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 non-bijective 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.

中文翻译：

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三角形网格上 12 个邻居的数字化旋转

存在各种几何变换，例如平移、旋转，它们在欧几里得空间中始终是双射。它们的数字对应物，即它们的数字化变体是在离散网格上定义的，因为如今我们的大多数图片都是数字化的。通常，这些转换的数字版本具有与原始连续变体不同的属性。旋转在欧几里得平面上是双射的，但在许多情况下，它们在数字网格上不是单射的，也不是满射的。由于这些变换在图像处理和图像处理中起着重要作用，因此发现它们的特性很重要。邻域运动地图是分析数字变换的工具，例如，通过局部双射性观点的旋转。在本文中，我们在三角形网格上展示了一个像素及其 12 个邻域的数字化旋转。特别地，相对于相应的主像素，例如边缘中点和角点，考虑不同的旋转中心。详细描述了所有局部双射和非双射旋转的角度。还表明，在邻域运动图中丢失像素的数量方面，三角形网格在某些情况下比方形网格表现出更好的性能。