Digital Geometry on the Dual of Some Semi-regular Tessellations

Saadat, MohammadReza; Nagy, Benedek

doi:10.1007/978-3-030-76657-3_20

Cited by 8 publications

(6 citation statements)

References 15 publications

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“…The coordinate system we provide here (and defined and presented first in Saadat & Nagy, 2021) is based on the Cartesian coordinate frame for the square grid; thus, we use pairs for addressing the tiles.…”

Section: Description Of the Tiling: Coordinate Systemmentioning

confidence: 99%

“…Some preliminary parts of the article were presented at the First International Conference on Discrete Geometry and Mathematical Morphology (DGMM 2021, Saadat & Nagy, 2021). The author also thanks the help of his PhD student M. R. Saadat for writing software tools.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…It should be noted that various coordinate systems have been invented for these grids, for comparisons of these systems, see for example (Comic, 2021a, 2021b). Moreover, in Comic (2021b) a coordinate system is suggested that is equivalent to the coordinate system we have coined in Saadat and Nagy (2021) and we also use in this article. On the other hand, the studies of the papers (Comic, 2021a, 2021b) are concentrated only on the coordinate systems and conversions between them, and thus, digital distances, properties of the grid and other features have not been addressed there.…”

Section: Introductionmentioning

confidence: 99%

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A digital geometry on the tetrakis square tiling—Distance and coarsening

Nagy

2023

Transactions in GIS

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There are various tessellations of the plane, including three regular and eight semi‐regular tilings. The square grid is self‐dual, and the hexagonal and triangular tilings are dual to each other. The semi‐regular tessellations are based on more than one type of regular tiles, while their dual tilings are based on a sole but not a regular tile. In various applications, including Geographical Information Systems, it is worth considering non‐regular grids instead of the most used square grid. In this article, we are interested in the dual of the semi‐regular truncated quadrille tiling, T(8,8,4), which is also known as the Khalimsky grid due to its connectedness structure. In our grid, which is called the tetrakis square or kisquadrille tiling, while it is denoted by D(8,8,4), we consider the right‐angled triangle regions of the usual two‐dimensional Khalimsky graph as tiles/pixels. We give an easy‐to‐use coordinate frame addressing the triangles of all the four different orientations. Neighbor relations are described mathematically based on this frame. Based on the shortest path algorithm, a closed formula is proven to compute the digital, that is, path‐based distance on this grid. Some properties of the distance function have also been studied. Hierarchical coarsening is a frequently used technique both in Geometric and Geographical Information Systems to rescale some parts of the map. The tetrakis square grid is apt for hierarchical coarsening, and thus, it can easily be used in image compression and multigrid and other related methods.

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Section: Description Of the Tiling: Coordinate Systemmentioning

confidence: 99%

Section: Acknowledgmentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A digital geometry on the tetrakis square tiling—Distance and coarsening

Nagy

2023

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“…We also recall here that distances on the hexagonal (Luczak & Rosenfeld, 1976;Nagy, 2002Nagy, , 2017, triangular (Nagy, 2002(Nagy, , 2014 and on some other (2D) grids, including some semi-regular grids and some other related structures, were computed: e.g. Khalimsky grid, also called truncated square tiling (Kova ´cs et al, 2017a), truncated hexagonal grid (Kova ´cs et al, 2021c), trihexagonal grid (Kova ´cs & Nagy, 2017b), Cairo pattern (Kova ´cs et al, 2021b) and some others (Saadat & Nagy, 2021). In 3D, the two basic neighborhood-based distances were computed on the f.c.c.…”

Section: Literature Reviewmentioning

confidence: 99%

Distances in the face-centered cubic crystalline structure applying operational research

Stomfai,

Kovács,

Nagy

et al. 2023

Acta Cryst Sect A

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The f.c.c. (face-centered cubic) grid is the structure of many crystals and minerals. It consists of four cubic lattices. It is supposed that there are two types of steps between two grid points. It is possible to step to one of the nearest neighbors of the same cubic lattice (type 1) or to step to one of the nearest neighbors of another cubic lattice (type 2). Steps belonging to the same type have the same length (weight). However, the two types have different lengths and thus may have different weights. This paper discusses the minimal path between any two points of the f.c.c. grid. The minimal paths are explicitly given, i.e. to obtain a minimal path one is required to perform only O(1) computations. The mathematical problem can be the model of different spreading phenomena in crystals having the f.c.c. structure.

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“…In this way the computation can be fast and efficient. We should note that there are various non‐traditional grids (Nagy, 2023a; Saadat & Nagy, 2021) having various applications in relation to GIS (de Sousa & Leitão, 2018; Nagy, 2023b; Wang et al., 2020). Triangular grids have been applied frequently, as on the one hand, the triangles are the simplest polygons, thus in various applications, it is very easy to work with them.…”

Section: Introductionmentioning

confidence: 99%

Weighted distances and distance transforms on the triangular tiling

Nagy

2023

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Digital geometry is a field in the intersection of discrete mathematics and geometry having various applications including geographical information systems (GIS). In digital spaces, in grids, distances can be defined based on steps in paths in somewhat similarly as in graph theory. However, the grids have more definite structures, thus one may obtain more concrete results, for example, close formulae, than on arbitrary graphs. In this article, the weighted (also called chamfer) distances, and based on them, the distance transform are investigated on the regular triangular grid. Three types of neighborhood relations are used on the grid, and therefore, three weights are used to define a distance function. Natural conditions are used on the weights such as they are positive and a larger step (in the usual and also in the Euclidean sense) cannot have a smaller weight than a smaller one. Some properties of the weighted distances are discussed; for example, they are proven to be metrics. We also give algorithms and formulae that compute the weighted distance of any point pair on a triangular grid. Algorithm for weighted distance transform is provided based on wave‐front propagation. Therefore, these new distance functions are ready for further applications in GIS, in image processing tasks, in computer vision, in graphics, in networking, and also in other applied fields.

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Digital Geometry on the Dual of Some Semi-regular Tessellations

Cited by 8 publications

References 15 publications

A digital geometry on the tetrakis square tiling—Distance and coarsening

A digital geometry on the tetrakis square tiling—Distance and coarsening

Distances in the face-centered cubic crystalline structure applying operational research

Weighted distances and distance transforms on the triangular tiling

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