973 research outputs found
The persistent cosmic web and its filamentary structure I: Theory and implementation
We present DisPerSE, a novel approach to the coherent multi-scale
identification of all types of astrophysical structures, and in particular the
filaments, in the large scale distribution of matter in the Universe. This
method and corresponding piece of software allows a genuinely scale free and
parameter free identification of the voids, walls, filaments, clusters and
their configuration within the cosmic web, directly from the discrete
distribution of particles in N-body simulations or galaxies in sparse
observational catalogues. To achieve that goal, the method works directly over
the Delaunay tessellation of the discrete sample and uses the DTFE density
computed at each tracer particle; no further sampling, smoothing or processing
of the density field is required.
The idea is based on recent advances in distinct sub-domains of computational
topology, which allows a rigorous application of topological principles to
astrophysical data sets, taking into account uncertainties and Poisson noise.
Practically, the user can define a given persistence level in terms of
robustness with respect to noise (defined as a "number of sigmas") and the
algorithm returns the structures with the corresponding significance as sets of
critical points, lines, surfaces and volumes corresponding to the clusters,
filaments, walls and voids; filaments, connected at cluster nodes, crawling
along the edges of walls bounding the voids. The method is also interesting as
it allows for a robust quantification of the topological properties of a
discrete distribution in terms of Betti numbers or Euler characteristics,
without having to resort to smoothing or having to define a particular scale.
In this paper, we introduce the necessary mathematical background and
describe the method and implementation, while we address the application to 3D
simulated and observed data sets to the companion paper.Comment: A higher resolution version is available at
http://www.iap.fr/users/sousbie together with complementary material.
Submitted to MNRA
Combinatorial Gradient Fields for 2D Images with Empirically Convergent Separatrices
This paper proposes an efficient probabilistic method that computes
combinatorial gradient fields for two dimensional image data. In contrast to
existing algorithms, this approach yields a geometric Morse-Smale complex that
converges almost surely to its continuous counterpart when the image resolution
is increased. This approach is motivated using basic ideas from probability
theory and builds upon an algorithm from discrete Morse theory with a strong
mathematical foundation. While a formal proof is only hinted at, we do provide
a thorough numerical evaluation of our method and compare it to established
algorithms.Comment: 17 pages, 7 figure
Operatori za multi-rezolucione komplekse Morza i Δelijske komplekse
The topic of the thesis is analysis of the topological structure of scalar fields and shapes represented through Morse and cell complexes, respectively. This is achieved by defining simplification and refinement operators on these complexes. It is shown that the defined operators form a basis for the set of operators that modify Morse and cell complexes. Based on the defined operators, a multi-resolution model for Morse and cell complexes is constructed, which contains a large number of representations at uniform and variable resolution.Π’Π΅ΠΌΠ° Π΄ΠΈΡΠ΅ΡΡΠ°ΡΠΈΡΠ΅ ΡΠ΅ Π°Π½Π°Π»ΠΈΠ·Π° ΡΠΎΠΏΠΎΠ»ΠΎΡΠΊΠ΅ ΡΡΡΡΠΊΡΡΡΠ΅ ΡΠΊΠ°Π»Π°ΡΠ½ΠΈΡ
ΠΏΠΎΡΠ° ΠΈ ΠΎΠ±Π»ΠΈΠΊΠ° ΠΏΡΠ΅Π΄ΡΡΠ°Π²ΡΠ΅Π½ΠΈΡ
Ρ ΠΎΠ±Π»ΠΈΠΊΡ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ° ΠΠΎΡΠ·Π° ΠΈ ΡΠ΅Π»ΠΈΡΡΠΊΠΈΡ
ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ°, ΡΠ΅Π΄ΠΎΠΌ. Π’ΠΎ ΡΠ΅ ΠΏΠΎΡΡΠΈΠΆΠ΅ Π΄Π΅ΡΠΈΠ½ΠΈΡΠ°ΡΠ΅ΠΌ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠ° Π·Π° ΡΠΈΠΌΠΏΠ»ΠΈΡΠΈΠΊΠ°ΡΠΈΡΡ ΠΈ ΡΠ°ΡΠΈΠ½Π°ΡΠΈΡΡ ΡΠΈΡ
ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ°. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ ΡΠ΅ Π΄Π° Π΄Π΅ΡΠΈΠ½ΠΈΡΠ°Π½ΠΈ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠΈ ΡΠΈΠ½Π΅ Π±Π°Π·Ρ Π·Π° ΡΠΊΡΠΏ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠ° Π½Π° ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠΈΠΌΠ° ΠΠΎΡΠ·Π° ΠΈ ΡΠ΅Π»ΠΈΡΡΠΊΠΈΠΌ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠΈΠΌΠ°. ΠΠ° ΠΎΡΠ½ΠΎΠ²Ρ Π΄Π΅ΡΠΈΠ½ΠΈΡΠ°Π½ΠΈΡ
ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠ° ΠΊΠΎΠ½ΡΡΡΡΠΈΡΠ°Π½ ΡΠ΅ ΠΌΡΠ»ΡΠΈ-ΡΠ΅Π·ΠΎΠ»ΡΡΠΈΠΎΠ½ΠΈ ΠΌΠΎΠ΄Π΅Π» Π·Π° ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ΅ ΠΠΎΡΠ·Π° ΠΈ ΡΠ΅Π»ΠΈΡΡΠΊΠ΅ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ΅, ΠΊΠΎΡΠΈ ΡΠ°Π΄ΡΠΆΠΈ Π²Π΅Π»ΠΈΠΊΠΈ Π±ΡΠΎΡ ΡΠ΅ΠΏΡΠ΅Π·Π΅Π½ΡΠ°ΡΠΈΡΠ° ΡΠ½ΠΈΡΠΎΡΠΌΠ½Π΅ ΠΈ Π²Π°ΡΠΈΡΠ°Π±ΠΈΠ»Π½Π΅ ΡΠ΅Π·ΠΎΠ»ΡΡΠΈΡΠ΅.Tema disertacije je analiza topoloΕ‘ke strukture skalarnih polja i oblika predstavljenih u obliku kompleksa Morza i Δelijskih kompleksa, redom. To se postiΕΎe definisanjem operatora za simplifikaciju i rafinaciju tih kompleksa. Pokazano je da definisani operatori Δine bazu za skup operatora na kompleksima Morza i Δelijskim kompleksima. Na osnovu definisanih operatora konstruisan je multi-rezolucioni model za komplekse Morza i Δelijske komplekse, koji sadrΕΎi veliki broj reprezentacija uniformne i varijabilne rezolucije
Operatori za multi-rezolucione komplekse Morza i Δelijske komplekse
The topic of the thesis is analysis of the topological structure of scalar fields and shapes represented through Morse and cell complexes, respectively. This is achieved by defining simplification and refinement operators on these complexes. It is shown that the defined operators form a basis for the set of operators that modify Morse and cell complexes. Based on the defined operators, a multi-resolution model for Morse and cell complexes is constructed, which contains a large number of representations at uniform and variable resolution.Π’Π΅ΠΌΠ° Π΄ΠΈΡΠ΅ΡΡΠ°ΡΠΈΡΠ΅ ΡΠ΅ Π°Π½Π°Π»ΠΈΠ·Π° ΡΠΎΠΏΠΎΠ»ΠΎΡΠΊΠ΅ ΡΡΡΡΠΊΡΡΡΠ΅ ΡΠΊΠ°Π»Π°ΡΠ½ΠΈΡ
ΠΏΠΎΡΠ° ΠΈ ΠΎΠ±Π»ΠΈΠΊΠ° ΠΏΡΠ΅Π΄ΡΡΠ°Π²ΡΠ΅Π½ΠΈΡ
Ρ ΠΎΠ±Π»ΠΈΠΊΡ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ° ΠΠΎΡΠ·Π° ΠΈ ΡΠ΅Π»ΠΈΡΡΠΊΠΈΡ
ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ°, ΡΠ΅Π΄ΠΎΠΌ. Π’ΠΎ ΡΠ΅ ΠΏΠΎΡΡΠΈΠΆΠ΅ Π΄Π΅ΡΠΈΠ½ΠΈΡΠ°ΡΠ΅ΠΌ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠ° Π·Π° ΡΠΈΠΌΠΏΠ»ΠΈΡΠΈΠΊΠ°ΡΠΈΡΡ ΠΈ ΡΠ°ΡΠΈΠ½Π°ΡΠΈΡΡ ΡΠΈΡ
ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ°. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ ΡΠ΅ Π΄Π° Π΄Π΅ΡΠΈΠ½ΠΈΡΠ°Π½ΠΈ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠΈ ΡΠΈΠ½Π΅ Π±Π°Π·Ρ Π·Π° ΡΠΊΡΠΏ ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠ° Π½Π° ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠΈΠΌΠ° ΠΠΎΡΠ·Π° ΠΈ ΡΠ΅Π»ΠΈΡΡΠΊΠΈΠΌ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠΈΠΌΠ°. ΠΠ° ΠΎΡΠ½ΠΎΠ²Ρ Π΄Π΅ΡΠΈΠ½ΠΈΡΠ°Π½ΠΈΡ
ΠΎΠΏΠ΅ΡΠ°ΡΠΎΡΠ° ΠΊΠΎΠ½ΡΡΡΡΠΈΡΠ°Π½ ΡΠ΅ ΠΌΡΠ»ΡΠΈ-ΡΠ΅Π·ΠΎΠ»ΡΡΠΈΠΎΠ½ΠΈ ΠΌΠΎΠ΄Π΅Π» Π·Π° ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ΅ ΠΠΎΡΠ·Π° ΠΈ ΡΠ΅Π»ΠΈΡΡΠΊΠ΅ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ΅, ΠΊΠΎΡΠΈ ΡΠ°Π΄ΡΠΆΠΈ Π²Π΅Π»ΠΈΠΊΠΈ Π±ΡΠΎΡ ΡΠ΅ΠΏΡΠ΅Π·Π΅Π½ΡΠ°ΡΠΈΡΠ° ΡΠ½ΠΈΡΠΎΡΠΌΠ½Π΅ ΠΈ Π²Π°ΡΠΈΡΠ°Π±ΠΈΠ»Π½Π΅ ΡΠ΅Π·ΠΎΠ»ΡΡΠΈΡΠ΅.Tema disertacije je analiza topoloΕ‘ke strukture skalarnih polja i oblika predstavljenih u obliku kompleksa Morza i Δelijskih kompleksa, redom. To se postiΕΎe definisanjem operatora za simplifikaciju i rafinaciju tih kompleksa. Pokazano je da definisani operatori Δine bazu za skup operatora na kompleksima Morza i Δelijskim kompleksima. Na osnovu definisanih operatora konstruisan je multi-rezolucioni model za komplekse Morza i Δelijske komplekse, koji sadrΕΎi veliki broj reprezentacija uniformne i varijabilne rezolucije
Chunk Reduction for Multi-Parameter Persistent Homology
The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among all the complexes quasi-isomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
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