Minkowski Functionals of Abell/ACO Clusters

We determine the Minkowski functionals for a sample of Abell/ACO clusters, 401 with measured and 16 with estimated redshifts. The four Minkowski functionals (including the void probability function and the mean genus) deliver a global description of the spatial distribution of clusters on scales from $10$ to 60\hMpc with a clear geometric interpretation. Comparisons with mock catalogues of N--body simulations using different variants of the CDM model demonstrate the discriminative power of the description. The standard CDM model and the model with tilted perturbation spectrum cannot generate the Minkowski functionals of the cluster data, while a model with a cosmological constant and a model with breaking of the scale invariance of perturbations (BSI) yield compatible results.Comment: 10 pages, 13 Postscript figures, uses epsf.sty and mn.sty (included), submitted to MNRA

Minkowski Functionals of SDSS galaxies I : Analysis of Excursion Sets

We present a first morphometric investigation of a preliminary sample from the SDSS of 154287 galaxies with apparent magnitude 14.5<m_r<17.5 and redshift 0.001<z<0.4. We measure the Minkowski Functionals, which are a complete set of morphological descriptors. To account for the complicated wedge--like geometry of the present survey data, we construct isodensity contour surfaces from the galaxy positions in redshift space and employ two complementary methods of computing the Minkowski Functionals. We find that the observed Minkowski Functionals for SDSS galaxies are consistent with the prediction of a Lambda--dominated spatially--flat Cold Dark Matter model with random--Gaussian initial conditions, within the cosmic variance estimated from the corresponding mock catalogue. We expect that future releases of the SDSS survey will allow us to distinguish morphological differences in the galaxy distribution with regard to different morphological type and luminosity ranges.Comment: 35 pages, 13 figures, accepted for publication in PASJ. For preprint with higher-resolution PS files, see http://www.a.phys.nagoya-u.ac.jp/~hikage/MFs/mf_sdss.ps.g

From 3D Models to 3D Prints: an Overview of the Processing Pipeline

Due to the wide diffusion of 3D printing technologies, geometric algorithms for Additive Manufacturing are being invented at an impressive speed. Each single step, in particular along the Process Planning pipeline, can now count on dozens of methods that prepare the 3D model for fabrication, while analysing and optimizing geometry and machine instructions for various objectives. This report provides a classification of this huge state of the art, and elicits the relation between each single algorithm and a list of desirable objectives during Process Planning. The objectives themselves are listed and discussed, along with possible needs for tradeoffs. Additive Manufacturing technologies are broadly categorized to explicitly relate classes of devices and supported features. Finally, this report offers an analysis of the state of the art while discussing open and challenging problems from both an academic and an industrial perspective.Comment: European Union (EU); Horizon 2020; H2020-FoF-2015; RIA - Research and Innovation action; Grant agreement N. 68044

Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

Multi-scale morphology of the galaxy distribution

Many statistical methods have been proposed in the last years for analyzing the spatial distribution of galaxies. Very few of them, however, can handle properly the border effects of complex observational sample volumes. In this paper, we first show how to calculate the Minkowski Functionals (MF) taking into account these border effects. Then we present a multiscale extension of the MF which gives us more information about how the galaxies are spatially distributed. A range of examples using Gaussian random fields illustrate the results. Finally we have applied the Multiscale Minkowski Functionals (MMF) to the 2dF Galaxy Redshift Survey data. The MMF clearly indicates an evolution of morphology with scale. We also compare the 2dF real catalog with mock catalogs and found that Lambda-CDM simulations roughly fit the data, except at the finest scale.Comment: 17 pages, 19 figures, accepted for publication in MNRA