Digitized Rotations of Closest Neighborhood on the Triangular Grid

Avkan, Aydın; Nagy, Benedek; Saadetoğlu, Müge

doi:10.1007/978-3-030-05288-1_5

Cited by 8 publications

(3 citation statements)

References 11 publications

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“…The authors of that paper extended the previous study of the digitized rigid motions from the square grid to the hexagonal grid in the framework of neighborhood motion maps. In this paper, extending the work presented in [3], we describe the neighborhood motion maps of the 12-neighborhood (that is the main pixel and its 12 neighbors) on the triangular grid. In [3], maps considering the closest neighborhood, that contains 3 neighbors, were presented.…”

Section: Introductionmentioning

confidence: 94%

“…In this paper, extending the work presented in [3], we describe the neighborhood motion maps of the 12-neighborhood (that is the main pixel and its 12 neighbors) on the triangular grid. In [3], maps considering the closest neighborhood, that contains 3 neighbors, were presented. In our approach, for brevity, we consider the digitized rigid motions having no translation, i.e., the translation vector is the null vector.…”

Section: Introductionmentioning

confidence: 94%

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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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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 94%

Digitized rotations of 12 neighbors on the triangular grid

Avkan

Nagy

Saadetoğlu

2020

Ann Math Artif Intell

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“…We have discovered various cases of translations on the triangular plane and a full characterization of them is given to three categories. In the future, we plan to analyze also other operations, e.g., rotations [14,15] mixed with translations and other more complex operations on the triangular grid. Our results can be applied not only in image processing and computer graphics, but also, in various engineering solutions connected to digital signal processing, camera sensors, etc.…”

Section: Characterizing Semi-bijective Translationsmentioning

confidence: 99%

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

Nagy

2023

Lecture Notes in Computer Science

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Digitized Rotations of Closest Neighborhood on the Triangular Grid

Cited by 8 publications

References 11 publications

Digitized rotations of 12 neighbors on the triangular grid

Digitized rotations of 12 neighbors on the triangular grid

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

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

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