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Categorification of Reeb Graphs

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Categorification of Reeb Graphs Categorification of Reeb Graphs Elizabeth Munch University of Minnesota :: Institute for Mathematics and Its Applications February 5, 2014 Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 1 / 46 What's the plan? Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 2 / 46 What's the plan? Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 2 / 46 What's the plan? Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 2 / 46 Original Reeb Graph construction Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 3 / 46 Original Reeb Graph construction Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 3 / 46 Original Reeb Graph Definition Definition Given topological space X with a function f : X ! R, say x ∼ y if x and y are in the same connected component of f −1(a). The Reeb graph of the function f is the space X= ∼ with the quotient topology. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 4 / 46 Morse Function Setting Let f be Morse with distinct critical values. critical values , Nodes Index of critical values , Type of node I Min/max , Deg 1 vertex I Saddle , Deg 3 vertex Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 5 / 46 Applications Handle removal, from Wood 3D shape matching, from et al, 2002 Hilaga et.al, 2001 Basis for homology groups, Shape skeletonization, from Dey et. al, 2013 from Biasotti et.al, 2008 Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 6 / 46 Smoothing Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 7 / 46 Smoothing Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 7 / 46 Other work Functional distortion distance [Bauer Ge Wang 2013] Topological simplification [Doraiswamy Natarajan 2012], [Ge et at 2011], [Pascucci et al 2007] Smoothing Goals Give a metric to compare Reeb graphs. Find a natural way to \smooth out" the Reeb graph to get rid of small holes arising from noise. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 7 / 46 Smoothing Goals Give a metric to compare Reeb graphs. Find a natural way to \smooth out" the Reeb graph to get rid of small holes arising from noise. Other work Functional distortion distance [Bauer Ge Wang 2013] Topological simplification [Doraiswamy Natarajan 2012], [Ge et at 2011], [Pascucci et al 2007] Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 7 / 46 Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 8 / 46 Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 8 / 46 Goal: Define a Reeb graph as a topological space. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 9 / 46 Generalized Reeb Graph Definition Let Γ be a 1D simplicial complex. Let f :Γ ! R. We call the pair (Γ; f ) a Reeb graph. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 10 / 46 Generalized Reeb Graph - S constructible Definition Let S ⊂ R be finite set of points. A Reeb graph (Γ; f ) is S-constructible if f takes vertices to points of S and edges to 1-cells homeomorphically. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 11 / 46 Generalized Reeb Graph - S constructible Definition Let S ⊂ R be finite set of points. A Reeb graph (Γ; f ) is S-constructible if f takes vertices to points of S and edges to 1-cells homeomorphically. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 11 / 46 Generalized Reeb Graphs - Comparing S-constructible Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 12 / 46 Note: For S ⊆ T , we can think of an S-constructible Reeb graph as a T -constructible Reeb graph. So, we can compare S and U-constructible Reeb graphs by thinking of them as S [ U-constructible. Generalized Reeb Graphs - Comparing S-constructible Definition A Reeb graph morphism between two S-constructible Reeb graphs (Γ; f ) and (Λ; g) is a continuous map α : jΓj ! jΛj such that ' jΓj / jΛj @@ ~ @@ ~~ f @@ ~~g @ ~~ R commutes. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 12 / 46 Generalized Reeb Graphs - Comparing S-constructible Definition A Reeb graph morphism between two S-constructible Reeb graphs (Γ; f ) and (Λ; g) is a continuous map α : jΓj ! jΛj such that ' jΓj / jΛj @@ ~ @@ ~~ f @@ ~~g @ ~~ R commutes. Note: For S ⊆ T , we can think of an S-constructible Reeb graph as a T -constructible Reeb graph. So, we can compare S and U-constructible Reeb graphs by thinking of them as S [ U-constructible. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 12 / 46 Generalized Reeb Graphs Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 13 / 46 Generalized Reeb Graphs Definition The set of Objects: Constructible Reeb graphs (Γ; f ) Morphisms: Reeb graph morphisms α : jΓj ! jΛj such that α jΓj / jΛj @@ ~ @@ ~~ f @@ ~~g @ ~~ R commutes. is a category which we will call ReebGraph. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 13 / 46 New Goal: Define a Reeb graph as a functor. Want: A method for comparing Reeb graphs. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 14 / 46 Want: A method for comparing Reeb graphs. New Goal: Define a Reeb graph as a functor. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 14 / 46 Category Theory 101 Category A category C consists of objects (X ; Y ; Z; etc) with morphisms (!) between them satisfying some \niceness" properties (associativity, identity). Objects Morphisms Set Sets Functions Open(R) U ⊂ R open ⊆ Int Open intervals (a; b) ⊂ R ⊆ Vect Vector spaces Linear Transformations Strat Reeb Graph (Γ; f ) Commutative triangle Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 15 / 46 Category Theory 102 Functors Functors are maps between categories which take objects to objects and morphisms to morphisms. X 7! F (X ) (f : X ! Y ) 7! (F [f ]: F [X ] ! F [Y ]) with some \niceness" properties (moving identity, associativity). Examples Hp : Top ! Vect π0 : Top ! Set Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 16 / 46 Definition A functor F : Open(R) ! Set is a cosheaf if for all open U ⊂ R and covering fUi g of U, F (U) is the colimit of the diagram p1 ` F (U \ U ) / ` F (U ) F (U) i j / i / p2 Definition A stalk of F at x, denoted Fx , is Fx = lim F (U) x2U Cosheaves Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 17 / 46 Definition A stalk of F at x, denoted Fx , is Fx = lim F (U) x2U Cosheaves Definition A functor F : Open(R) ! Set is a cosheaf if for all open U ⊂ R and covering fUi g of U, F (U) is the colimit of the diagram p1 ` F (U \ U ) / ` F (U ) F (U) i j / i / p2 Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 17 / 46 Cosheaves Definition A functor F : Open(R) ! Set is a cosheaf if for all open U ⊂ R and covering fUi g of U, F (U) is the colimit of the diagram p1 ` F (U \ U ) / ` F (U ) F (U) i j / i / p2 Definition A stalk of F at x, denoted Fx , is Fx = lim F (U) x2U Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 17 / 46 Constructible Cosheaves Definition A cosheaf F is compactly supported if the set of x which have nonempty stalks Fx is compact. A cosheaf is S-constructible if it is compactly supported and I \ S = J \ S implies F [I ⊂ J]: F (I ) ! F (J) is an isomorphism. Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 18 / 46 How do you compare functors? Definition A natural transformation ' : F ) G between functors F ; G : C!D is a set of maps 'X for each X 2 C such that F [f ] F (X ) / F (Y ) 'X 'Y G[f ] G(X ) / G(Y ) commutes for any morphism f : X ! Y Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 19 / 46 Categorical Reeb Graph Definition Cosh(R) consists of Objects: Constructible cosheaves F : Open(R) ! Set Morphisms: Natural transformations Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 20 / 46 Review Topological - ReebGraph Constructible 1D Functorial - Cosh(R) simplicial complexes Constructible Cosheaves (Γ; f ), f :Γ ! R Maps between them Maps between them play given by natural nice with the original transformations function Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 21 / 46 Given Reeb graph (Γ; f ), f : X ! R, construct the functor f −1 π0 Open( ) / Open( ) / Top / Set R X 7 F The Reeb functor and construction Reeb ReebGraph / Cosh(R) Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 22 / 46 The Reeb functor and construction Reeb ReebGraph / Cosh(R) Given Reeb graph (Γ; f ), f : X ! R, construct the functor f −1 π0 Open( ) / Open( ) / Top / Set R X 7 F Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 22 / 46 Given constructible cosheaf F : Open(R) ! Set, construct the topological space Points: (t; p) for p 2 Germt (F ). Open sets: BU;c = f(t; p) j t 2 U; πU (p) = cg [Display locale of a cosheaf - Funk 1995] The other functor and construction Reeb / ReebGraph Cosh(R) Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 23 / 46 Given constructible cosheaf F : Open(R) ! Set, construct the topological space Points: (t; p) for p 2 Germt (F ). Open sets: BU;c = f(t; p) j t 2 U; πU (p) = cg [Display locale of a cosheaf - Funk 1995] The other functor and construction Reeb ReebGraph / Cosh( ) o R Beer Liz Munch (IMA) Reeb Graphs - SAMSI February 5, 2014 23 / 46 The other functor and construction Reeb ReebGraph / Cosh( ) o R Beer Given constructible cosheaf F : Open(R) ! Set, construct the topological space Points: (t; p) for p 2 Germt (F ).
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