1,443 research outputs found

    Ascending and descending regions of a discrete Morse function

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    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

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    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

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    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

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    The Morse-Smale complex of a function ff decomposes the sample space into cells where ff 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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