Connectivity forests for homological analysis of digital volumes

In this paper, we provide a graph-based representation of the homology (information related to the different “holes” the object has) of a binary digital volume. We analyze the digital volume AT-model representation [8] from this point of view and the cellular version of the AT-model [5] is precisely described here as three forests (connectivity forests), from which, for instance, we can straightforwardly determine representative curves of “tunnels” and “holes”, classify cycles in the complex, computing higher (co)homology operations,... Depending of the order in which we gradually construct these trees, tools so important in Computer Vision and Digital Image Processing as Reeb graphs and topological skeletons appear as results of pruning these graphs

Cup products on polyhedral approximations of 3D digital images

Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combinatorial structure of Q(I) and obtain a homeomorphic cellular complex P(I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H *(P(I)). The cup product encodes important geometrical information not captured by the cohomology groups. Consequently, the ring structure of H *(P(I)) is a finer topological invariant. The algorithm proposed here can be applied to compute cup products on any polyhedral approximation of an object embedded in 3-space

Using membrane computing for obtaining homology groups of binary 2D digital images

Membrane Computing is a new paradigm inspired from cellular communication. Until now, P systems have been used in research areas like modeling chemical process, several ecosystems, etc. In this paper, we apply P systems to Computational Topology within the context of the Digital Image. We work with a variant of P systems called tissue-like P systems to calculate in a general maximally parallel manner the homology groups of 2D images. In fact, homology computation for binary pixel-based 2D digital images can be reduced to connected component labeling of white and black regions. Finally, we use a software called Tissue Simulator to show with some examples how these systems wor

The accretion disk in the post period-minimum cataclysmic variable SDSS J080434.20+510349.2

This study of SDSS0804 is primarily concerned with the double-hump shape in the light curve and its connection with the accretion disk in this bounce-back system. Time-resolved photometric and spectroscopic observations were obtained to analyze the behavior of the system between superoutbursts. A geometric model of a binary system containing a disk with two outer annuli spiral density waves was applied to explain the light curve and the Doppler tomography. Observations were carried out during 2008-2009, after the object's magnitude decreased to V~17.7(0.1) from the March 2006 eruption. The light curve clearly shows a sinusoid-like variability with a 0.07 mag amplitude and a 42.48 min periodicity, which is half of the orbital period of the system. In Sept. 2010, the system underwent yet another superoutburst and returned to its quiescent level by the beginning of 2012. This light curve once again showed a double-humps, but with a significantly smaller ~0.01mag amplitude. Other types of variability like a "mini-outburst" or SDSS1238-like features were not detected. Doppler tomograms, obtained from spectroscopic data during the same period of time, show a large accretion disk with uneven brightness, implying the presence of spiral waves. We constructed a geometric model of a bounce-back system containing two spiral density waves in the outer annuli of the disk to reproduce the observed light curves. The Doppler tomograms and the double-hump-shape light curves in quiescence can be explained by a model system containing a massive >0.7Msun white dwarf with a surface temperature of ~12000K, a late-type brown dwarf, and an accretion disk with two outer annuli spirals. According to this model, the accretion disk should be large, extending to the 2:1 resonance radius, and cool (~2500K). The inner parts of the disk should be optically thin in the continuum or totally void.Comment: 12 pages, 15 figures, accepted for publication in A&

Incremental-Decremental Algorithm for Computing AT-Models and Persistent Homology

In this paper, we establish a correspondence between the incremental algorithm for computing AT-models [8,9] and the one for computing persistent homology [6,14,15]. We also present a decremental algorithm for computing AT-models that allows to extend the persistence computation to a wider setting. Finally, we show how to combine incremental and decremental techniques for persistent homology computation

Homological tree-based strategies for image analysis

Homological characteristics of digital objects can be obtained in a straightforward manner computing an algebraic map φ over a finite cell complex K (with coefficients in the finite field F2={0,1}) which represents the digital object [9]. Computable homological information includes the Euler characteristic, homology generators and representative cycles, higher (co)homology operations, etc. This algebraic map φ is described in combinatorial terms using a mixed three-level forest. Different strategies changing only two parameters of this algorithm for computing φ are presented. Each one of those strategies gives rise to different maps, although all of them provides the same homological information for K. For example, tree-based structures useful in image analysis like topological skeletons and pyramids can be obtained as subgraphs of this forest

The role of scattered trees and habitat diversity for biodiversity of Iberian dehesas

PosterWe studied 10 dehesas of CW Spain (40º 00’-10’ N, 06º 10’-20’ W), mapping every habitat according to a standardized protocol developed by the European BioBio project. We defined 35 habitat types, with 19 habitat types (split in 85 plots) per dehesa, on average. In one randomly selected plot per habitat type diversity of the four taxa, plants, bees, spiders and earthworms, were assessed. In total, 450 plant species (average of 189 per farm and 36 per habitat), 63 bee species (17.6 and 3.2), 130 spider species (43.8 and 7.4), and 17 earthworm species ( 7.8 and 2.5) were recorded. In each taxa, only some species were very abundant, while most of the species were found only in few farms/habitats. A high proportion of species (ca. 40%) were observed only in just one habitat per farm, indicating that farm biodiversity strongly depends on the habitat diversity. The analysis of unique and shared species among habitats revealed that every habitat contribute significantly to farm biodiversity. By contrast, species richness was poorly explained by the presence of scattered trees, whereas the combination of wood pastures and open pastures was a significant predictor. Summarizing, our extensive survey showed that diversity of the four taxa was strongly related to the existence of a wide mosaic of habitats, including non-productive habitats and linear elements, which harbor a disproportionate number of species compared to the low area occupied. Moreover, these habitats harbor a high number of exclusive species. As a next step, the importance of the spatial arrangement of main and non-productive habitats for biodiversity at the farm and landscape levels, need to be checked

Invariant Representative Cocycles of Cohomology Generators using Irregular Graph Pyramids

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns `quantities' to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. An extension to obtain scanning and rotation invariant cocycles is given.Comment: Special issue on Graph-Based Representations in Computer Visio