE 18

MOTIVATION

  • Planar domain
  • Big collection of trajectories
  • Problem
  • Build catalogue of ‘common’ trajectories
  • Solution
  • Cluster the trajectories
  • Pick a good representative for each cluster
slide-19
SLIDE 19

MOTIVATION

  • Planar domain
  • Big collection of trajectories
  • Problem
  • Build catalogue of ‘common’ trajectories
  • Solution
  • Cluster the trajectories
  • Pick a good representative for each cluster
slide-20
SLIDE 20

MOTIVATION

  • Planar domain
  • Big collection of trajectories
  • Problem
  • Build catalogue of ‘common’ trajectories
  • Solution
  • Cluster the trajectories
  • Pick a good representative for each cluster
  • ... how did we do that second step?
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SLIDE 21

PROBLEM STATEMENT (IN MILDLY VAGUE TERMS)

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SLIDE 22

PROBLEM STATEMENT (IN MILDLY VAGUE TERMS)

  • Input: a set of ‘similar’ trajectories
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SLIDE 23

PROBLEM STATEMENT (IN MILDLY VAGUE TERMS)

  • Same start point s and end point t
  • Input: a set of ‘similar’ trajectories
slide-24
SLIDE 24

PROBLEM STATEMENT (IN MILDLY VAGUE TERMS)

  • Same start point s and end point t
  • Sort of the same shape
  • Input: a set of ‘similar’ trajectories
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SLIDE 25

PROBLEM STATEMENT (IN MILDLY VAGUE TERMS)

  • Same start point s and end point t
  • Sort of the same shape
  • Input: a set of ‘similar’ trajectories
  • Output: a representative trajectory
slide-26
SLIDE 26

PROBLEM STATEMENT (IN MILDLY VAGUE TERMS)

  • Same start point s and end point t
  • Sort of the same shape
  • Input: a set of ‘similar’ trajectories
  • Output: a representative trajectory
  • Should also go from s to t
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SLIDE 27

PROBLEM STATEMENT (IN MILDLY VAGUE TERMS)

  • Same start point s and end point t
  • Sort of the same shape
  • Input: a set of ‘similar’ trajectories
  • Output: a representative trajectory
  • Should also go from s to t
  • Shape should represent the whole set of

input trajectories

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SLIDE 28

PROBLEM STATEMENT (IN MILDLY VAGUE TERMS)

  • Same start point s and end point t
  • Sort of the same shape
  • Input: a set of ‘similar’ trajectories
  • Output: a representative trajectory
  • Should also go from s to t
  • Shape should represent the whole set of

input trajectories

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SLIDE 29

DATA REPRESENTATIVES

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SLIDE 30

DATA REPRESENTATIVES

  • Set of numbers: {1, 1, 2, 5, 6}
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SLIDE 31

DATA REPRESENTATIVES

  • Set of numbers: {1, 1, 2, 5, 6}
  • Average/mean: 3
  • Median: 2
  • Modal: 1
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SLIDE 32

DATA REPRESENTATIVES

  • Set of numbers: {1, 1, 2, 5, 6}
  • Average/mean: 3
  • Median: 2
  • Modal: 1
  • Set of points in the plane
slide-33
SLIDE 33

DATA REPRESENTATIVES

  • Set of numbers: {1, 1, 2, 5, 6}
  • Average/mean: 3
  • Median: 2
  • Modal: 1
  • Set of points in the plane
  • Average/mean of x- and y-coordinates
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SLIDE 34

DATA REPRESENTATIVES

  • Set of numbers: {1, 1, 2, 5, 6}
  • Average/mean: 3
  • Median: 2
  • Modal: 1
  • Set of points in the plane
  • Average/mean of x- and y-coordinates
  • Median of x- and y-coordinates
slide-35
SLIDE 35

DATA REPRESENTATIVES

  • Set of numbers: {1, 1, 2, 5, 6}
  • Average/mean: 3
  • Median: 2
  • Modal: 1
  • Set of points in the plane
  • Average/mean of x- and y-coordinates
  • Median of x- and y-coordinates
  • Centre of smallest enclosing circle
slide-36
SLIDE 36

DATA REPRESENTATIVES

  • Set of numbers: {1, 1, 2, 5, 6}
  • Average/mean: 3
  • Median: 2
  • Modal: 1
  • Set of points in the plane
  • Average/mean of x- and y-coordinates
  • Median of x- and y-coordinates
  • Centre of smallest enclosing circle
  • Pick representative from input?
slide-37
SLIDE 37

DATA REPRESENTATIVES

  • Set of numbers: {1, 1, 2, 5, 6}
  • Average/mean: 3
  • Median: 2
  • Modal: 1
  • Set of points in the plane
  • Average/mean of x- and y-coordinates
  • Median of x- and y-coordinates
  • Centre of smallest enclosing circle
  • Pick representative from input?
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SLIDE 38

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

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SLIDE 39

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

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SLIDE 40

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
slide-41
SLIDE 41

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
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SLIDE 42

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
slide-43
SLIDE 43

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
slide-44
SLIDE 44

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
slide-45
SLIDE 45

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
  • Yes?
slide-46
SLIDE 46

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
  • Yes?
slide-47
SLIDE 47

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
  • Yes?
  • Which trajectory do we pick?
slide-48
SLIDE 48

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
  • Yes?
  • Which trajectory do we pick?
slide-49
SLIDE 49

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
  • Yes?
  • Which trajectory do we pick?
  • There may not be any good representative!
slide-50
SLIDE 50

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
  • Yes?
  • Which trajectory do we pick?
  • There may not be any good representative!
  • Use pieces of different trajectories
slide-51
SLIDE 51

TRAJECTORIES: PICK REPRE– SENTATIVE FROM INPUT?

  • No?
  • Parameterised mean trajectory
  • May interfere with environment!
  • Yes?
  • Which trajectory do we pick?
  • There may not be any good representative!
  • Use pieces of different trajectories
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SLIDE 52

II DEFINING THE MEDIAN

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SLIDE 53

SIMPLE MEDIAN

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SLIDE 54

SIMPLE MEDIAN

  • For x-monotone trajectories...
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SLIDE 55

SIMPLE MEDIAN

  • For x-monotone trajectories...
  • Take the median at each x-coordinate
slide-56
SLIDE 56

SIMPLE MEDIAN

  • For x-monotone trajectories...
  • Take the median at each x-coordinate
slide-57
SLIDE 57

SIMPLE MEDIAN

  • For x-monotone trajectories...
  • Take the median at each x-coordinate
  • Result: the n/2-level
slide-58
SLIDE 58

SIMPLE MEDIAN

  • For x-monotone trajectories...
  • Take the median at each x-coordinate
  • Result: the n/2-level
  • Start in the middle, switch at each crossing
slide-59
SLIDE 59

SIMPLE MEDIAN

  • For x-monotone trajectories...
  • Take the median at each x-coordinate
  • Result: the n/2-level
  • Start in the middle, switch at each crossing
  • For arbitrary trajectories...
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SLIDE 60

SIMPLE MEDIAN

  • For x-monotone trajectories...
  • Take the median at each x-coordinate
  • Result: the n/2-level
  • Start in the middle, switch at each crossing
  • For arbitrary trajectories...
  • Why not do the same thing?
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SLIDE 61

SIMPLE MEDIAN

  • For x-monotone trajectories...
  • Take the median at each x-coordinate
  • Result: the n/2-level
  • Start in the middle, switch at each crossing
  • For arbitrary trajectories...
  • Why not do the same thing?
  • We call this the simple median of a

set of curves

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SLIDE 62

SHORTCUTS

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SLIDE 63

SHORTCUTS

  • Problem
slide-64
SLIDE 64

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

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SLIDE 65

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

slide-66
SLIDE 66

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

slide-67
SLIDE 67

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

  • Solution
slide-68
SLIDE 68

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

  • Solution
  • Plant a tree
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SLIDE 69

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

  • Solution
  • Place a pole
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SLIDE 70

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

  • Solution
  • Place a pole
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SLIDE 71

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

  • Solution
  • Place a pole
  • Require the median to go around it
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SLIDE 72

SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

  • Solution
  • Place a pole
  • Require the median to go around it
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SLIDE 73
  • ... how do we steer the median

correctly around these poles? SHORTCUTS

  • Problem
  • Simple median may miss large parts of the

input trajectories

  • Solution
  • Place a pole
  • Require the median to go around it
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SLIDE 74

HOMOTOPY

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SLIDE 75

HOMOTOPY

  • Ingredients
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SLIDE 76

HOMOTOPY

  • Ingredients
  • One punctured plane
slide-77
SLIDE 77

HOMOTOPY

  • Ingredients
  • One punctured plane
slide-78
SLIDE 78

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
slide-79
SLIDE 79

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
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SLIDE 80

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
slide-81
SLIDE 81

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
slide-82
SLIDE 82

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

slide-83
SLIDE 83

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

  • ... the concatenation is a closed curve that

can be contracted to a point

slide-84
SLIDE 84

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

  • ... the concatenation is a closed curve that

can be contracted to a point

slide-85
SLIDE 85

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

  • ... the concatenation is a closed curve that

can be contracted to a point

slide-86
SLIDE 86

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

  • ... the concatenation is a closed curve that

can be contracted to a point

slide-87
SLIDE 87

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

  • ... the concatenation is a closed curve that

can be contracted to a point

slide-88
SLIDE 88

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

  • ... the concatenation is a closed curve that

can be contracted to a point

slide-89
SLIDE 89

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

  • ... the concatenation is a closed curve that

can be contracted to a point

slide-90
SLIDE 90

HOMOTOPY

  • Ingredients
  • One punctured plane
  • Two points s and t in the plane
  • Two continuous curves from s to t
  • The curves are homotopic if ...
  • ... one can be smoothly transformed into

the other

  • ... the concatenation is a closed curve that

can be contracted to a point

slide-91
SLIDE 91

COVERING SPACE

slide-92
SLIDE 92

COVERING SPACE

  • Consider a point in a space E
slide-93
SLIDE 93

COVERING SPACE

  • Consider a point in a space E
slide-94
SLIDE 94

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

slide-95
SLIDE 95

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

slide-96
SLIDE 96

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

slide-97
SLIDE 97

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

slide-98
SLIDE 98

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

  • The resulting space E′ is called the

covering space of E

slide-99
SLIDE 99

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

  • The resulting space E′ is called the

covering space of E

slide-100
SLIDE 100

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

  • The resulting space E′ is called the

covering space of E

slide-101
SLIDE 101

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

  • The resulting space E′ is called the

covering space of E

slide-102
SLIDE 102

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

  • The resulting space E′ is called the

covering space of E

slide-103
SLIDE 103

COVERING SPACE

  • Consider a point in a space E
  • Make a copy of the point for each

homotopically different way to reach it

  • The resulting space E′ is called the

covering space of E

  • E′ is simply connected
slide-104
SLIDE 104

HOMOTOPIC MEDIAN

slide-105
SLIDE 105

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
slide-106
SLIDE 106

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
slide-107
SLIDE 107

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
slide-108
SLIDE 108

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
slide-109
SLIDE 109

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
slide-110
SLIDE 110

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
slide-111
SLIDE 111

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
slide-112
SLIDE 112

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
slide-113
SLIDE 113

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
  • The homotopic median is just the simple

median in the covering space

slide-114
SLIDE 114

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
  • The homotopic median is just the simple

median in the covering space

slide-115
SLIDE 115

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
  • The homotopic median is just the simple

median in the covering space

  • For non-homotopic trajectories...
slide-116
SLIDE 116

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
  • The homotopic median is just the simple

median in the covering space

  • For non-homotopic trajectories...
slide-117
SLIDE 117

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
  • The homotopic median is just the simple

median in the covering space

  • For non-homotopic trajectories...
  • Endpoint t is not the same!
slide-118
SLIDE 118

HOMOTOPIC MEDIAN

  • For homotopic trajectories...
  • Lift the trajectories into the covering space
  • Ignore crossings that are no longer there
  • The homotopic median is just the simple

median in the covering space

  • For non-homotopic trajectories...
  • Endpoint t is not the same!
  • It doesn’t work
slide-119
SLIDE 119

PLACING POLES

slide-120
SLIDE 120

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

slide-121
SLIDE 121

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

slide-122
SLIDE 122

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Place poles in...
slide-123
SLIDE 123

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
slide-124
SLIDE 124

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
slide-125
SLIDE 125

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
slide-126
SLIDE 126

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
slide-127
SLIDE 127

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
  • Both at the same time?
slide-128
SLIDE 128

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
  • Both at the same time?
slide-129
SLIDE 129

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
  • Both at the same time?
  • Not uniquely defined!
slide-130
SLIDE 130

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
  • Both at the same time?
  • Not uniquely defined!
slide-131
SLIDE 131

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
  • Both at the same time?
  • Not uniquely defined!
slide-132
SLIDE 132

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
  • Both at the same time?
  • Not uniquely defined!
slide-133
SLIDE 133

PLACING POLES

  • Given a set of trajectories, can we

compute a reasonable set of poles?

  • Big faces?
  • Place poles in...
  • Faces where they preserve the homotopy?
  • Both at the same time?
  • Not uniquely defined!
slide-134
SLIDE 134

III RESULTS

slide-135
SLIDE 135

COMPUTING THE MEDIAN

slide-136
SLIDE 136

COMPUTING THE MEDIAN

  • n: Number of input vertices
  • h: Number of input poles
  • A: Complexity of arrangement of input
  • k: Complexity of output trajectory
slide-137
SLIDE 137

COMPUTING THE MEDIAN

  • Simple median
  • O(n2 log n) time
  • O((n + k)α(n) log n) time
  • n: Number of input vertices
  • h: Number of input poles
  • A: Complexity of arrangement of input
  • k: Complexity of output trajectory
slide-138
SLIDE 138

COMPUTING THE MEDIAN

  • Simple median
  • O(n2 log n) time
  • Homotopic median
  • O((n

√ h + k)α(n) log n + h1+ε + A) time

  • O(n2+ε) time
  • O((n + k)α(n) log n) time
  • n: Number of input vertices
  • h: Number of input poles
  • A: Complexity of arrangement of input
  • k: Complexity of output trajectory
slide-139
SLIDE 139

EXPERIMENTAL SETUP

slide-140
SLIDE 140

EXPERIMENTAL SETUP

  • Trajectory generator
slide-141
SLIDE 141

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-142
SLIDE 142

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-143
SLIDE 143

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-144
SLIDE 144

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-145
SLIDE 145

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-146
SLIDE 146

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-147
SLIDE 147

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-148
SLIDE 148

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-149
SLIDE 149

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-150
SLIDE 150

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-151
SLIDE 151

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-152
SLIDE 152

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-153
SLIDE 153

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
slide-154
SLIDE 154

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
slide-155
SLIDE 155

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
slide-156
SLIDE 156

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
slide-157
SLIDE 157

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
slide-158
SLIDE 158

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
  • Discards non-homotopic trajectories
slide-159
SLIDE 159

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
  • Discards non-homotopic trajectories
slide-160
SLIDE 160

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
  • Discards non-homotopic trajectories
  • Measures of interest
slide-161
SLIDE 161

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
  • Discards non-homotopic trajectories
  • Measures of interest
slide-162
SLIDE 162

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
  • Discards non-homotopic trajectories
  • Measures of interest
  • Angular change
slide-163
SLIDE 163

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
  • Discards non-homotopic trajectories
  • Measures of interest
  • Angular change
  • Complexity
slide-164
SLIDE 164

EXPERIMENTAL SETUP

  • Trajectory generator
  • Random walk towards a series of waypoints
  • Simple or self-intersecting trajectories
  • Pole generator
  • Places poles in faces that are large enough
  • Discards non-homotopic trajectories
  • Measures of interest
  • Angular change
  • Complexity
  • Length
slide-165
SLIDE 165

EXPERIMENTAL RESULTS

slide-166
SLIDE 166

EXPERIMENTAL RESULTS

  • Complexity
  • No self-intersections
  • Self-intersections

S H

345% 319% 343% 207%

slide-167
SLIDE 167

EXPERIMENTAL RESULTS

  • Complexity
  • Length
  • No self-intersections
  • Self-intersections
  • No self-intersections
  • Self-intersections

S H

345% 319% 343% 207% 96% 99% 96% 51%

slide-168
SLIDE 168

EXPERIMENTAL RESULTS

  • Complexity
  • Length
  • Angular change
  • No self-intersections
  • Self-intersections
  • No self-intersections
  • Self-intersections
  • No self-intersections
  • Self-intersections

S H

345% 319% 343% 207% 96% 99% 96% 51% 539% 468% 466% 381%

slide-169
SLIDE 169

CONCLUSION

slide-170
SLIDE 170

CONCLUSIONS

slide-171
SLIDE 171

CONCLUSIONS

  • This work
slide-172
SLIDE 172

CONCLUSIONS

  • This work
  • Two definitions for median trajectory
  • Efficient algorithms for computing them
  • Quantitative evaluation on generated data
slide-173
SLIDE 173

CONCLUSIONS

  • This work
  • Future work
  • Two definitions for median trajectory
  • Efficient algorithms for computing them
  • Quantitative evaluation on generated data
slide-174
SLIDE 174

CONCLUSIONS

  • This work
  • Future work
  • More intelligent automatic pole placement?
  • Evaluation on real-world data?
  • Understand better what makes a us accept

a certain curve as a good median

  • Two definitions for median trajectory
  • Efficient algorithms for computing them
  • Quantitative evaluation on generated data
slide-175
SLIDE 175

THANK YOU!