152 research outputs found
Optimal topological simplification of discrete functions on surfaces
We solve the problem of minimizing the number of critical points among all
functions on a surface within a prescribed distance {\delta} from a given input
function. The result is achieved by establishing a connection between discrete
Morse theory and persistent homology. Our method completely removes homological
noise with persistence less than 2{\delta}, constructively proving the
tightness of a lower bound on the number of critical points given by the
stability theorem of persistent homology in dimension two for any input
function. We also show that an optimal solution can be computed in linear time
after persistence pairs have been computed.Comment: 27 pages, 8 figure
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
A topological classification of convex bodies
The shape of homogeneous, generic, smooth convex bodies as described by the
Euclidean distance with nondegenerate critical points, measured from the center
of mass represents a rather restricted class M_C of Morse-Smale functions on
S^2. Here we show that even M_C exhibits the complexity known for general
Morse-Smale functions on S^2 by exhausting all combinatorial possibilities:
every 2-colored quadrangulation of the sphere is isomorphic to a suitably
represented Morse-Smale complex associated with a function in M_C (and vice
versa). We prove our claim by an inductive algorithm, starting from the path
graph P_2 and generating convex bodies corresponding to quadrangulations with
increasing number of vertices by performing each combinatorially possible
vertex splitting by a convexity-preserving local manipulation of the surface.
Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist,
this algorithm not only proves our claim but also generalizes the known
classification scheme in [36]. Our expansion algorithm is essentially the dual
procedure to the algorithm presented by Edelsbrunner et al. in [21], producing
a hierarchy of increasingly coarse Morse-Smale complexes. We point out
applications to pebble shapes.Comment: 25 pages, 10 figure
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