Non-traditional 2D Grids in Combinatorial Imaging – Advances and Challenges

Nagy, Benedek

doi:10.1007/978-3-031-23612-9_1

Cited by 8 publications

(4 citation statements)

References 95 publications

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“…We should mention one of the main advantages of dual semi‐regular grids over the semi‐regular ones: in the former, the tiles are of the same size, which gives the idea to use them as image grids: the same‐size pixels give more pleasant pictures than pictures on grids without this property. On the other hand, the symmetric properties of the semi‐regular and dual semi‐regular grids are very similar to the symmetric properties of the regular grids (Conway et al, 2008; Nagy, 2022).…”

Section: Introductionsupporting

confidence: 59%

“…By our terminology, a two‐dimensional grid is a periodic tessellation/tiling of the plane that is based on one or more geometric shapes, referred to as tiles, without overlaps and gaps. Since our world is generally not rectangular, various non‐traditional grids and tilings have significant importance in various disciplines (see, e.g., Nagy, 2022; Radványi, 2002), in ancient and modern arts, in mosaics, floor tilings, etc. Thus, when one changes the scope from continuous to digital, there could be many other choices than the square/rectangular grid.…”

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: Introductionsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

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

Nagy

2023

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“…Each of the dual grids of the semi-regular grids is built up by only one tile which is not a regular polygon and appears in various orientations. Some of the semi-regular grids, and also some of their dual grids, have various advantages over the square grid (Nagy, 2023). Digital geometry is discussed by Conway et al (2008), Kiselman (2022) and Klette & Rosenfeld (2004).…”

Section: Introductionmentioning

confidence: 99%

A digital distance on the kisrhombille tiling

Kablan,

Vizvári,

Nagy

2024

Acta Cryst Sect A

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The kisrhombille tiling is the dual tessellation of one of the semi-regular tessellations. It consists of right-angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons.

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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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Non-traditional 2D Grids in Combinatorial Imaging – Advances and Challenges

Cited by 8 publications

References 95 publications

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

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

A digital distance on the kisrhombille tiling

Weighted distances and distance transforms on the triangular tiling

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