Digital geometry is a field in the intersection of discrete mathematics and geometry having various applications including geographical information systems (GIS). In digital spaces, in grids, distances can be defined based on steps in paths in somewhat similarly as in graph theory. However, the grids have more definite structures, thus one may obtain more concrete results, for example, close formulae, than on arbitrary graphs. In this article, the weighted (also called chamfer) distances, and based on them, the distance transform are investigated on the regular triangular grid. Three types of neighborhood relations are used on the grid, and therefore, three weights are used to define a distance function. Natural conditions are used on the weights such as they are positive and a larger step (in the usual and also in the Euclidean sense) cannot have a smaller weight than a smaller one. Some properties of the weighted distances are discussed; for example, they are proven to be metrics. We also give algorithms and formulae that compute the weighted distance of any point pair on a triangular grid. Algorithm for weighted distance transform is provided based on wave-front propagation. Therefore, these new distance functions are ready for further applications in GIS, in image processing tasks, in computer vision, in graphics, in networking, and also in other applied fields.

中文翻译：

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三角形平铺上的加权距离和距离变换

数字几何是离散数学和几何的交叉领域，具有各种应用，包括地理信息系统（GIS）。在数字空间中，在网格中，距离可以根据路径中的步骤来定义，这与图论中有些类似。然而，网格具有更明确的结构，因此可以获得比任意图更具体的结果，例如封闭公式。在本文中，研究了加权（也称为倒角）距离，并基于它们在规则三角形网格上进行距离变换。网格上使用三种邻域关系，因此使用三个权重来定义距离函数。自然条件用于权重，例如它们是正的，并且较大步长（在通常情况下以及在欧几里得意义上）不能具有比较小步长更小的权重。讨论了加权距离的一些性质；例如，它们被证明是度量。我们还给出了计算三角形网格上任何点对的加权距离的算法和公式。给出了基于波前传播的加权距离变换算法。因此，这些新的距离函数已准备好在 GIS、图像处理任务、计算机视觉、图形、网络以及其他应用领域中进一步应用。