Non-bijective translations on the triangular plane

Abuhmaidan, Khaled; Nagy, Benedek

doi:10.1109/sami.2018.8324836

Cited by 7 publications

(5 citation statements)

References 14 publications

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“…Concluding remarks close the paper. This paper is a continuation and extension of our earlier conference paper [11] where non-bijective and bijective translations were presented. Here we also separate two cases of the bijective ones.…”

Section: Introductionmentioning

confidence: 83%

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

Abuhmaidan¹,

Nagy

2019

Mathematics

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The triangular plane is the plane which is tiled by the regular triangular tessellation. The underlying discrete structure, the triangular grid, is not a point lattice. There are two types of triangle pixels. Their midpoints are assigned to them. By having a real-valued translation of the plane, the midpoints of the triangles may not be mapped to midpoints. This is the same also on the traditional square grid. However, the redigitized result on the square grid always gives a bijection (gridpoints of the square grid are mapped to gridpoints in a bijective way). This property does not necessarily hold on to the triangular plane, i.e., the redigitized translated points may not be mapped to the original points by a bijection. In this paper, we characterize the translation vectors that cause non bijective translations. Moreover, even if a translation by a vector results in a bijection after redigitization, the neighbor pixels of the original pixels may not be mapped to the neighbors of the resulting pixel, i.e., a bijective translation may not be digitally ‘continuous’. We call that type of translation semi-bijective. They are actually bijective but do not keep the neighborhood structure, and therefore, they seemingly destroy the original shape. We call translations strongly bijective if they are bijective and also the neighborhood structure is kept. Characterizations of semi- and strongly bijective translations are also given.

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Section: Introductionmentioning

confidence: 83%

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

Abuhmaidan¹,

Nagy

2019

Mathematics

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“…This extension is needed and helpful for various applications, where the grid points are not necessarily mapped to grid points, e.g., arbitrary angled rotations, zooming or interpolation of images. We should also mention translations of images [25] since the triangular grid is not a point lattice. Mathematical morphology operators are also based on local translations [21].…”

Section: Discussionmentioning

confidence: 99%

A Continuous Coordinate System for the Plane by Triangular Symmetry

Nagy¹,

Abuhmaidan²

2019

Symmetry

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The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this paper, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems.The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [−1, 1], which gives many other vital properties of this coordinate system. Symmetry 2019, 11, 191 2 of 17 of the grid. In Figure 1a, the first coordinate value is ascending right-upwardly, the second values are ascending into the right-downward direction, and the third one is ascending into the left-upward direction [7,8]. We should mention that this system could be seen as the extension of the oblique coordinate system using two independent integer values [9] by concerning the third value to obtain zero-sum for every triplet. The digital distance based on the neighborhood relation is computed in Reference [9]. Since the vectors describing the grid are not orthogonal, some geometric descriptions based on Cartesian coordinates are not very clear. However, to simplify the expressions of the constrained three-dimensional coordinate system is recommended. We should also mention that 0-sum triplets allowing real numbers were used in Reference [8] to describe rotations (that may not map the hexagonal grid to itself). In this way, a useful digitization operator is found. Her's system was mentioned and used in References [10,11] for various imaging-related disciplines.Symmetry 2019, 11, 191 3 of 17

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“…In our approach, for brevity, we consider the digitized rigid motions having no translation, i.e., the translation vector is the null vector. We note here that translations on the triangular grid are interesting themselves, since some of them are not bijective, and they are described in [2]. In the earlier mentioned works on the topic on other grids, e.g., in [28,29], the center of the rotation was always the midpoint of a pixel.…”

Section: Introductionmentioning

confidence: 94%

“…Another way to continue the research is to combine rotations with other operations, e.g., translations (described in [2]).…”

Section: Digitized Rotations Of 12 Neighbors On the Triangular Gridmentioning

confidence: 99%

“…This is impossible to do with hexagonal pixels (hexels). Various image processing techniques and algorithms are also performed on the triangular grid: tomography is considered in [22][23][24], digital distances are computed in [18], digital circles and disks are characterized in [12,17,21], mathematical morphology is presented in [1,30], digital translations are described [2], thinning algorithms and skeletonization are presented [8][9][10][11] and straight lines are characterized in [6]. While in the hexagonal grid every hexel has six neighbors, in the square grid each pixel has (four closest, so-called cityblock neighbors and altogether) eight (so called chessboard) neighbors, in the triangular grid a trixel has twelve neighbors (including three closest ones).…”

Section: Introductionmentioning

confidence: 99%

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Digitized rotations of 12 neighbors on the triangular grid

Avkan

Nagy

Saadetoğlu

2020

Ann Math Artif Intell

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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 nonbijective 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.

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Non-bijective translations on the triangular plane

Cited by 7 publications

References 14 publications

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

A Continuous Coordinate System for the Plane by Triangular Symmetry

Digitized rotations of 12 neighbors on the triangular grid

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