Abstract
Recently, operations research, especially linear integer-programming, is used in various grids to find optimal paths and, based on that, digital distance. The 4 and higher-dimensional body-centered-cubic grids is the nD (n≥4) equivalent of the 3D body-centered cubic grid, a well-known grid from solid state physics. These grids consist of integer points such that the parity of all coordinates are the same: either all coordinates are odd or even. A popular type digital distance, the chamfer distance, is used which is based on chamfer paths. There are two types of neighbors (closest same parity and closest different parity point-pairs), and the two weights for the steps between the neighbors are fixed. Finding the minimal path between two points is equivalent to an integer-programming problem. First, we solve its linear programming relaxation. The optimal path is found if this solution is integer-valued. Otherwise, the Gomory-cut is applied to obtain the integer-programming optimum. Using the special properties of the optimization problem, an optimal solution is determined for all cases of positive weights. The geometry of the paths are described by the Hilbert basis of the non-negative part of the kernel space of matrix of steps.
Highlights
In digital geometry, by modeling the world on a grid, path-based distances are frequently used [1,2]
The face-centered cubic (FCC) and body-centered cubic (BCC) lattices are very important non-traditional grids appearing in nature
While the FCC grid is obtained from the cubic grid by adding a point to the center of each square face of the unit cubes, in the BCC
Summary
By modeling the world on a grid, path-based distances are frequently used [1,2]. One of the main advantages of the BCC grid is, that there are only face neighbor voxels, i.e., if two Voronoi bodies of the grid share at least one point on their boundary, they share a full face (either a hexagon or a square) In this way, in the BCC grid, the topological paradox mentioned above cannot occur. Concerning path-based (in grids, they are called digital) distances, one of the simplest, but on the other hand, very practical and well applicable choices is to use chamfer distances [3]. These distances are, weighted distances based on various (positive) weights assigned to steps to various types of neighbors.
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