1,443 research outputs found
Ascending and descending regions of a discrete Morse function
We present an algorithm which produces a decomposition of a regular cellular
complex with a discrete Morse function analogous to the Morse-Smale
decomposition of a smooth manifold with respect to a smooth Morse function. The
advantage of our algorithm compared to similar existing results is that it
works, at least theoretically, in any dimension. Practically, there are
dimensional restrictions due to the size of cellular complexes of higher
dimensions, though. We prove that the algorithm is correct in the sense that it
always produces a decomposition into descending and ascending regions of the
critical cells in a finite number of steps, and that, after a finite number of
subdivisions, all the regions are topological discs. The efficiency of the
algorithm is discussed and its performance on several examples is demonstrated.Comment: 23 pages, 12 figure
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
The persistent cosmic web and its filamentary structure II: Illustrations
The recently introduced discrete persistent structure extractor (DisPerSE,
Soubie 2010, paper I) is implemented on realistic 3D cosmological simulations
and observed redshift catalogues (SDSS); it is found that DisPerSE traces
equally well the observed filaments, walls, and voids in both cases. In either
setting, filaments are shown to connect onto halos, outskirt walls, which
circumvent voids. Indeed this algorithm operates directly on the particles
without assuming anything about the distribution, and yields a natural
(topologically motivated) self-consistent criterion for selecting the
significance level of the identified structures. It is shown that this
extraction is possible even for very sparsely sampled point processes, as a
function of the persistence ratio. Hence astrophysicists should be in a
position to trace and measure precisely the filaments, walls and voids from
such samples and assess the confidence of the post-processed sets as a function
of this threshold, which can be expressed relative to the expected amplitude of
shot noise. In a cosmic framework, this criterion is comparable to friend of
friend for the identifications of peaks, while it also identifies the connected
filaments and walls, and quantitatively recovers the full set of topological
invariants (Betti numbers) {\sl directly from the particles} as a function of
the persistence threshold. This criterion is found to be sufficient even if one
particle out of two is noise, when the persistence ratio is set to 3-sigma or
more. The algorithm is also implemented on the SDSS catalogue and used to locat
interesting configurations of the filamentary structure. In this context we
carried the identification of an ``optically faint'' cluster at the
intersection of filaments through the recent observation of its X-ray
counterpart by SUZAKU. The corresponding filament catalogue will be made
available online.Comment: A higher resolution version is available at
http://www.iap.fr/users/sousbie together with complementary material (movie
and data). Submitted to MNRA
Statistical Inference using the Morse-Smale Complex
The Morse-Smale complex of a function decomposes the sample space into
cells where is increasing or decreasing. When applied to nonparametric
density estimation and regression, it provides a way to represent, visualize,
and compare multivariate functions. In this paper, we present some statistical
results on estimating Morse-Smale complexes. This allows us to derive new
results for two existing methods: mode clustering and Morse-Smale regression.
We also develop two new methods based on the Morse-Smale complex: a
visualization technique for multivariate functions and a two-sample,
multivariate hypothesis test.Comment: 45 pages, 13 figures. Accepted to Electronic Journal of Statistic
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