Large Scale Data Analysis - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 326

(a) (b)

Fig. 27.3 A Morse complex of a set of maxima before a and after b merging the red and yellow
stable manifolds. The persistence of the merge is the difference in function value between the circled
critical point pair

Fig. 27.4 Stablemanifolds of a late stagemixing surface in a Raleigh-Taylor instabilities at: a About
1% persistence where several large scale bubbles are over-segmented; b About 2 5% persistence

which reduces the over segmentation but still shows some artifacts; and c About 5% persistence
where the segmentation matches the human intuition

.

1-Skeleton of the Morse-Smale complex The Morse complex is the collection of
stable manifolds of all critical points. The unstable manifold of a critical point is
the collection of integral lines originating at that critical point. A function is Morse-
Smale if its stable and unstable manifolds intersect only transversally. The cells of
the Morse-Smale complex are formed by the intersections of stable and unstable
manifolds. The set of 0- and 1-dimensional cells ( nodes and arcs ) of the complex
are known as the 1-skeleton , describing the connectivity structure of gradient flow
of a function. Each node has an associated stable and unstable manifold. Each arc
connects two nodes, and additionally stores its geometric embedding (the integral
line it represents).
Ahierarchy ofMorse-Smale complexes is created by repeated cancellation of criti-
cal points in order of persistence. These cancellations aremanifested in the 1-skeleton
as the creation of new arcs, removal of arcs, and removal of a pair of nodes [ 9 ]. Each
new arc has geometric embedding that is the merging of the embeddings of three
removed arcs, as in Fig. 27.5 . However, when reconstructing the geometry of an arc,
only children that appear an odd number of times in the hierarchy should be used,
since the ones that appear an even number of times correspond to a path that doubles
back on itself. Therefore, when computing statistics on arcs in a hierarchy, this rule
must be enforced.