🗺️Morse Theory Unit 11 – Reeb Graphs and the Topology of Level Sets

Key Concepts and Definitions

• Reeb graph a topological structure that captures the connectivity of level sets of a scalar function defined on a manifold
• Level set the set of points in a manifold where a scalar function takes on a specific value
• Critical point a point on a manifold where the gradient of a scalar function vanishes (includes local minima, local maxima, and saddle points)
  • Local minimum a point where the function value is lower than all nearby points
  • Local maximum a point where the function value is higher than all nearby points
  • Saddle point a point where the function increases in some directions and decreases in others
• Morse function a smooth function on a manifold with non-degenerate critical points
• Contour the boundary of a level set, often used in 2D visualizations
• Topology the study of properties that are preserved under continuous deformations (stretching, twisting, etc.) but not tearing or gluing
• Homeomorphism a continuous bijection with a continuous inverse, preserving topological properties

Historical Context and Development

• Reeb graphs introduced by Georges Reeb in 1946 as a tool for studying the topology of smooth manifolds
• Early applications focused on understanding the structure of level sets and critical points of functions on surfaces and 3D manifolds
• In the 1990s, Reeb graphs gained attention in computer graphics and visualization communities for their ability to capture shape information and simplify complex data
• Advancements in computational topology and Morse theory led to efficient algorithms for constructing and analyzing Reeb graphs
  • Sweep-line algorithm a method for constructing Reeb graphs by tracking changes in level set topology as the function value increases
  • Randomized algorithm a faster approach that samples function values and builds the Reeb graph incrementally
• Recent years have seen a surge in applications of Reeb graphs in fields such as data analysis, machine learning, and topological data analysis

Fundamental Principles of Reeb Graphs

• Reeb graphs encode the evolution of level set topology as the function value changes
• Each point in the Reeb graph represents a connected component of a level set
• Edges in the Reeb graph represent the connectivity between level set components
  • An edge is created when two components merge or a single component splits
• The Reeb graph is a simplified representation of the original manifold, capturing its essential topological features
• Critical points of the function correspond to nodes in the Reeb graph
  • Local minima and maxima appear as leaf nodes (nodes with only one incident edge)
  • Saddle points appear as internal nodes with multiple incident edges
• The Reeb graph is invariant under continuous deformations of the manifold that preserve the function values

Topology of Level Sets: The Basics

• Level sets provide a way to study the behavior of a function on a manifold
• For a scalar function $f: M \rightarrow \mathbb{R}$, the level set at value $c$ is defined as $L_c = \{x \in M | f(x) = c\}$
• The topology of a level set can change only at critical points of the function
  • At a local minimum, a new connected component appears
  • At a local maximum, a connected component disappears
  • At a saddle point, connected components can merge or split
• Between critical points, the topology of the level sets remains unchanged (homeomorphic)
• The preimage theorem relates the topology of a level set to the topology of the manifold and the function
  • If $c$ is a regular value (not a critical value), then $L_c$ is a submanifold of $M$ with codimension 1
• Morse theory provides a powerful framework for studying the topology of level sets and their relationships to critical points

Constructing and Analyzing Reeb Graphs

• Reeb graph construction algorithms aim to efficiently capture the evolution of level set topology
• The sweep-line algorithm sorts critical points by function value and processes them in ascending order
  • At each critical point, the algorithm updates the Reeb graph structure based on the type of critical point (minimum, maximum, or saddle)
  • The algorithm maintains a set of active contours and their connectivity
• The randomized algorithm samples function values and builds the Reeb graph incrementally
  • It maintains a union-find data structure to track connected components of level sets
  • The algorithm updates the Reeb graph as it encounters critical points during the sampling process
• Reeb graph simplification techniques reduce the complexity of the graph while preserving its essential topological features
  • Edge contraction merges adjacent nodes and updates the graph structure accordingly
  • Persistence-based simplification removes features with small persistence (difference in function values between critical points)
• Reeb graph comparison and matching algorithms enable the analysis of similarities and differences between Reeb graphs
  • Graph edit distance measures the cost of transforming one Reeb graph into another through node and edge operations
  • Functional distortion distance quantifies the dissimilarity between Reeb graphs based on the distortion of function values

Applications in Data Analysis and Visualization

• Reeb graphs provide a compact and informative representation of scalar fields and shapes
• In scientific visualization, Reeb graphs are used to analyze and explore complex datasets
  • Identifying and tracking features in time-varying data (e.g., vortices in fluid simulations)
  • Segmenting and labeling regions of interest in medical imaging data (e.g., organs in CT scans)
• Reeb graphs facilitate level-of-detail rendering and progressive transmission of 3D models
  • Decomposing a model into meaningful parts based on the Reeb graph structure
  • Prioritizing the transmission and rendering of important features
• In topological data analysis, Reeb graphs are used to study the shape and connectivity of high-dimensional datasets
  • Extracting topological features and summaries from point cloud data
  • Identifying and comparing clusters and substructures in the data
• Reeb graphs have applications in computer graphics, including shape matching, retrieval, and morphing
  • Comparing and aligning 3D shapes based on their Reeb graph representations
  • Generating smooth interpolations between shapes by manipulating their Reeb graphs

Connection to Morse Theory

• Morse theory provides a powerful framework for studying the relationship between the topology of a manifold and the critical points of a smooth function defined on it
• Reeb graphs can be seen as a discrete analog of Morse theory, capturing the essential topological information of a scalar field
• The Morse lemma states that near a non-degenerate critical point, a Morse function can be locally approximated by a quadratic form
  • The index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point
  • The index determines the local behavior of the level sets near the critical point (0 for minima, 1 for saddles, 2 for maxima in 2D)
• The Morse inequalities relate the number of critical points of each index to the Betti numbers (ranks of homology groups) of the manifold
  • $\beta_0 \leq m_0, \beta_1 \leq m_1, \beta_2 \leq m_2$, where $m_i$ is the number of critical points of index $i$
• Morse-Smale complexes partition the manifold into regions based on the gradient flow between critical points
  • Each region consists of points whose gradient flow originates at a specific minimum and terminates at a specific maximum
  • Reeb graphs can be seen as a simplified representation of Morse-Smale complexes, capturing the connectivity of regions

Advanced Topics and Current Research

• Multi-resolution Reeb graphs extend the concept to capture topological features at different scales
  • Constructing a hierarchy of Reeb graphs by simplifying the function or the underlying manifold
  • Enabling multi-scale analysis and visualization of complex datasets
• Reeb spaces generalize Reeb graphs to higher-dimensional functions (e.g., vector fields)
  • Capturing the topology of fibers (preimages of points in the codomain)
  • Analyzing the relationships between multiple scalar fields on a manifold
• Persistent homology combines Reeb graphs with persistence diagrams to study the stability and robustness of topological features
  • Quantifying the significance of features based on their persistence across different function values
  • Providing a multi-scale representation of the topology of a scalar field
• Discrete Morse theory extends the concepts of Morse theory to discrete structures (e.g., simplicial complexes)
  • Defining discrete analogs of critical points, gradient flows, and Morse functions
  • Enabling the computation of Reeb graphs and Morse-Smale complexes on discrete datasets
• Topological signatures and descriptors derived from Reeb graphs are used for shape analysis and comparison
  • Reeb graph-based shape descriptors (e.g., extended Reeb graphs, augmented Reeb graphs)
  • Topological persistence-based signatures (e.g., extended persistence diagrams)
• Current research focuses on efficient algorithms, scalability to large datasets, and applications in various domains
  • Parallel and distributed algorithms for Reeb graph computation and simplification
  • Interactive exploration and visualization of Reeb graphs and their associated data
  • Integration of Reeb graphs with machine learning techniques for feature extraction and classification