Digitized rotations of 12 neighbors on the triangular grid

Avkan, Aydın; Nagy, Benedek; Saadetoğlu, Müge

doi:10.1007/s10472-019-09688-w

Cited by 7 publications

(2 citation statements)

References 29 publications

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“…The information loss which is an important factor at various transformations, e.g., rotations of digital images is coming when the result is re-digitized, which operation, of course, causes certain information loss in many cases either or both on the square and on the triangular grids. Some preliminary studies on the digital rotations comparing these grids can be found in [ 40 ].…”

Section: Discussionmentioning

confidence: 99%

Vector Arithmetic in the Triangular Grid

Abuhmaidan¹,

Aldwairi

Nagy

2021

Entropy

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Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

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Section: Discussionmentioning

confidence: 99%

Vector Arithmetic in the Triangular Grid

Abuhmaidan¹,

Aldwairi

Nagy

2021

Entropy

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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 12 neighbors on the triangular grid

Cited by 7 publications

References 29 publications

Vector Arithmetic in the Triangular Grid

Vector Arithmetic in 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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