The Experts below are selected from a list of 5973 Experts worldwide ranked by ideXlab platform
Benedek Nagy - One of the best experts on this subject based on the ideXlab platform.
-
On disks of the Triangular Grid: An application of optimization theory in discrete geometry
Discrete Applied Mathematics, 2020Co-Authors: Gergely Kovacs, Benedek Nagy, Bela VizvariAbstract:Abstract Chamfer (or weighted) distances are popular digital distances used in various Grids. They are based on the weights assigned to steps to various neighborhoods. In the Triangular Grid there are three usually used neighbor relations, consequently, chamfer distances based on three weights are used. A chamfer (or digital) disk of a Grid is the set of the pixels which have distance from the origin that is not more than a given finite bound called radius. These disks are well known and well characterized on the square Grid. Using the two basic (i.e., the cityblock and the chessboard) neighbors, the convex hull of a disk is always an octagon (maybe degenerated). Recently, these disks have been defined on the Triangular Grid; their shapes have a great variability even with the traditional three type of neighbors, but their complete characterization is still missing. Chamfer balls are convex hulls of integer points that lie in polytopes defined by linear inequalities, and thus can be computed through a linear integer programming approach. Generally, the integer hull of a polyhedral set is the convex hull of the integer points of the set. In most of the cases, for example when the set is bounded, the integer hull is a polyhedral set, as well. The integer hull can be determined in an iterative way by Chvatal cuts. In this paper, sides of the chamfer disks are determined by the inequalities with their Chvatal rank 1. The most popular coordinate system of the Triangular Grid uses three coordinates. By giving conditions depending only a coordinate, the embedding hexagons of the shapes are obtained. These individual bounds are described completely by Chvatal cuts. They also give the complete description of some disks. Further inequalities having Chvatal rank 1 are also discussed.
-
Digitized rotations of 12 neighbors on the Triangular Grid
Annals of Mathematics and Artificial Intelligence, 2020Co-Authors: Aydın Avkan, Benedek Nagy, Müge SaadetoğluAbstract:There are various geometric transformations, e.g., translations, rotations, which are always bijections in the Euclidean space. Their digital counterpart, i.e., their digitized variants are defined on discrete Grids, since most of our pictures are digital nowadays. Usually, these digital versions of the transformations have different properties than the original continuous variants have. Rotations are bijective on the Euclidean plane, but in many cases they are not injective and not surjective on digital Grids. Since these transformations play an important role in image processing and in image manipulation, it is important to discover their properties. Neighborhood motion maps are tools to analyze digital transformations, e.g., rotations by local bijectivity point of view. In this paper we show digitized rotations of a pixel and its 12-neighbors on the Triangular Grid. In particular, different rotation centers are considered with respect to the corresponding main pixel, e.g. edge midpoints and corner points. Angles of all locally bijective and non-bijective rotations are described in details. It is also shown that the Triangular Grid shows better performance in some cases than the square Grid regarding the number of lost pixels in the neighborhood motion map.
-
On the Number of Shortest Weighted Paths in a Triangular Grid
Mathematics, 2020Co-Authors: Benedek Nagy, Bashar KhassawnehAbstract:Counting the number of shortest paths in various graphs is an important and interesting combinatorial problem, especially in weighted graphs with various applications. We consider a specific infinite graph here, namely the honeycomb Grid. Changing to its dual, the Triangular Grid, paths between triangle pixels (we abbreviate this term to trixels) are counted. The number of shortest weighted paths between any two trixels of the Triangular Grid is discussed. For each trixel, there are three different types of neighbor trixels, 1-, 2- and 3-neighbours, depending the Euclidean distance of their midpoints. When considering weighted distances, the positive values α, β and γ are assigned to the ‘steps’ to various neighbors. We gave formulae for the number of shortest weighted paths between any two trixels in various cases by the respective weight values. The results are nicely connected to various numbers well-known in combinatorics, e.g., to binomial coefficients and Fibonacci numbers.
-
on the angles of change of the neighborhood motion maps on the Triangular Grid
International Symposium on Image and Signal Processing and Analysis, 2019Co-Authors: Aydın Avkan, Benedek Nagy, Müge SaadetoğluAbstract:Rotations of digital images are important geometric transformations which are considered in various digital Grids. These digitized rotations have different properties than the analogous Euclidean rotations. On the Triangular Grid, we consider the digitized rotations of a pixel (referred as the main pixel) together with its closest neighbor pixels. Given a main pixel, of a certain distance dfrom the origin, we calculate the angles of the rotations where the neighborhood motion map of the main pixel changes. The neighborhood motion map changes when the respective locations of the closest neighbors change. We also differentiate the cases when the neighborhood motion map is injective and not injective. The former case is connected to the bijective digital rotations, while the latter case is connected to the cases when some image information is lost (in the neighborhood of the main pixel).
-
digitized rotations of closest neighborhood on the Triangular Grid
International Workshop on Combinatorial Image Analysis, 2018Co-Authors: Aydın Avkan, Benedek Nagy, Müge SaadetoğluAbstract:Rigid motions on the plane play an important role in image processing and in image manipulation. They have many properties including the bijectivity and the isometry. On the other hand, digitized rigid motions may fail to satisfy this injectivity or surjectivity properties. Pluta et al. investigated digitized rigid motions locally on the square Grid and the hexagonal Grid by using neighborhood motion maps. In this paper we show digitized rigid rotations of a pixel and its closest neighbors on the Triangular Grid. In particular, different rotation centers are considered with respect to the corresponding main pixel, e.g. edge midpoints and corner points. Angles of all bijective and non-bijective rotations are proven for rotations described above.
Yaohsin Hwang - One of the best experts on this subject based on the ideXlab platform.
-
stability and accuracy analyses for the incompressible navier stokes equations on the staggered Triangular Grid
Numerical Heat Transfer Part B-fundamentals, 1997Co-Authors: Yaohsin HwangAbstract:Abstract Stability and accuracy analyses for the discretized incompressible Navier-Stokes equations on the staggered Triangular Grid are performed in this article. The discretized equations are derived by a finite-volume procedure with a term-by-term consideration of flow convection, pressure gradient, and diffusion contributions, which results in a highly comprehensible solution procedure. The staggered Grid, where pressure is calculated at the cell center and velocity on the cell faces, is adopted to eliminate the pressure checkerboard problem, which may occur in a nonstaggered Grid. Stability study based on Fourier analysis provides the mathematical explanation for the prevention of the unphysical pressure field. Accuracy of the discretized equations is depicted with the truncation error term derived from Taylor series expansion. A previous formulation is proven to be conditionally consistent, and therefore a modified version is proposed to provide an unconditionally consistent discretization regardles...
-
calculations of incompressible flow on a staggered Triangular Grid part ii applications
Numerical Heat Transfer Part B-fundamentals, 1995Co-Authors: Yaohsin HwangAbstract:Abstract In this article, applicability of the solution procedure on a staggered Triangular Grid is shown by solving several test problems. The computational results are also compared with those by other numerical analyses or available experimental data. It is shown that the proposed method is a feasible tool to calculate incompressible flow field with arbitrary boundaries. Meanwhile, several computational parameters in the solution procedure are considered to assess the numerical performance of the present formulation by soaring two more complicated problems. These computational parameters include a linear system equation solver, inner iteration number, and underrelaxation factors. From the numerical experiments, it is demonstrated that, with suitably chosen computational parameters, this suggested method is quite robust in solving incompressible flow over a wide range of Reynolds numbers.
-
calculations of incompressible flow on a staggered Triangular Grid part i mathematical formulation
Numerical Heat Transfer Part B-fundamentals, 1995Co-Authors: Yaohsin HwangAbstract:Abstract A novel staggered Triangular Grid system and the associated solution procedure to simulate incompressible flow field is proposed in this article. In this Grid system, pressure is calculated at the centroid of the Triangular cell and velocity components at the faces. The difference governing equations are derived using a general finite-volume procedure, which is based on term-by-term consideration of flow convection, diffusion, and source contributions. These terms are cast to reflect the physical meaning of various flow processes, thus resulting in a highly comprehensible solution procedure. A standard SIMPLE pressure correction method is employed. From the solution procedure described, it is seen that the present formulation can be easily extended to apply on meshes with arbitrarily convex polygons.
A. Ellgardt - One of the best experts on this subject based on the ideXlab platform.
-
A Single Polarized Triangular Grid Tapered-Slot Array Antenna
IEEE Transactions on Antennas and Propagation, 2009Co-Authors: A. Ellgardt, A. WikstromAbstract:A Triangular Grid single polarized tapered-slot array antenna for radar applications is studied. Compared with a rectangular Grid an equilateral Triangular Grid allows a larger unit cell without any onset of grating lobes. Since single polarized tapered-slots in Triangular Grids support guided modes, which cause scan blindness, the increase in unit cell size is smaller than the optimal 15%. The design presented in the paper is capable of scan angles out to 60° from broadside in the E and H planes. To improve the match over the radar band a local minimum in the active reflection coefficient is positioned at the most critical scan direction, resulting in a reflection coefficient that is less than -12 dB in the X-band. To reduce the radar cross section for the cross-polarization an absorbing layer is positioned above the ground plane, which affects some of the guided modes that lead to scan blindnesses. An experimental antenna with 16 × 16 elements was built, and it was found that the H-plane performance for large scan angles for the finite antenna deviates more than expected from the infinite array approximation. Otherwise both mutual coupling measurements and embedded element patterns agrees well with the numerical results.
-
effects on scan blindnesses of an absorbing layer covering the ground plane in a Triangular Grid single polarized tapered slot array
IEEE Antennas and Propagation Society International Symposium, 2008Co-Authors: A. EllgardtAbstract:Tapered-slot phased arrays can be capable of wide-band and wide-scan performance, and are therefore useful in numerous applications. Some applications require the antenna to have a low radar cross section. However, a single-polarized tapered-slot antenna will have a strong backscattered field for an orthogonally polarized wave due to the ground plane. To reduce the backscattered field an absorber can be positioned above the ground plane. The effect of the absorber is small compared to elements for which the ground plane distance is an important design parameter. Single-polarized tapered-slot antennas form a parallel plate waveguide structure where the cutoff frequency is determined by the H-plane spacing and substrate thickness. The cutoff frequency condition is approximately the same as the condition for a grating lobe free antenna, which requires a H-plane spacing that is less than lambda/2. Therefore, for a wide-scan array, waves with the same polarization as the elements are cutoff by the parallel plate structure for the operational frequencies and the absorbing layer will have almost no affect on the performance. The parallel plate structure can support surface waves and these waves will lead to scan blindness in the E-plane [1, 2, 3]. The blindness described in [3] occurs in Triangular Grid arrays when they are scanned in the E-plane and moves the field intensity towards the ground plane where the absorbing layer can reduce the backscattered field. Therefore, the absorber mainly alters the performance when the element does not function as intended.
Brian Y. Sun - One of the best experts on this subject based on the ideXlab platform.
-
Word-Representability of Face Subdivisions of Triangular Grid Graphs
Graphs and Combinatorics, 2016Co-Authors: Herman Z. Chen, Sergey Kitaev, Brian Y. SunAbstract:A graph $$G=(V,E)$$G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if $$(x,y)\in E$$(x,y)źE. A Triangular Grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a Triangular Grid graph is replacing some of its cells by plane copies of the complete graph $$K_4$$K4. Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of Grid graphs related to convex polyominoes, we characterize word-representable face subdivisions of Triangular Grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any face subdivision of boundary triangles in the Sierpinski gasket graph is word-representable.
-
Word-representability of subdivisions of Triangular Grid graphs
arXiv: Combinatorics, 2015Co-Authors: Zongqing Chen, Sergey Kitaev, Brian Y. SunAbstract:A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$. A Triangular Grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A subdivision of a Triangular Grid graph is replacing some of its cells by plane copies of the complete graph $K_4$. Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of Grid graphs related to convex polyominoes, we characterize word-representable subdivisions of Triangular Grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any subdivision of boundary triangles in the Sierpi\'{n}ski gasket graph is word-representable.
Gavin Walker - One of the best experts on this subject based on the ideXlab platform.
-
Finite volume approximation of multidimensional aggregation population balance equation on Triangular Grid
Mathematics and Computers in Simulation, 2020Co-Authors: Mehakpreet Singh, R Singh, Sukhjit Singh, Gagandeep Singh, Gavin WalkerAbstract:Abstract The present work shows the first ever implementation of two-order moments conserving finite volume scheme (FVS) for approximating a multidimensional aggregation population balance equations (PBE’s) on a structured Triangular Grid. This scheme is based on preservation of the zeroth and conservation of the first order moments. Our main aim is to demonstrate the ability of the FVS to adapt the structured Triangular Grid well, hence, improves the accuracy of number density function as well as various order moments. The numerical results obtained by the FVS on a Triangular Grid are compared with the cell average technique. The comparison is also enhanced to illustrate that the FVS with a Triangular Grid provides the numerical results with higher precision and at lesser computational time as compared to the FVS with a rectangular Grid. Additionally, we also study the mixing state of a bicomponent population of clusters (granules) characterized by the normalized variance of excess solute, χ , a parameter that measures the deviation of the composition of each granule from the overall mean. It is shown that the accuracy of the total variance of the excess solute improves when a Triangular Grid is used in place of a rectangular Grid.
-
finite volume approximation of multidimensional aggregation population balance equation on Triangular Grid
Mathematics and Computers in Simulation, 2020Co-Authors: Mehakpreet Singh, R Singh, Sukhjit Singh, Gagandeep Singh, Gavin WalkerAbstract:Abstract The present work shows the first ever implementation of two-order moments conserving finite volume scheme (FVS) for approximating a multidimensional aggregation population balance equations (PBE’s) on a structured Triangular Grid. This scheme is based on preservation of the zeroth and conservation of the first order moments. Our main aim is to demonstrate the ability of the FVS to adapt the structured Triangular Grid well, hence, improves the accuracy of number density function as well as various order moments. The numerical results obtained by the FVS on a Triangular Grid are compared with the cell average technique. The comparison is also enhanced to illustrate that the FVS with a Triangular Grid provides the numerical results with higher precision and at lesser computational time as compared to the FVS with a rectangular Grid. Additionally, we also study the mixing state of a bicomponent population of clusters (granules) characterized by the normalized variance of excess solute, χ , a parameter that measures the deviation of the composition of each granule from the overall mean. It is shown that the accuracy of the total variance of the excess solute improves when a Triangular Grid is used in place of a rectangular Grid.
-
finite volume approximation of nonlinear agglomeration population balance equation on Triangular Grid
Journal of Aerosol Science, 2019Co-Authors: Mehakpreet Singh, Hamza Y Ismail, R Singh, Ahmad B Albadarin, Gavin WalkerAbstract:Abstract In this present work, a finite volume scheme for approximating a multidimensional nonlinear agglomeration population balance equation on a regular Triangular Grid is developed. The finite volume schemes developed in literature are restricted to a rectangular Grid [43]. However, the accuracy and efficiency of finite volume scheme can be enhanced by considering Triangular Grids. The Triangular Grid is generated using the concept of ‘Voronoi Partitioning’ and ‘Delaunay Triangulation’. To test the accuracy and efficiency of the scheme on a Triangular Grid, the numerical results are compared with the sectional method, namely Cell Average Technique [38] for various analytically tractable kernels. The results reveal that the finite volume scheme on a Triangular Grid is computationally less expensive and predicts the number density function along with the different order moments more accurately than the cell average technique. Furthermore, the numerical comparison is extended by comparing the finite volume scheme on a rectangular Grid. It also demonstrates that the finite volume scheme with a regular Triangular Grid computes the numerical results more accurately and efficiently than the finite volume scheme with a rectangular Grid.
