1. Topic 8 Reeb Graphs and Contour Trees Yusu Wang Computer Science and Engineering Dept The Ohio State University

2. Introduction  Given a topological space X and function f: X  R  Level set at value a :  X a := { x  X | f(x) = a}  A contour at value a :  a connected component of X a  Reeb graph R f (X) of X w.r.t. f:  continuous collapsing of each contour of f to a point  A continuous surjection  : X  R f (X) s.t,  (x) =  (y) if and only if x and y is in the same contour

3. More on Reeb Graph  Reeb graph R f (X) is an abstract graph  Imagine sweeping X in increasing order of f  Track the changes in  -th homology of level sets  i.e, changes in contours  Node:  where changes happen  Arc:  evolution of a single contour

4. Applications  Handle removal  Skelentonize a shape  Shape matching Courtesy of Wood et al. 2002 Courtesy of Biosotti et al. 2008 Courtesy of Hilaga et al. 2001

5. Applications of Reeb Graphs, cont.  Handle / Tunnel loops computation  [Dey, Fan, Wang, Siggraph 2013]  Learning geometric graphs from point cloud data  [Ge, Safa, Belkin, Wang, NIPS 2011], [Chazal, Sun, SoCG 2014]  Mapper: a generalization of Reeb graph for data analysis  [Singh, Mémoli, Carlsson, PBS 2007] Courtesy of Nicolau et al., PNAS 2011

6. Homology Relations Lemma

7. Contour Trees The inverse is not true.

8. Contour Trees

9.  Reeb graphs  Special case of the Reeb graph for a Morse function defined on a manifold.  Features / simplification / PL setting  One application  Contour trees  Contour trees and variants  PL setting / computation  One application

10. Critical Points  Given an m-manifold M and f: M  R,  A point p  M is critical if gradient of f vanishes at p  A critical point is non-degenerate  if it has non-degenerate Hessian  For every non-degenerate critical point

11. Critical Points cont.  For non-degenerate critical points  Suppose M is 2-manifold max saddlemin

12. Local View  For d-dimensional manifold M  An open neighborhood B r (p) of p  B r (p) is an m-dimensional open ball  Consider the boundary of the closure of B r (p) Courtesy of Edelsbrunner 2006

13. Morse Function  Function f: M  R is Morse if  No critical point of f is degenerate  No two critical points share the same value

14. Reeb Graphs for Morse Function  If M is an d-manifold and f: M  R a Morse function  Degree  nodes:  Minimum or maximum  Degree 3 nodes:   -saddles (merging forks) or (d-  )-saddles (splitting forks)  Degree 2 nodes:  All other nodes Not all 1-saddle or (d-1)-saddle necessarily corresponds to a degree-3 node

15. M is a 2-Manifold The Reeb graph of a Morse function on a connected, orientable 2-manifold of genus g has g loops. The Reeb graph of a Morse function on a connected, non-norientable 2-manifold of genus g has at most g/2 loops.

16. horizontal vertical [Dey and W, DCG2012]

17. Stability / Informative

18.  Reeb graphs  Special case of the Reeb graph for a Morse function defined on a manifold.  Features / simplification / PL setting  One application  Contour trees  Contour trees and variants  PL setting / computation  One application

19. Features in Reeb Graphs

20. A (Very) Simplified Illustration birth-time death- time (c, e) c e (b, f) b f

21.  Captured by Extended / Zigzag persistence:  [Cohen-Steiner, Edelsbrunner and Harer, FoCM 2009]  [Carlsson, de Silva, Morozov, SoCG 2009] Loop Features birth-time death- time (c, e) c e (b, f) b f h d (d,h)

22. Features in Reeb Graphs  Two types of features  Branching features  Loop features

23. Two Types of Features  Branching features  min/saddle, max/saddle features  tree-saddles:  saddles paired with a min or max by persistence algorithm  Union/find data structure to find pairings for them in O(n log n) time  loop features  loop-saddles: non-tree-saddles.  How to compute it efficiently?

24. Pairing of Loop-saddles  relation to extended persistence  (x, y) is paired, if  y is the lowest hi-pt of any cycle containing x as low-pt  x is the highest low-pt of any cycle containing y as high-pt Time complexity for computing pairing: Contour tree: O(n log n) Reeb graph: O(n log 2 n)

25. Persistence-based Simplification  Simplifying features based on their persistence value  Branching features  Loop featuers

26. PL Setting  PL function f defined on simplicial complex K  f is decided by function values on the vertices V of K  only 2-skelenton (V,E,T) of K matters  Reeb graph R f (X) can be computed in O(m log n) time  m: number of vertices, edges, and triangles of X,  n: number of vertices

27.  Reeb graphs  Special case of the Reeb graph for a Morse function defined on a manifold.  Featuers / simplification / PL setting  One application  Contour trees  Contour trees and variants  PL setting / computation  One application

28. Geometric Graphs Reconstruction  Geometric graphs  River / road networks, root systems, blood vessels, particle trajectories …  Problem statement:  Given a set of points P sampled on / around a hidden geometric graph, reconstruct the graph from P  [Ge, Safa, Belkin, Wang, NIPS 2011]

29. Related Work  Principal curves  [Hastie and Stuetzle 84, 89], and many followups  Self-consistent curves passing through the middle of a data cloud  Principle graph  [Kegl et al., 2002]  2D images  Principle graphs / surfaces  [Ozertem and Erdogmus 2011]  High-dimensional, but does not guarantee output is a graph  Metric graph learning  [Aanjaneya et al., 2011 ]  With certain theoretical guarantee! But sensitive to parameters

30. New Approach  Use the Reeb graph  Robust, always recover a graph structure  Natural simplification algorithm  Simple and efficient  Certain theoretical guarantee

31. Reeb Graph  Given a topological space X and function f: X  R  Level set at value a :  X a := { x  X | f(x) = a}  A contour at value a :  a connected component of X a  Reeb graph R f (X) of X w.r.t. f:  continuous collapsing of each contour of f to a point  A continuous surjection  : X  R f (X) s.t,  (x) =  (y) if and only if x and y is in the same contour 

32. Intuition  Given a graph G, fix any base point b, consider the shortest path distance (to b) function f  Reeb graph R f (G) same as G  Given a graph-like structure X, fix any base point b, consider the shortest distance function f to b  Reeb graph R f (X) captures underlying graph b v

33. Main Issues  Discrete points sampled from a hidden graph-like structure  Use (Vietoris-)Rips complex to connect points to a simplicial complex K  Approximate geodesic function  Compute Reeb graph from K 

34. Rips Complex

35. Algorithm Overview  Input: Given a set of input points P  Presumably P sampled around a hidden geometric graph-like structure.  Construct Rips Complex K=R r (P) for appropriate parameter r  Choose a base point b, compute shortest distance to all other vertices in K as a PL function f  Compute Reeb graph w.r.t. f  Simplify Reeb graph (to remove noise) if necessary

36.  Question :  Relation between the Reeb graph from Rips complex with that of the hidden structure  [Dey and W, DCG 2012] b

37. Simplification  Branches and Loops simplificaiton noisy branch noisy loop d b c d e a b c a e a b c d e f a b c d e f

38. Post-processing

39. Examples

40. More Examples

41. 3D singular surface reconstruction  [Dey, Ge,Que,Safa,Wang,Wang, SGP2012] input output

42. More Examples

43. More Examples – Speech Data  Projected in 3D for visualization

44.  Reeb graphs  Special case of the Reeb graph for a Morse function defined on a manifold.  Features / simplification / PL setting  One application  Contour trees  Contour trees and variants  PL setting / computation  One application

45. Simple Summaries of Scalar Fields

48. Another View of Join Tree

49. Simple Summaries of Scalar Fields

51. Join, Split and Contour Trees

52. Relations Contour tree contains full information of join and split trees!

53. A little advertisement …

54. Computation

55.  Reeb graphs  Special case of the Reeb graph for a Morse function defined on a manifold.  Features / simplification / PL setting  One application  Contour trees  Contour trees and variants  PL setting / computation  One application

56. Visualizing Protein Energy Landscape  Protein folding / molecular interaction / dynamics simulations  Huge amount of simulation data are routinely generated  Given a massive set of molecular simulation data:  Goal: would like a way to visually explore, navigate data

57. Molecular Simulation Data

58. [Wolynes et al., Folding and Design 1996] Energy Landscape  Energy landscape:  X : protein conformational space  Energy function E: X -> R  Important to understand molecular dynamics  Several theories,  e.g, smoothed energy landscape view (funnel theory), [Bryngelson et al, 1995], [Onuchic, Luthey- Schulten, Wolynes, 1997]

59. [Wolynes et al., Folding and Design 1996] Energy Landscape  Energy landscape:  X : protein conformational space  Energy function E: X -> R  Important to understand molecular dynamics  Several theories, e.g, smoothed energy landscape view (funnel theory), [Bryngelson et al, 1995], [Onuchic, Luthey- Schulten, Wolynes, 1997]

60. Visualizing Energy Landscape  Energy landscape  A high dimensional scalar field  Need to project to low dimensions  Using certain reaction coordinates  can be hard to choose the right reaction coordinates  Using dimensionality reduction  data can be intrinsically very high-dimensional [Zhou, PNAS 2003] [Das et al, PNAS 2006]

61.  Dimensionality reduction on survivin data set:  Most peaks/valleys are artifacts of distance distortion Question: When intrinsic dimension is far larger than 2 or 3, when distance distortion is not avoidable, how to visualize a scalar field in low dimensional space?

62. [Zhou, PNAS 2003] [Das et al, PNAS 2006] Visualizing Energy Landscape  Which key features?  Main idea:  Instead of focusing on geometry, focus on other key features  Create a 2D terrain metaphor preserving such information Contour trees  Which key features?  Contour evolution, peak/valley connection  Volume preservation

63. Interpretation for Contour Trees

64. Scalar Field on the Plane  For a Morse function defined on R 2  A contour is a simple closed curve  except for level sets corresponding to critical values  Any two contours in R 2 :  by Jordan curve theorem, either their bounded regions in R 2 are disjoint, or one contains the other.

65. Containment Relations  Consider two consecutive points, t and u, on contour tree and their corresponding contours C t and C u  Use arrow to indicate containment relations ?

66. Evolution of Contours  A merging (down-fork) saddle Case 1 Case 2 The arrows (of containment relations) have to be consistent.

67. Key Observations  Once we choose a root for the contour tree  induces a unique set of consistent edge directions a unique containment relation a topological configuration for a terrain in 2D

68. Topological Classification of 2D Terrains  Theorem  Given a contour tree with n nodes, there are exactly 2n-1 distinct configurations of 2D terrain models in the plane  n of them:  generated by choosing any tree node as the root  n-1of them:  choosing an arbitrary point from each tree edge as the root

69. Generating One Terrain Layout  Given a configuration u

70. Visual Analysis Platform  Other augmentation to facilitate data exploration  Global and local simplification  Display other measurements / augmentation on terrain  Callback system

71. Terrain Simplification  Simplification of contour tree  Persistence pairing for contour tree nodes  min-saddle or max-saddle  Collapse all ``branches” with low persistence  Regions corresponding to collapsed branches are merged in the terrain layout

72. Example  REMD simulation data of survivin protein  Energy landscape metaphor overlayed with alpha-helix formation fraction as color

73. Examples

74.  Compare the spaces sampled by the three methods: REMD ~20K REMD ~200K NPT-MD 437

75. Examples  Fraction of native contacts formed Monomer A Monomer B