Abstract
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.
Highlights
Digital images consist of a set of pixels
We can say that the hexagonal and triangular grids are valid alternatives of the traditional grid both in computer graphics and in digital image processing [1], in some cases, they could lead to better results than the square grid
The discretized translations in the square and the hexagonal grids are always bijective, but they are not always bijective in the triangular grid. This is because when a grid-vector is taken from any gridpoint, it will always end up at a certain gridpoint, i.e., these two grids are point lattices
Summary
Digital images consist of a (finite) set of pixels. both the (re)presentations of the images and the possible operations on them are strictly connected to the underlying grid. We can say that the hexagonal and triangular grids are valid alternatives of the traditional grid both in computer graphics and in digital image processing [1], in some cases, they could lead to better results than the square grid. Are valid alternatives of the traditional grid bothdiscrete in computer the image qualityand or triangular losing some By their different properties, and graphics and in digital image processing [1], in some cases, they could lead to better results continuous transformations yield two very different theories. Translations of images are such losing some information.” By their different properties, discrete and continuous transformations yield basic and frequently used operations and usually do not garner any attention alone. We separate two cases of the bijective ones
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