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

Dark Energy Accretion onto black holes in a cosmic scenario

In this paper we study the accretion of dark energy onto a black hole in the cases that dark energy is equipped with a positive cosmological constant and when the space-time is described by a Schwarzschild-de Sitter metric. It is shown that, if confronted with current observational data, the results derived when no cosmological constant is present are once again obtained in both cases.Comment: 7 pages, 3 figure

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

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

Advanced homology computation of digital volumes via cell complexes

Given a 3D binary voxel-based digital object V, an algorithm for computing homological information for V via a polyhedral cell complex is designed. By homological information we understand not only Betti numbers, representative cycles of homology classes and homological classification of cycles but also the computation of homology numbers related additional algebraic structures defined on homology (coproduct in homology, product in cohomology, (co)homology operations,...). The algorithm is mainly based on the following facts: a) a local 3D-polyhedrization of any 2×2×2 configuration of mutually 26-adjacent black voxels providing a coherent cell complex at global level; b) a description of the homology of a digital volume as an algebraic-gradient vector field on the cell complex (see Discrete Morse Theory [5], AT-model method [7,5]). Saving this vector field, we go further obtaining homological information at no extra time processing cost