5,390 research outputs found
Generalized topological simplification of scalar fields on surfaces
pre-printWe present a combinatorial algorithm for the general topological simplification of scalar fields on surfaces. Given a scalar field f, our algorithm generates a simplified field g that provably admits only critical points from a constrained subset of the singularities of f, while guaranteeing a small distance ||f - g||â for data-fitting purpose. In contrast to previous algorithms, our approach is oblivious to the strategy used for selecting features of interest and allows critical points to be removed arbitrarily. When topological persistence is used to select the features of interest, our algorithm produces a standard Ï”-simplification. Our approach is based on a new iterative algorithm for the constrained reconstruction of sub- and sur-level sets. Extensive experiments show that the number of iterations required for our algorithm to converge is rarely greater than 2 and never greater than 5, yielding O(n log(n)) practical time performances. The algorithm handles triangulated surfaces with or without boundary and is robust to the presence of multi-saddles in the input. It is simple to implement, fast in practice and more general than previous techniques. Practically, our approach allows a user to arbitrarily simplify the topology of an input function and robustly generate the corresponding simplified function. An appealing application area of our algorithm is in scalar field design since it enables, without any threshold parameter, the robust pruning of topological noise as selected by the user. This is needed for example to get rid of inaccuracies introduced by numerical solvers, thereby providing topological guarantees needed for certified geometry processing. Experiments show this ability to eliminate numerical noise as well as validate the time efficiency and accuracy of our algorithm. We provide a lightweight C++ implementation as supplemental material that can be used for topological cleaning on surface meshes
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
SU(5)-invariant decomposition of ten-dimensional Yang-Mills supersymmetry
The N=1,d=10 superYang-Mills action is constructed in a twisted form, using
SU(5)-invariant decomposition of spinors in 10 dimensions. The action and its
off-shell closed twisted scalar supersymmetry operator Q derive from a
Chern-Simons term. The action can be decomposed as the sum of a term in the
cohomology of Q and of a term that is Q-exact. The first term is a fermionic
Chern-Simons term for a twisted component of the Majorana-Weyl gluino and it is
related to the second one by a twisted vector supersymmetry with 5 parameters.
The cohomology of Q and some topological observables are defined from descent
equations. In this SU(5)<SO(10)$ invariant decomposition, the N=1, d=10 theory
is determined by only 6 supersymmetry generators, as in the twisted N=4, d=4
theory. There is a superspace with 6 twisted fermionic directions, with
solvable constraints.Comment: 10 page
Generalized Attractors in Five-Dimensional Gauged Supergravity
In this paper we study generalized attractors in N=2 gauged supergravity
theory in five dimensions coupled to arbitrary number of hyper, vector and
tensor multiplets. We look for attractor solutions with constant anholonomy
coefficients. By analyzing the equations of motion we derive the attractor
potential. We further show that the generalized attractor potential can be
obtained from the fermionic shifts. We study some simple examples and show that
constant anholonomy gives rise to homogeneous black branes in five dimensions.Comment: 30 pages, no figures,V3 minor revisions, to appear in JHE
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