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FPTAS for Minimizing the Earth Mover’s Distance Under Rigid Transformations and Related Problems

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Abstract

In this paper, we consider the problem (denoted as EMDRT) of minimizing the earth mover’s distance between two sets of weighted points A and B in \({\mathbb {R}}^{d}\) under rigid transformation. EMDRT is an important problem in both theory and applications and has received considerable attentions in recent years. Previous research on this problem has resulted in only constant factor approximations and it has been an open problem for a long time to achieve PTAS solution. In this paper, we present the first FPTAS algorithm for EMDRT. Our algorithm runs roughly in \(O((nm)^{d+2}(\log nm)^{2d})\) time (which is close to a lower bound on any PTAS for this problem), where n and m are the sizes of A and B, respectively. Our result is based on several new techniques, such as the Sequential Orthogonal Decomposition and Optimum Guided Base, and can be extended to several related problems, such as the problem of earth mover’s distance under similarity transformation and the alignment problem, to achieve FPTAS for each of them.

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Notes

  1. Note that [17, 23] also analyze the rigid transformations in a step-by-step fashion, but their methods are only for unweighted point-sets.

  2. We just need to determine which point belongs to the base from B, rather than specify the position of each point.

  3. This lemma might have been studied in some other papers or presented in other forms; for self-completeness, we present our proof here.

  4. In \({\mathbb {R}}^d\), a j-dimensional flat for any \(j<d\) is a j-dimensional subspace translated by some vector.

  5. Actually, another reason causing stop is that \(q_{b(l)}\) locates on the flat \(span\{q_{b(1)}, \ldots , \) \(q_{b(l-1)}\}\) for some \(2\le l\le d\). If that happens, recalling the Algorithm Base-Selection, we know that the whole B would locate on the \((l-2)\)-dimensional flat. In this case we can assume that \({ SOD}(B, R, f)\) does not stop in the following \(d-l+1\) steps, but the corresponding \(d-l+1\) rotation angles are all zeros.

  6. Two point sets are congruent if one can be transformed to completely coincide with the other by some rigid transformation.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Michigan State University, East Lansing, MI, USA

    Hu Ding

  2. Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY, USA

    Jinhui Xu

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  1. Hu Ding
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  2. Jinhui Xu
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Correspondence to Hu Ding.

Additional information

This research was supported in part by NSF under Grants CCF-1422324, IIS- 1422591, IIS-1115220, and CNS-1547167. A preliminary version of this paper has appeared in the 21st European Symposium on Algorithms (ESA 2013).

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Ding, H., Xu, J. FPTAS for Minimizing the Earth Mover’s Distance Under Rigid Transformations and Related Problems. Algorithmica 78, 741–770 (2017). https://doi.org/10.1007/s00453-016-0173-4

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