Mathematical Morphology on the Triangular Grid: The Strict Approach

Abdalla, Mohsen; Nagy, Benedek

doi:10.1137/19m128017x

Cited by 10 publications

(3 citation statements)

References 17 publications

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“…Thus, when one changes the scope from continuous to digital, there could be many other choices than the square/rectangular grid. Digital geometries of the other two regular grids, the triangular and hexagonal grids are also widely used and applied (Abdalla & Nagy, 2020;Brimkov & Barneva, 2005;Deutsch, 1972;Gaspar et al, 2009;Her, 1995;Kardos & Palágyi, 2015, 2017Lukic & Nagy, 2019;Matej et al, 1998Matej et al, , 1999Middleton & Sivaswamy, 2005;Nagy, 2002Nagy, , 2004aNagy, , 2004bNagy, , 2007Nagy, , 2014Nagy & Lukic, 2016Nagy & Moisi, 2017;Nagy & Strand, 2011). They have various advantages over the square grid as, for example, they have more directions of symmetry axes.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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

Nagy

2023

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“…Also, the theory of distances based on neighborhood sequences on the triangular grid is well developed (Nagy, 2003a(Nagy, , 2007b. The triangular grid appears recently in other image processing related articles, see, for example, Abdalla and Nagy (2020), Kardos and Palágyi (2015),…”

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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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“…Step 2: By applying the morphological closing operator [39], the unclosed borders of parts are retrieved and connected. Since the cross-sections of the experimental group are obtained from the 3D point cloud data, the obtained cross-sections may have some small cracks and do not form a closed geometry.…”

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Experimental Study and GRNN Modeling of Shrinkage Characteristics for Wax Patterns of Gas Turbine Blades Considering the Influence of Complex Structures

et al. 2023

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With the continuous increase in power demand in aerospace, shipping, electricity, and other industries, a series of manufacturing requirements such as high precision, complex structure, and thin wall have been put forward for gas turbines. Gas turbine blades are the key parts of the gas turbine. Their manufacturing accuracy directly affects the fuel economy of the gas turbine. Thus, how to improve the manufacturing accuracy of gas turbine blades has always been a hot research topic. In this study, we perform a quantitative study on the correlation between process parameters and the overall wax pattern shrinkage of gas turbine blades in the wax injection process. A prediction model based on a generalized regression neural network (GRNN) is developed with the newly defined cross-sectional features consisting of area, area ratio, and some discrete point deviations. In the qualitative analysis of the cross-sectional features, it is concluded that the highest accuracy of the wax pattern is obtained for the fourth group of experiments, which corresponds to a holding pressure of 18 bars, a holding time of 180 s, and an injection temperature of 62 °C. The prediction model is trained and tested based on small experimental data, resulting in an average RE of 1.5% for the area, an average RE of 0.58% for the area ratio, and a maximum MSE of less than 0.06 mm2 for discrete point deviations. Experiments show that the GRNN prediction model constructed in this study is relatively accurate, which means that the shrinkage of the remaining major investment casting procedures can also be modeled and controlled separately to obtain turbine blades with higher accuracy.

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Mathematical Morphology on the Triangular Grid: The Strict Approach

Cited by 10 publications

References 17 publications

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

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

Weighted distances and distance transforms on the triangular tiling

Experimental Study and GRNN Modeling of Shrinkage Characteristics for Wax Patterns of Gas Turbine Blades Considering the Influence of Complex Structures

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