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Chamfer distances on the isometric grid: a structural description of minimal distances based on linear programming approach

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Abstract

Chamfer distances on the isometric grid are considered. A new method to compute the chamfer distances based on linear optimization is presented. In the LP model the starting pixel is the Origin, that is a triangle of the grid having co-ordinates (0, 0, 0). The co-ordinates of the end pixel of the path give the right-hand side of the model. The variables are the used numbers of the elementary steps. Each type of an elementary step has a uniquely defined weight. Our operational research approach determines the optimal paths as basic feasible solutions of a linear programming problem. We give directed graphs with feasible bases as nodes and arcs with conditions on the used weights such that the simplex method of linear programming may step from one feasible basis to another feasible basis. Thus, the possible course of the simplex method can be followed and the optimal bases can easily be captured. Thus, the final result of the analysis is an O(1) checking of the feasibility and optimality conditions. The optimal bases are summarized in a theorem which is the consequence of the general theory of linear programming. The method can be applied for other grids, but it needs to be adjusted for the particular grid.

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Authors and Affiliations

  1. Edutus University, Tatabánya, Hungary

    Gergely Kovács

  2. Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, via Mersin-10, Famagusta, North Cyprus, Turkey

    Benedek Nagy

  3. Department of Industrial Engineering, Eastern Mediterranean University, via Mersin-10, Famagusta, North Cyprus, Turkey

    Béla Vizvári

Authors
  1. Gergely Kovács
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  2. Benedek Nagy
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  3. Béla Vizvári
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Correspondence to Gergely Kovács.

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Kovács, G., Nagy, B. & Vizvári, B. Chamfer distances on the isometric grid: a structural description of minimal distances based on linear programming approach. J Comb Optim 38, 867–886 (2019). https://doi.org/10.1007/s10878-019-00425-x

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  • DOI: https://doi.org/10.1007/s10878-019-00425-x

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