opology Based Feature Extraction from 3D Scalar Fields

Transcription

1 opology Based Feature Extraction from 3D Scalar Fields Attila Gyulassy Vijay Natarajan, Peer-Timo Bremer, Bernd Hamann, Valerio Pascucci Institute for Data Analysis and Visualization, UC Davis Lawrence Livermore National Laboratory

2 Motivation Scalar functions are ubiquitous in scientific computing Global analysis structure needed Morse-Smale complex an efficient solution

3 Motivation Electron probability density in a Hydrogen atom

4 Motivation Why have a topology based approach? Level of control Interact with features directly

5 Previous Work Reeb Graphs Trace isocontour/isosurface compontents Contour trees for simply connected domains Feature removal (Carr 2004)(Wood 2004) Morse-Smale complex Persistence/simplification, 3D complex (Edelsbrunner et al 2002, 2003) Topological hierarchy (Bremer 2004) Pascucci et al 2005 Bremer et al 2004

6 Our Approach Combinatorial algorithm Compute Morse-Smale complex in 3D Simplify topology representation Cancellation of pairs of critical points Characterization of transformation non-trivial extension from 2D for 1-saddle 2-saddle cancellation

7 Critical Points Let ƒ be a scalar valued function whose critical points are not degenerate. We call ƒ a Morse function, and in the neighborhood of a critical point p, the function can be represented as ƒ(p) = 0 ƒ(x, y, z) = ƒ(p) ± x 2 ± y 2 ± z 2 Regular Minimum 1-Saddle 2-Saddle Maximum Index 0 Index 1 Index 2 Index 3

8 Integral lines An integral line is a maximal path that agrees with the gradient of at every point

9 Descending and Ascending manifolds Descending Manifold D(p) = {p} { x x є l, dest(l) = p} Ascending Manifold A(p) = {p} { x x є l, orig(l) = p} 3- Manifold 1-Manifold 3-Manifold 1-Manifold 2-Manifold 0-Manifold 2-Manifold 0-Manifold Maximum 2-Saddle 1-Saddle Minimum

10 What is the Morse-Smale Complex? The intersection of all descending and ascending manifolds. D(p) A(q), for all pairs p,q of ƒ Any cell in the complex has the property that all integral lines in that cell share an origin and a destination. The Morse-Smale complex is a segmentation of the domain that clusters integral lines that share a common origin and destination.

11 Morse-Smale complex 1D Example

12 Morse-Smale complex 1D Example minimum maximum

13 Morse-Smale complex 1D Example minimum maximum

14 Morse-Smale complex 2D Example

15 Morse-Smale complex 2D Example

16 Morse-Smale complex 2D Example

17 Morse-Smale complex 2D Example

18 Morse-Smale complex 2D Example

19 Morse-Smale complex 2D Example

20 Morse-Smale complex 3D Example Bump function

21 Morse-Smale complex 3D Example Bump function - critical points

22 Morse-Smale complex 3D Example Bump function - critical points - 1-manifolds

23 Morse-Smale complex 3D Example Bump function - critical points - 1-manifolds - Ascending 3-Manifold

24 Morse-Smale complex 3D Example Bump function - critical points - 1-manifolds - Descending 3-Manifold

25 Morse-Smale complex 3D Example Bump function - critical points - 1-manifolds - MS complex

26 Morse-Smale Complex 3D - Cells The cells of different dimensions are crystals, quads, arcs, and nodes.

27 Building the MS Complex Difficult to compute complex directly We build a complex for a similar function every original data point becomes a minimum

28 Building the MS Complex Difficult to compute complex directly We build a complex for a similar function every original data point becomes a minimum

29 Building the MS Complex Difficult to compute complex directly We build a complex for a similar function every original data point becomes a minimum Original data points

30 Building the MS Complex Guaranteed a combinatorially correct Morse-Smale complex The function converted to a Morse function Multi-saddles flat regions Original data points

31 Building the MS Complex Too many features Complex has no meaning Need for simplification Initially number of nodes equal to data size Large overhead Original data points

32 Topology-based Simplification Index Lemma Critical points can be created or destroyed in pairs with index that differs by one. Morse-Smale complex pairs critical points Remove pairs in order of persistence Absolute difference in function value in pairs that are removed Low persistence = small feature

33 Topology-based Simplification

34 Topology-based Simplification Remove extra critical points Correct Morse-Smale complex within small error bound

35 Simplification in 3D MS complex Combinatorial algorithm Find lowest persistence pair Apply atomic cancellation operation

36 Simplification in 3D MS complex Two kinds of cancellations Saddle extremum Minimum 1-saddle dual to 2-saddle maximum 1-Saddle 2-saddle

37 Simplification - Saddle-Extremum Cancel Saddle-Extremum pair Removes bump from function Merge three critical points into one. Based on merging two 3-Manifolds

38 Simplification - Saddle-Extremum

39 Simplification - Saddle-Extremum Simplification - Saddle-Extremum

40 Simplification - Saddle-Extremum Simplification - Saddle-Extremum

41 Simplification - Saddle-Extremum Simplification - Saddle-Extremum

42 Simplification - Saddle-Extremum Simplification - Saddle-Extremum

43 Simplification - Saddle-Extremum Simplification - Saddle-Extremum

44 Simplification - Saddle-Extremum Update the Morse-Smale complex New destination for arcs Quads are merged Destroy a ring of crystals

45 Simplification of 1-Saddle 2-Saddle Pair Cancel 1-Saddle 2-Saddle pair Removes tunnel from level sets Effect a separation between extrema

46 Simplification 1-Saddle 2-Saddle Simplification of 1-Saddle 2-Saddle Pair

47 Simplification 1-Saddle 2-Saddle Simplification of 1-Saddle 2-Saddle Pair

48 Simplification 1-Saddle 2-Saddle Simplification of 1-Saddle 2-Saddle Pair

49 Simplification 1-Saddle 2-Saddle Simplification of 1-Saddle 2-Saddle Pair

50 Simplification 1-Saddle 2-Saddle Simplification of 1-Saddle 2-Saddle Pair

51 Simplification 1-Saddle 2-Saddle Simplification of 1-Saddle 2-Saddle Pair

52 Simplification 1-Saddle 2-Saddle Simplification of 1-Saddle 2-Saddle Pair

53 Simplification of 1-Saddle 2-Saddle Pair Update the Morse-Smale complex New cells Contract quads in the fan around the 1-2 pair

54 Simplification Constraints Some configurations of the complex do not allow simplification Saddle-Extremum obstruction Saddle-Saddle obstruction A crystal must always have at least one 1-saddle and one 2-saddle

55 Synthetic Data 64x64x64 -diminishing features Results

56 Synthetic Data 64x64x64 -diminishing features Results

57 Neghip 64x64x64 spatial probability distribution of electrons in a high potential protein molecule Results

58 Neghip 64x64x64 spatial probability distribution of electrons in a high potential protein molecule Results

59 Bonsai 64x64x64 CT scan Results

60 Bonsai 64x64x64 CT scan Results

61 Conclusions We present Robust combinatorial algorithm for computing the Morse-Smale complex in 3D Complete characterization of atomic simplification operations Results of simplification on volumetric data

62 Future work Efficient construction of MS complex using incremental methods Display of quads and crystals Smoothing of gradient lines Numerical realization of simplification Applications to larger data Extension to 4D data

63 Thank You! ucdavis.edu Institute for Data Analysis and Visualization, University of California, Davis and Lawrence Livermore National Laboratory

64 Simplification 1-Saddle 2-Saddle

65 Topology-based Simplification Electron probability density in a Hydrogen atom

66 Topology-Based Simplification

67 Simplification 1-Saddle 2-Saddle

68 Simplification 1-Saddle 2-Saddle

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70 Topology Background Brief Overview

71 Toy Data 64x64x64 -diminishing features Results

72 1D Simplification Example Remove small features by canceling critical point pairs minimum maximum

73 1D Simplification Example The local change leaves the rest of the domain untouched minimum maximum

74 Results Results generated on Intel Xeon 3GHz desktop with 2GB memory Straightforward implementation of construction limits data size

75 Outline Motivation Topology Background Morse-Smale complex Topology-based Simplification Results Future Work

76 Topology Background Manifolds Descending 3-manifold of a maximum p max is the set of points whose integral lines terminate at p max. The ascending 3-manifold of a minimum p min is the set of points whose integral lines originate at p min.

77 Topology Background Manifolds The ascending 2-manifold of a 1-saddle p s1 is the set of points whose integral lines originate at p s1. The descending 2-manifold of a 2-saddle p s2 is the set of points whose integral lines terminate at p s2.

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80 Overview Complex Data Noise Quantization error Data exploration Visualization Simplification

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83 Simplification 1-Saddle 2-Saddle

84 Morse-Smale Complex 3D Arcs An arc of the Morse-Smale complex is an integral line that originates at an index n critical point, and terminates at an index n+1 critical point.

85 Introduction Feature identification through topological simplification Identify topological structure Remove small features directly Introduce saddle-saddle cancellation Combinatorial algorithm Atomic simplification operations Feature identification/ordering

86 Morse-Smale Complex 3D Quads An quad of the Morse-Smale complex is a cell whose integral lines that originates at an index n critical point, and terminate at an index n+2 critical point.

87 Simplification 1-Saddle 2-Saddle Remove a 1-saddle 2-saddle pair Apply 2D cancellation restricted to each pair of 2-manifolds Enforce a separation between existing extrema