2,023 research outputs found

    Repairing 3D binary images using the BCC grid with a 4-valued combinatorial coordinate system

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    A 3D binary image I is called well-composed if the set of points in the topological boundary of the cubes in I is a 2-manifold. Repairing a 3D binary image is a process which produces a well composed image (or a polyhedral complex) from the non-well-composed image I.We propose here to repair 3D images by associating the Body-Centered Cubic grid (BCC grid) to the cubical grid. The obtained polyhedral complex is well composed, since two voxels in the BCC grid either share an entire face or are disjoint. We show that the obtained complex is homotopy equivalent to the cubical complex naturally associated with the image I.To efficiently encode and manipulate the BCC grid, we present an integer 4-valued combinatorial coordinate system that addresses cells of all dimensions (voxels, faces, edges and vertices), and allows capturing all the topological incidence and adjacency relations between cells by using only integer operations.We illustrate an application of this coordinate system on two tasks related with the repaired image: boundary reconstruction and computation of the Euler characteristic

    Digital objects in rhombic dodecahedron grid

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    Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system

    Surface-Based Computation of the Euler Characteristic in the BCC Grid

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    As opposed to the 3D cubic grid, the body-centered cubic (BCC) grid has some favorable topological properties: each set of voxels in the grid is a 3-manifold, with 2-manifold boundary. Thus, the Euler characteristic of an object O in this grid can be computed as half of the Euler characteristic of its boundary ∂O . We propose three new algorithms to compute the Euler characteristic in the BCC grid with this surface-based approach: one based on (critical point) Morse theory and two based on the discrete Gauss–Bonnet theorem. We provide a comparison between the three new algorithms and the classic approach based on counting the number of cells, either of the 3D object or of its 2D boundary surface

    Distance-based skeletonization on the BCC grid

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    Strand proposed a distance-based thinning algorithm for computing surface skeletons on the body-centered cubic (BCC) grid. In this paper, we present two modified versions of this algorithm that are faster than the original one, and less sensitive to the visiting order of points in the sequential thinning phase. In addition, a novel algorithm capable of producing curve skeletons is also reported

    Axiomatic Digital Topology

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    The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF space is isomorphic to an abstract cell complex. The paper demonstrates that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the "maximal" one, corresponding to 3n\"i1 neighbors. Even these (a, b)-adjacencies have important limitations and drawbacks. The most important one is that they are applicable only to binary images. The way of easily using LF spaces in computer imagery on standard orthogonal grids containing only pixels or voxels and no cells of lower dimensions is suggested

    AFLOW-SYM: Platform for the complete, automatic and self-consistent symmetry analysis of crystals

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    Determination of the symmetry profile of structures is a persistent challenge in materials science. Results often vary amongst standard packages, hindering autonomous materials development by requiring continuous user attention and educated guesses. Here, we present a robust procedure for evaluating the complete suite of symmetry properties, featuring various representations for the point-, factor-, space groups, site symmetries, and Wyckoff positions. The protocol determines a system-specific mapping tolerance that yields symmetry operations entirely commensurate with fundamental crystallographic principles. The self consistent tolerance characterizes the effective spatial resolution of the reported atomic positions. The approach is compared with the most used programs and is successfully validated against the space group information provided for over 54,000 entries in the Inorganic Crystal Structure Database. Subsequently, a complete symmetry analysis is applied to all 1.7++ million entries of the AFLOW data repository. The AFLOW-SYM package has been implemented in, and made available for, public use through the automated, ab-initio\textit{ab-initio} framework AFLOW.Comment: 24 pages, 6 figure

    One More Step Towards Well-Composedness of Cell Complexes over nD Pictures

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    An nD pure regular cell complex K is weakly well-composed (wWC) if, for each vertex v of K, the set of n-cells incident to v is face-connected. In previous work we proved that if an nD picture I is digitally well composed (DWC) then the cubical complex Q(I) associated to I is wWC. If I is not DWC, we proposed a combinatorial algorithm to “locally repair” Q(I) obtaining an nD pure simplicial complex PS(I) homotopy equivalent to Q(I) which is always wWC. In this paper we give a combinatorial procedure to compute a simplicial complex PS(¯I) which decomposes the complement space of |PS(I)| and prove that PS(¯I) is also wWC. This paper means one more step on the way to our ultimate goal: to prove that the nD repaired complex is continuously well-composed (CWC), that is, the boundary of its continuous analog is an (n − 1)- manifold.Ministerio de Economía y Competitividad MTM2015-67072-
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