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 Reeb graphs Summary Outline 1 Reeb graphs 2 Piecewise linear functions 3 Constructing Reeb graphs

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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) Isosurface extraction Reeb graph

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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 Isosurface extraction Reeb graph

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of contours Given a continuous map f : X → R , the level sets form a partition of topological space X

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of contours

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of contours Given a continuous map f : X → R , the level sets form a partition of 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of contours

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of contours Given a continuous map f : X → R , the level sets form a partition of 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of contours

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of Reeb graphs Definition The Reeb graph of f is the set of contours R ( f ) .

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of Reeb graphs Definition The Reeb graph of f is the set of contours R ( f ) .

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of Reeb graphs We started with only the continuous map f . Where does the Reeb graph fit in? f X R π ψ

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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: f X R π ψ R ( f )

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Definition of Reeb graphs Let π : R ( f ) → R be the unique map whose composition with ψ is f : f X R π ψ R ( f ) π ( ψ ( x )) = f ( x )

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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. f X R π ψ R ( f )

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Properties of Reeb graphs Connected components Function ψ : X → R ( f ) maps components to components

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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: β 0 ( R ( f )) = β 0 ( X )

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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: β 1 ( R ( f )) ≤ β 1 ( X )

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Properties of Reeb graphs Contour tree From these properties: β 0 ( R ( f )) = β 0 ( X ) β 1 ( R ( f )) ≤ β 1 ( 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary 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

Reeb graphs Motivation Piecewise linear functions Definitions Constructing Reeb graphs Properties of Reeb graphs Summary Reeb graphs of Morse functions Reeb graphs of Morse functions