SLIDE 1
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary
Reeb Graphs and Piecewise Linear Functions Koen Klaren Eindhoven - - PowerPoint PPT Presentation
Reeb Graphs and Piecewise Linear Functions Koen Klaren Eindhoven - - PowerPoint PPT Presentation
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Reeb Graphs and Piecewise Linear Functions Koen Klaren Eindhoven University of Technology Seminar Algorithms, 2018 Reeb graphs Piecewise linear functions Constructing
SLIDE 2
SLIDE 3
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Outline
1 Reeb graphs
Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
2 Piecewise linear functions 3 Constructing Reeb graphs
SLIDE 4
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Motivation for Reeb graphs
Make the structure of a function explicit by visualizing the evolution of the components of the level set
Reeb graph
SLIDE 5
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Motivation for Reeb graphs
Make the structure of a function explicit by visualizing the evolution of the components of the level set Accelerate extraction of isosurfaces (level sets)
Reeb graph Isosurface extraction
SLIDE 6
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Motivation for Reeb graphs
Make the structure of a function explicit by visualizing the evolution of the components of the level set Accelerate extraction of isosurfaces (level sets) Applications in medical imaging and other areas of science and engineering
Reeb graph Isosurface extraction
SLIDE 7
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Outline
1 Reeb graphs
Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
2 Piecewise linear functions 3 Constructing Reeb graphs
SLIDE 8
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of contours
Given a continuous map f : X → R, the level sets form a partition
- f topological space X
SLIDE 9
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of contours
SLIDE 10
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of contours
Given a continuous map f : X → R, the level sets form a partition
- f topological space X
Definition Two points x, y ∈ X are called equivalent if they belong to a common component in a level set of f
SLIDE 11
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of contours
SLIDE 12
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of contours
Given a continuous map f : X → R, the level sets form a partition
- f topological space X
Definition Two points x, y ∈ X are called equivalent if they belong to a common component in a level set of f Definition Defined equivalence classes are the contours of f
SLIDE 13
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of contours
SLIDE 14
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of Reeb graphs
Definition The Reeb graph of f is the set of contours R(f).
SLIDE 15
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of Reeb graphs
Definition The Reeb graph of f is the set of contours R(f).
SLIDE 16
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of Reeb graphs
We started with only the continuous map f. Where does the Reeb graph fit in? X R
f ψ π
SLIDE 17
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of Reeb graphs
Let ψ: X → R(f) be the function that maps a point x to the contour ψ(x) that contains it: X R R(f)
f ψ π
SLIDE 18
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of Reeb graphs
Let π: R(f) → R be the unique map whose composition with ψ is f: X R R(f)
f ψ π
π(ψ(x)) = f(x)
SLIDE 19
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Definition of Reeb graphs
This speeds up construction of a level set, f−1(a). We can use π−1(a) to find contours r ∈ R(f) that form the level set. X R R(f)
f ψ π
SLIDE 20
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Outline
1 Reeb graphs
Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
2 Piecewise linear functions 3 Constructing Reeb graphs
SLIDE 21
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Properties of Reeb graphs
Connected components
Function ψ: X → R(f) maps components to components
SLIDE 22
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Properties of Reeb graphs
Connected components
Function ψ: X → R(f) maps components to components Expressed in terms of Betti numbers, we get: β0(R(f)) = β0(X)
SLIDE 23
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Properties of Reeb graphs
1-dimensional connectivity
In some cases the Reeb graph reflects the 1-dimensional connectivity of the space
SLIDE 24
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Properties of Reeb graphs
1-dimensional connectivity
In some cases the Reeb graph reflects the 1-dimensional connectivity of the space A 1-cycle in R(f) is called a loop The preimage of a loop is non-contractible in X Different loops correspond to non-homologous 1-cycles in X
SLIDE 25
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Properties of Reeb graphs
1-dimensional connectivity
In some cases the Reeb graph reflects the 1-dimensional connectivity of the space A 1-cycle in R(f) is called a loop The preimage of a loop is non-contractible in X Different loops correspond to non-homologous 1-cycles in X Expressed in terms of Betti numbers, we get: β1(R(f)) ≤ β1(X)
SLIDE 26
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Properties of Reeb graphs
Contour tree
From these properties: β0(R(f)) = β0(X) β1(R(f)) ≤ β1(X) it follows that if X is contractible then R(f) is a tree, independent of the function f Because of this R(f) is sometimes called a contour tree
SLIDE 27
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Outline
1 Reeb graphs
Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
2 Piecewise linear functions 3 Constructing Reeb graphs
SLIDE 28
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Reeb graphs of Morse functions
More can be said if X = M is a manifold of dimension d ≥ 2 and f : M → R is a Morse function
SLIDE 29
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Reeb graphs of Morse functions
More can be said if X = M is a manifold of dimension d ≥ 2 and f : M → R is a Morse function Recall that each point r ∈ R(f) is the image of a contour in M Call r a node if it is the image of a critical point under ψ The other points form arcs that connect the nodes
SLIDE 30
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Reeb graphs of Morse functions
SLIDE 31
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Reeb graphs of Morse functions
More can be said if X = M is a manifold of dimension d ≥ 2 and f : M → R is a Morse function Recall that each point r ∈ R(f) is the image of a contour in M Call r a node if it is the image of a critical point under ψ The other points form arcs that connect the nodes By definition of Morse functions, this implies a bijection between the critical points of f and the nodes of R(f)
SLIDE 32
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
Minimum starts a contour and thus corresponds to a degree
- ne node
SLIDE 33
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
SLIDE 34
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
Minimum starts a contour and thus corresponds to a degree
- ne node
Index 1 saddle that merges two contours into one corresponds to a degree three node
SLIDE 35
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
SLIDE 36
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
Minimum starts a contour and thus corresponds to a degree
- ne node
Index 1 saddle that merges two contours into one corresponds to a degree three node Index d − 1 saddle that splits a contour into two corresponds to a degree three node
SLIDE 37
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
SLIDE 38
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
Minimum starts a contour and thus corresponds to a degree
- ne node
Index 1 saddle that merges two contours into one corresponds to a degree three node Index d − 1 saddle that splits a contour into two corresponds to a degree three node Maximum closes a contour and thus corresponds to a degree
- ne node
SLIDE 39
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
SLIDE 40
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definitions Properties of Reeb graphs Reeb graphs of Morse functions
Bijection between critical points and nodes
Minimum starts a contour and thus corresponds to a degree
- ne node
Index 1 saddle that merges two contours into one corresponds to a degree three node Index d − 1 saddle that splits a contour into two corresponds to a degree three node Maximum closes a contour and thus corresponds to a degree
- ne node
All other critical points correspond to nodes of degree two
SLIDE 41
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Outline
1 Reeb graphs 2 Piecewise linear functions
Motivation Definition of PL functions Lower star filtration PL Morse functions
3 Constructing Reeb graphs
SLIDE 42
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Motivation for piecewise linear (PL) functions
Functions are rarely smooth in practical situations
SLIDE 43
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Motivation for piecewise linear (PL) functions
Functions are rarely smooth in practical situations Instead we have non-smooth functions that approximate smooth ones
SLIDE 44
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Motivation for piecewise linear (PL) functions
Functions are rarely smooth in practical situations Instead we have non-smooth functions that approximate smooth ones Thus we introduce piecewise linear (PL) functions
SLIDE 45
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Outline
1 Reeb graphs 2 Piecewise linear functions
Motivation Definition of PL functions Lower star filtration PL Morse functions
3 Constructing Reeb graphs
SLIDE 46
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Definition of PL functions
Let K be a simplicial complex with real values specified at all
- vertices. Using linear extension over the simplices, we obtain a
piecewise linear (PL) function f : |K| → R.
SLIDE 47
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Genericity
Assume that f is generic, meaning that all vertices have distinct function values
SLIDE 48
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Genericity
Assume that f is generic, meaning that all vertices have distinct function values Now the vertices can be ordered by increasing function value: f(u1) < f(u2) < . . . < f(un)
SLIDE 49
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Genericity
Assume that f is generic, meaning that all vertices have distinct function values Now the vertices can be ordered by increasing function value: f(u1) < f(u2) < . . . < f(un) Let Ki be the full subcomplex defined by the first i vertices
SLIDE 50
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Outline
1 Reeb graphs 2 Piecewise linear functions
Motivation Definition of PL functions Lower star filtration PL Morse functions
3 Constructing Reeb graphs
SLIDE 51
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Lower star filtration
Recall that the star of a vertex ui is the set of cofaces of ui in K Definition The lower star is the subset of simplices in the star of ui for which ui is the vertex with maximum function value: St ui = {σ ∈ St ui | x ∈ σ ⇒ f(x) ≤ f(ui)}
SLIDE 52
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Lower star filtration
By assumption of genericity, each simplex has a unique maximum vertex and thus belongs to a unique lower star
SLIDE 53
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Lower star filtration
By assumption of genericity, each simplex has a unique maximum vertex and thus belongs to a unique lower star It follows that the lower stars partition K. Also, Ki is the union of the first i lower stars.
SLIDE 54
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Lower star filtration
By assumption of genericity, each simplex has a unique maximum vertex and thus belongs to a unique lower star It follows that the lower stars partition K. Also, Ki is the union of the first i lower stars. This nested sequence of subcomplexes is thus called the lower star filtration of f: ∅ = K0 ⊆ K1 ⊆ . . . ⊆ Kn = K
SLIDE 55
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Outline
1 Reeb graphs 2 Piecewise linear functions
Motivation Definition of PL functions Lower star filtration PL Morse functions
3 Constructing Reeb graphs
SLIDE 56
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
Lower link
Recall that the link of a vertex ui is the set of simplices in the closed star that do not belong to the star Definition The lower link is the set of simplices in the closed lower star that do not belong to the lower star: Lk ui = {σ ∈ Lk ui | x ∈ σ ⇒ f(x) < f(ui)}
SLIDE 57
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
PL critical points
As we go from Ki−1 to Ki we attach the closed lower star of ui, gluing it along the lower link to the complex Ki−1
SLIDE 58
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
PL critical points
As we go from Ki−1 to Ki we attach the closed lower star of ui, gluing it along the lower link to the complex Ki−1 Homology of the lower link is used to classify vertices
SLIDE 59
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
PL critical points
Regular
Call ui a PL regular vertex if its lower link is non-empty but homologically trivial
SLIDE 60
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
PL critical points
Regular Critical (q = 0)
Call ui a PL regular vertex if its lower link is non-empty but homologically trivial Call ui a simple PL critical vertex of index q if its lower link has the reduced homology of the (q − 1)-sphere
SLIDE 61
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
PL critical points
Regular Critical (q = 0) Critical (q = 1)
Call ui a PL regular vertex if its lower link is non-empty but homologically trivial Call ui a simple PL critical vertex of index q if its lower link has the reduced homology of the (q − 1)-sphere
SLIDE 62
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
PL critical points
Regular Critical (q = 0) Critical (q = 1) Critical (q = 2)
Call ui a PL regular vertex if its lower link is non-empty but homologically trivial Call ui a simple PL critical vertex of index q if its lower link has the reduced homology of the (q − 1)-sphere
SLIDE 63
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Motivation Definition of PL functions Lower star filtration PL Morse functions
PL Morse functions
Definition We call a PL function f : |K| → R on a manifold a PL Morse function if:
1 Each vertex is either PL regular or simple PL critical 2 The function values of the vertices are distinct
SLIDE 64
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Outline
1 Reeb graphs 2 Piecewise linear functions 3 Constructing Reeb graphs
Sketch of the algorithm Running time
SLIDE 65
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Constructing a Reeb graph
Problem: Construct a Reeb graph of a function on a 2-manifold. We assume M is triangulated and the function f : M → R is PL Morse.
SLIDE 66
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Constructing a Reeb graph
Problem: Construct a Reeb graph of a function on a 2-manifold. We assume M is triangulated and the function f : M → R is PL Morse. Remember we can order the vertices of M: f(u1) < f(u2) < . . . < f(un)
SLIDE 67
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Constructing a Reeb graph
Problem: Construct a Reeb graph of a function on a 2-manifold. We assume M is triangulated and the function f : M → R is PL Morse. Remember we can order the vertices of M: f(u1) < f(u2) < . . . < f(un) We can use a plane sweep algorithm!
SLIDE 68
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Constructing a Reeb graph
Problem: Construct a Reeb graph of a function on a 2-manifold. We assume M is triangulated and the function f : M → R is PL Morse. Remember we can order the vertices of M: f(u1) < f(u2) < . . . < f(un) We can use a plane sweep algorithm! Status is the set of contours at the current level Events take place at each of the vertices
SLIDE 69
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Event handling
Four cases:
1 ui is a minimum: add a degree one node to the Reeb graph.
Start a new arc associated with the new contour.
SLIDE 70
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Event handling
SLIDE 71
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Event handling
Four cases:
1 ui is a minimum: add a degree one node to the Reeb graph.
Start a new arc associated with the new contour.
2 ui is a regular vertex: update the associated contour in the
status.
SLIDE 72
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Event handling
SLIDE 73
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Event handling
Four cases:
1 ui is a minimum: add a degree one node to the Reeb graph.
Start a new arc associated with the new contour.
2 ui is a regular vertex: update the associated contour in the
status.
3 ui is a saddle: three different cases depending on the index.
Usually merge/split contours in the status and add a degree three node to the Reeb graph.
SLIDE 74
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Event handling
SLIDE 75
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Event handling
Four cases:
1 ui is a minimum: add a degree one node to the Reeb graph.
Start a new arc associated with the new contour.
2 ui is a regular vertex: update the associated contour in the
status.
3 ui is a saddle: three different cases depending on the index.
Usually merge/split contours in the status and add a degree three node to the Reeb graph.
4 ui is a maximum: Remove contour from the status and add a
degree one node to the Reeb graph.
SLIDE 76
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Event handling
SLIDE 77
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Outline
1 Reeb graphs 2 Piecewise linear functions 3 Constructing Reeb graphs
Sketch of the algorithm Running time
SLIDE 78
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Running time
Implement the contour status using cyclic lists in a balanced search tree
SLIDE 79
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Running time
Implement the contour status using cyclic lists in a balanced search tree This data structure supports all required operations in O(log2 m) time, where m is the number of edges in M
SLIDE 80
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary Sketch of the algorithm Running time
Running time
Implement the contour status using cyclic lists in a balanced search tree This data structure supports all required operations in O(log2 m) time, where m is the number of edges in M Thus the algorithm constructs a Reeb graph in O(m log2 m) time
SLIDE 81
Reeb graphs Piecewise linear functions Constructing Reeb graphs Summary