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The edit distance for Reeb graphs of surfaces
Barbara Di Fabio1,3, Claudia Landi2,3
1Dipartimento di Matematica, Universita di Bologna2Dipartimento di Scienze e Metodi dell’Ingegneria, Universita di Modena e Reggio
Emilia3ARCES, Universita di Bologna
[email protected], [email protected]
Discrete, Computational and Algebraic TopologyNovember 10-14, 2014
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Outline
• Background on Reeb graphs
• State-of-the-art in Reeb graphs comparison
• Edit Distance between Reeb graphs of surfaces◦ combinatorial definition;◦ stability property;◦ optimality.
• Relationships with other stable metrics
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Background on Reeb graphs
Definition
Let X be a topological space and f : X → R a continuous function.For every p,q ∈ X , p ∼ q whenever p,q belong to the same connectedcomponent of f −1(f (p)). The quotient space X/∼f is known as theReeb graph associated with f .
[Reeb, 1946]: If f : M → R
is a simple Morse functionthen Rf = M /∼f is a finitesimplicial complex of dimen-sion 1.
[Shinagawa-Kunii-Kergosien, 1991]: Surface coding based on Morsetheory.3 of 18
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State-of-the-art in Reeb graphs comparison
[Hilaga-Shinagawa-Kohmura-Kunii, 2001]: Similarity betweenpolyhedral models is calculated by comparing Multiresolutional ReebGraphs constructed based on geodesic distance.
• Define similarity sim(P ,Q) between two nodes P ,Q weighted ontheir attributes
• Nodes with maximal similarity are paired according to rulesintroduced to ensure that topological consistency is preserved whenmatching nodes.
• The similarity between two MRGs is the sum of all nodesimilarities:
SIM(R ,S) = ∑m∈R,n∈S
sim(m,n)
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State-of-the-art in Reeb graphs comparison
[Biasotti-Marini-Spagnuolo-Falcidieno, 2006]: Comparison ofExtended Reeb Graphs is based on a relaxed version of the notion ofbest common subgraph.
• A distance function d between two nodes v1 and v2 involves nodeand edge attributes.
• The distance measure between two graphs G1 and G2 is defined by
D(G1,G2) = 1− ∑v∈G
(1−d(ψ1(v),ψ2(v)))
max(|G1|, |G2|)where G is the common sub-graph between G1 and G2, and ψ1
and ψ2 are the sub-graph isomorphisms from G to G1 and from Gto G2.
• Heuristics are used to improve quality of the results andcomputational time
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State-of-the-art in Reeb graphs comparison
[Di Fabio-L. 2012]: Edit distance for Reeb graphs of curves endowedwith simple Morse functions
v1v1v1v1
v2 v3
v4v4v4v4
v5v5
v6v6
v7v7v7v8v8v8
u1 u1 u1
u2u3
u4 u4 u4
u5 u5
u6 u6
(B)(B)
(D) (D)
(D)(D)(D) (R)
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State-of-the-art in Reeb graphs comparison
[Di Fabio-L. 2012]: Edit distance for Reeb graphs of curves endowedwith simple Morse functions
v1v1v1v1
v2 v3
v4v4v4v4
v5v5
v6v6
v7v7v7v8v8v8
u1 u1 u1
u2u3
u4 u4 u4
u5 u5
u6 u6
(B)(B)
(D) (D)
(D)(D)(D) (R)
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State-of-the-art in Reeb graphs comparison
[Di Fabio-L. 2012]: Edit distance for Reeb graphs of curves endowedwith simple Morse functions
v1v1v1v1
v2 v3
v4v4v4v4
v5v5
v6v6
v7v7v7v8v8v8
u1 u1 u1
u2u3
u4 u4 u4
u5 u5
u6 u6
(B)(B)
(D) (D)
(D)(D)(D) (R)
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State-of-the-art in Reeb graphs comparison
[Di Fabio-L. 2012]: Edit distance for Reeb graphs of curves endowedwith simple Morse functions
v1v1v1v1
v2 v3
v4v4v4v4
v5v5
v6v6
v7v7v7v8v8v8
u1 u1 u1
u2u3
u4 u4 u4
u5 u5
u6 u6
(B)(B)
(D) (D)
(D)(D)(D) (R)
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State-of-the-art in Reeb graphs comparison
[Di Fabio-L. 2012]: Edit distance for Reeb graphs of curves endowedwith simple Morse functions
v1v1v1v1
v2 v3
v4v4v4v4
v5v5
v6v6
v7v7v7v8v8v8
u1 u1 u1
u2u3
u4 u4 u4
u5 u5
u6 u6
(B)(B)
(D) (D)
(D)(D)(D) (R)
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State-of-the-art in Reeb graphs comparison
[Di Fabio-L. 2012]: Edit distance for Reeb graphs of curves endowedwith simple Morse functions
v1v1v1v1
v2 v3
v4v4v4v4
v5v5
v6v6
v7v7v7v8v8v8
u1 u1 u1
u2u3
u4 u4 u4
u5 u5
u6 u6
(B)(B)
(D) (D)
(D)(D)(D) (R)
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State-of-the-art in Reeb graphs comparison
[Di Fabio-L. 2012]: Edit distance for Reeb graphs of curves endowedwith simple Morse functions
v1v1v1v1
v2 v3
v4v4v4v4
v5v5
v6v6
v7v7v7v8v8v8
u1 u1 u1
u2u3
u4 u4 u4
u5 u5
u6 u6
(B)(B)
(D) (D)
(D)(D)(D) (R)
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State-of-the-art in Reeb graphs comparison
[Bauer-Ge-Wang, 2014]: Functional distorsion distance
• Compares Reeb graphs Rf and Rg as topological spaces
• measures the minimum distortion in the values of f and g inducedby maps Φ : Rf → Rg and Ψ : Rg → Rf
• stability property for tame functions on the same space
• more discriminative than the bottleneck distance
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Edit distance for Reeb graphs of surfaces
• M is a connected, closed, orientable, smooth surface of genus g;
• f : M → R is a simple Morse function;
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Edit distance for Reeb graphs of surfaces
• M is a connected, closed, orientable, smooth surface of genus g;
• f : M → R is a simple Morse function;
• there is a bijective correspondence between critical points of f andvertices of Γf .
M Γf
v1v2
v3v4v5
v6v7
v8v9
v10f
a1a2a3a4a5
a6a7a8a9a10
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Edit distance for Reeb graphs of surfaces
• M is a connected, closed, orientable, smooth surface of genus g;
• f : M → R is a simple Morse function;
• each v ∈ V (Γf ) is equipped with the value of f at thecorresponding critical point.
M (Γf , �f )
a1a2
a3a4a5
a6a7
a8a9
a10f
a1a2a3a4a5
a6a7a8a9a10
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Elementary deformations, inverses, and their costs
�f (v1)�f (v1)�f (v1)�f (v1)
�g (u1)
�g (u1)�g (u2)
�g (u2)
�f (v2)�f (v2)�f (v2)�f (v2)
(B)(B)
(D)(D)
• Birth (B):
c(T ) =|�g (u1)− �g (u2)|
2.
• Death (D):
c(T ) =|�f (u1)− �f (u2)|
2.
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Elementary deformations, inverses, and their costs
�f (v1)
�g (u1)�g (u2)
�f (v2)
�f (u1)�f (u2)
�f (v3) �g (v3)�g (v4)�f (v4)
�g (v1)�g (v2)
�f (v5) �g (v5)
�g (v6)�f (v6)
(R)
• Relabeling (R):
c(T ) = maxv∈V (Γf )
|�f (v)− �g (v)|.
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Elementary deformations, inverses, and their costs
�f (v1)
�f (v1)�f (v1)
�f (v1)
�g (u1) �g (u1)�g (u2)
�g (u2)�f (u1)
�f (u1)
�f (u2)�f (u2)
�f (v4)
�f (v4)�f (v4)
�f (v4)�f (v5)
�f (v5)�f (v5)
�f (v5)�f (v6)
�f (v6)�f (v6)
�f (v6)
(K1)(K1)
�f (v1)�f (v1)
�g (u1)
�g (u2)
�f (v2)�f (v2)
�f (u1)
�f (u2)
�f (v3)�f (v3)�f (v4)�f (v4)
(K2)
(K3)
• (Ki ), with i = 1,2,3:
c(T ) = max{|�f (u1)− �g (u1)|, |�f (u2)− �g (u2)|}.9 of 18
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Deformations, inverses, and their costs
• A deformation of (Γf , �f ) is a finite ordered sequenceT = (T1,T2, . . . ,Tr ) of elementary deformations such that Ti is anelementary deformation of Ti−1Ti−2 · · ·T1(Γf , �f ) for everyi = 1, . . . , r .
• c(T ) =r
∑i=1
c(Ti).
• The inverse deformation of T is T−1 = (T−1r , . . . ,T−1
1 ).Clearly, T−1(Γg , �g ) = T−1
1 · · ·T−1r (Γg , �g )� (Γf , �f ), and
c(T−1) = c(T ).
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Connecting Reeb graphs by deformations
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Connecting Reeb graphs by deformations
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Connecting Reeb graphs by deformations
(K3)
T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K3)
T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K3) (D)
T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K3) (D)
T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K3) (K3)(D)
T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K3) (K3)(D)
T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K3) (K3)(D)
(D)
T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K3) (K3)(D)
(D)
T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K1)
(K3) (K3)(D)
(D)
T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K1)
(K3) (K3)(D)
(D)
T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(K1)
(K3) (K3)
(K3)
(D)
(D)
T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(R)
(K1)
(K3) (K3)
(K3)
(D)
(D)
T4T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(R)
(K1)(K3)
(K3) (K3)(D)
(D)
T4T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(R)
(K1)
(K2)(K3) (K3)
(K3)
(D)
(D)
T5T4T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(R)
(K1)
(K2)(K3) (K3)
(K3)
(D)
(D)
T5T4T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(R)
(K1)
(K1)(K2)(K3) (K3)
(K3)
(D)
(D)
T6T5T4T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(R)
(K1)
(K1)(K2)(K3) (K3)
(K3)
(D)
(D)
T6T5T4T3T2T1(Γf , �f )
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Connecting Reeb graphs by deformations
(R)
(K1)
(K1)(K2)(K3) (K3)
(K3)
(D)
(D)
(B)
T7T6T5T4T3T2T1(Γf , �f ) = (Γg , �g )
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Connecting Reeb graphs by deformations
(Γf ,�f ) (Γg ,�g )
(R)
(K1)
(K1)(K2)(K3) (K3)
(K3)
(D)
(D)
(B)
⇒ T = (T1, . . . ,T7) ∈ T ((Γf , �f ),(Γg , �g ))
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The edit distance
Definition
For every two labeled Reeb graphs (Γf , �f ) and (Γg , �g ), we set
d((Γf , �f ),(Γg , �g )) = infT∈T ((Γf ,�f ),(Γg ,�g ))
c(T ).
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The edit distance
Definition
For every two labeled Reeb graphs (Γf , �f ) and (Γg , �g ), we set
d((Γf , �f ),(Γg , �g )) = infT∈T ((Γf ,�f ),(Γg ,�g ))
c(T ).
Definition
(Γf , �f )∼= (Γg , �g ), if there exists an edge-preserving bijectionΦ : V (Γf )→ V (Γg ) such that �f (v) = �g (Φ(v)) for all v ∈ V (Γf ).
Theorem
d is a pseudo-metric on isomorphism classes of labeled Reeb graphs.
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Stability property
Theorem
d((Γf , �f ),(Γg , �g ))≤ ‖f −g‖∞.
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Stability property
Theorem
d((Γf , �f ),(Γg , �g ))≤ ‖f −g‖∞.
An example in which d((Γf , �f ),(Γg , �g ))≤ a
2:
c1+ ac1
c2+ ac2
c3+ ac3
b2 b2
b1 b1
(Γf , �f ) (Γg , �g )f g
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Stability property (sketch of the proof)
F 0
F 1α
F 1β
Let F = C∞(M ,R) = F 0∪F 1α ∪F 1
β ∪ . . ., with• F 0 = simple Morse functions;• F 1
α = simple functions with exactly one degenerate critical point;• F 1
β =Morse functions with exactly one complicate point.14 of 18
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Stability property (sketch of the proof)
F 0
F 1α
F 1β
f
g
Let f ,g ∈ F 0. We want to find the relationship betweend((Γf , �f ),(Γg , �g )) and ‖f −g‖∞.
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Stability property (sketch of the proof)
F 0
F 1α
F 1β
f
g
f1
g1
There exist f1,g1 ∈ F 0 arbitrarily near to f ,g , resp., for which thepath h(λ ) = (1−λ )f1+λg1, λ ∈ [0,1], is such that• h(λ ) belongs to F 0∪F 1 for every λ ∈ [0,1];• h(λ ) is transversal to F 1.14 of 18
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Stability property (sketch of the proof)
F 0
F 1α
F 1β
f
g
f1
g1
h1
h2
A linear path between two functions h1, h2 in the same connectedcomponent of F 0 corresponds to deformations of type (R) with costless than ‖h1−h2‖∞.14 of 18
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Stability property (sketch of the proof)
F 0
F 1α
F 1β
f
g
f1
g1
hh1
h2
A linear path between two functions h1, h2 across F 1α corresponds to
deformations of type (B) or (D) with cost less than ‖h1−h2‖∞.
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Stability property (sketch of the proof)
(B)
(D)
hh1 h2
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Stability property (sketch of the proof)
F 0
F 1α
F 1β
f
g
f1
g1
hh1
h2
A linear path between two functions h1, h2 across F 1β correspond to a
deformation of type (R) or (Ki ), i = 1,2,3 with cost less than‖h1−h2‖∞.14 of 18
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Stability property (sketch of the proof)
p p pqq qq qq
(R)
hh1 h2
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Stability property (sketch of the proof)
pp pqq q
(K1)
hh1 h2
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Stability property (sketch of the proof)
pp pppp q qq
(K2)
(K3)
hh1 h2
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Stability property (sketch of the proof)
pp pp p
qq qq qq
(R)
hh1 h2
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Stability property (sketch of the proof)
pp pppp
q qq
(R)
hh1 h2
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Optimality of the edit distance
Theorem
d((Γf , �f ),(Γg , �g )) = infξ∈Diff(M )
‖f −g ◦ξ‖∞.
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Optimality of the edit distance
Theorem
d((Γf , �f ),(Γg , �g )) = infξ∈Diff(M )
‖f −g ◦ξ‖∞.
Theorem (Cagliari, Di Fabio, L., Forum Mathematicum)
δ ([f ], [g ]) := infξ∈Diff(M )
‖f −g ◦ξ‖∞ is a metric on classes of simple
Morse functions of surfaces up to composition with diffeomorphisms.
Corollary
d is a metric on isomorphism classes of labeled Reeb graphs.
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Relationship with the bottleneck distance
Corollary
Let Df ,Dg denote the persistence diagrams of f ,g, and dB thebottleneck distance. It holds that dB(Df ,Dg )≤ d((Γf , �f ),(Γg , �g ))and the inequality may be strict.
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Relationship with the bottleneck distance
Corollary
Let Df ,Dg denote the persistence diagrams of f ,g, and dB thebottleneck distance. It holds that dB(Df ,Dg )≤ d((Γf , �f ),(Γg , �g ))and the inequality may be strict.
00112233445566
Df = Dg7
77f g(Γf , �f ) (Γg , �g )
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Relationship with the functional distortion distance
Corollary
Let Rf ,Rg denote the Reeb spaces of f ,g, and dFD the functionaldistortion distance. It holds that dFD(Rf ,Rg )≤ d((Γf , �f ),(Γg , �g ))and the inequality may be strict.
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Relationship with the functional distortion distance
Corollary
Let Rf ,Rg denote the Reeb spaces of f ,g, and dFD the functionaldistortion distance. It holds that dFD(Rf ,Rg )≤ d((Γf , �f ),(Γg , �g ))and the inequality may be strict.
bb
dd
c1
c2
c1+ a
c2+ a
gf (Γf , �f ) (Γg , �g
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To do
• Generalization to the piecewise-linear case
• Generalization to the comparison of non-diffeomorphic surfaces
• Algorithm
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To do
• Generalization to the piecewise-linear case
• Generalization to the comparison of non-diffeomorphic surfaces
• Algorithm
Thank you for your attention!Preprint: http://arxiv.org/abs/1411.1544
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