On Chamfer Distances on the Square and Body-Centered Cubic Grids: An Operational Research Approach

Gergely Kovács,Benedek Nagy,Béla Vizvári,Gergely Stomfai,Neşet Deni̇Z Turgay,Mohammad Yazdi

doi:10.1155/2021/5582034

Abstract

Linear programming is used to solve optimization problems. Thus, finding a shortest path in a grid is a good target to apply linear programming. In this paper, specific bipartite grids, the square and the body-centered cubic grids are studied. The former is represented as a “diagonal square grid” having points with pairs of either even or pairs of odd coordinates (highlighting the bipartite feature). Therefore, a straightforward generalization of the representation describes the body-centered cubic grid in 3D. We use chamfer paths and chamfer distances in these grids; therefore, weights for the steps between the closest neighbors and steps between the closest same type points are fixed, and depending on the weights, various paths could be the shortest one. The vectors of the various neighbors form a basis if they are independent, and their number is the same as the dimension of the space studied. Depending on the relation of the weights, various bases could give the optimal solution and various steps are used in the shortest paths. This operational research approach determines the optimal paths as basic feasible solutions of a linear programming problem. A directed graph is given containing the feasible bases as nodes and arcs with conditions on the used weights such that the simplex method may step from one feasible basis to another one. Thus, the optimal bases can be determined, and they are summarized in two theorems. If the optimal solution is not integer, then the Gomory cut is applied and the integer optimal solution is reached after only one Gomory iteration. Chamfer distances are frequently used in image processing and analysis as well as graphics-related subjects. The body-centered cubic grid, which is well-known in solid state physics, material science, and crystallography, has various applications in imaging and graphics since less samples are needed to represent the signal in the same quality than on the cubic grid. Moreover, the body-centered cubic grid has also a topological advantage over the cubic grid, namely, the neighbor Voronoi cells always share a full face.

Highlights

Grids have a significant role in crystallography, material science, and solid state physics and in various engineering applications, e.g., in image processing and in digital geometry [1]. ere is a large variety of ways to tessellate the plane and the three dimensional space
Two of the nontraditional grids in 3D are the body-centered Cubic (BCC) and the face-centered cubic (FCC) grids. ey provide structures of various solid materials, and they play crucial importance in crystallography. ey are alternative to the cubic grid in various engineering applications. ese grids have many application areas, e.g., image processing [12], visualization [13], and topology preserving digitization [14]
In the hexagonal grid, there is only one usual neighborhood relation among the hexagon-shaped tiles/pixels, and two neighbor hexagons share a full side. e BCC grid generalizes this property to 3D, i.e., both of the used types of neighbor relations are among voxels such that either a full hexagon face or a square face is shared between the neighbor voxels. is topological advantage of the BCC grid can be used, e.g., by combinatorial coordinate systems in boundary tracking [15]

Summary

Introduction

Grids have a significant role in crystallography, material science, and solid state physics and in various engineering applications, e.g., in image processing and in digital geometry [1]. ere is a large variety of ways to tessellate the plane and the three dimensional space. E hexagonal grid has a useful symmetric coordinate frame, and every pixel, called hexel [5], has six neighbors. E BCC grid generalizes this property to 3D, i.e., both of the used types of neighbor relations are among voxels such that either a full hexagon face or a square face is shared between the neighbor voxels. The BCC grid is proven to have some better properties in applications than the cubic grid [16,17,18] highlighting the importance of our study. In electronic imaging, it is important what kind of mesh we scan the object according to. The BCC grid plays an important role in the latest chapters of material science [20]

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Conclusion