Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms

Rinat Ben-Avraham; Matthias Henze; Rafel Jaume; Balázs Keszegh; Orit E. Raz; Micha Sharir; Igor Tubis

doi:10.1007/s00453-017-0326-0

Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms

Rinat Ben-Avraham, Matthias Henze^*, Rafel Jaume, Balázs Keszegh, Orit E. Raz, Micha Sharir, Igor Tubis

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We consider the problem of minimizing the RMS distance (sum of squared distances between pairs of points) under translation between two point sets A and B, in the plane, with m= | B| ≪ n= | A| , in the partial-matching setup, in which each point in B is matched to a distinct point in A. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision D_B _, _A of the plane and derive improved bounds on its complexity. Specifically, we show that this complexity is O(n²m^3.5(eln m+ e) ^m) , so it is only quadratic in |A|. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time.

Original language	American English
Pages (from-to)	2400-2421
Number of pages	22
Journal	Algorithmica
Volume	80
Issue number	8
DOIs	https://doi.org/10.1007/s00453-017-0326-0
State	Published - 1 Aug 2018
Externally published	Yes

Bibliographical note

Funding Information:
Acknowledgements Rinat Ben-Avraham was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Matthias Henze was supported by ESF EUROCORES programme EuroGIGA-VORONOI, (DFG): RO 2338/5-1. Rafel Jaume was supported by “Obra Social La Caixa” and the DAAD. Balázs Keszegh was supported by Hungarian National Science Fund (OTKA), under Grants PD 108406, NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003), NK 78439, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the DAAD. Orit E. Raz was supported by Grant 892/13 from the Israel Science Foundation. Micha Sharir was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli

Funding Information:
Centers for Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski– MINERVA Center for Geometry at Tel Aviv University. Igor Tubis was supported by the Deutsch Institute. A preliminary version of this paper has appeared in [5].

Funding Information:
Rinat Ben-Avraham was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Matthias Henze was supported by ESF EUROCORES programme EuroGIGA-VORONOI, (DFG): RO 2338/5-1. Rafel Jaume was supported by ?Obra Social La Caixa? and the DAAD. Bal?zs Keszegh was supported by Hungarian National Science Fund (OTKA), under Grants PD 108406, NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003), NK 78439, by the J?nos Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the DAAD. Orit E. Raz was supported by Grant 892/13 from the Israel Science Foundation. Micha Sharir was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (Center No.?4/11), and by the Hermann Minkowski?MINERVA Center for Geometry at Tel Aviv University. Igor Tubis was supported by the Deutsch Institute. A preliminary version of this paper has appeared in?[5 ].

Publisher Copyright:
© 2017, Springer Science+Business Media New York.

Keywords

Convex subdivision
Local minimum
Partial matching
RMS distance
Shape matching

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10.1007/s00453-017-0326-0

Cite this

@article{6942171999c649af92fec13de8faafd1,

title = "Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms",

abstract = "We consider the problem of minimizing the RMS distance (sum of squared distances between pairs of points) under translation between two point sets A and B, in the plane, with m= | B| ≪ n= | A| , in the partial-matching setup, in which each point in B is matched to a distinct point in A. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision DB , A of the plane and derive improved bounds on its complexity. Specifically, we show that this complexity is O(n2m3.5(eln m+ e) m) , so it is only quadratic in |A|. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time.",

keywords = "Convex subdivision, Local minimum, Partial matching, RMS distance, Shape matching",

author = "Rinat Ben-Avraham and Matthias Henze and Rafel Jaume and Bal{\'a}zs Keszegh and Raz, {Orit E.} and Micha Sharir and Igor Tubis",

note = "Funding Information: Acknowledgements Rinat Ben-Avraham was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Matthias Henze was supported by ESF EUROCORES programme EuroGIGA-VORONOI, (DFG): RO 2338/5-1. Rafel Jaume was supported by “Obra Social La Caixa” and the DAAD. Bal{\'a}zs Keszegh was supported by Hungarian National Science Fund (OTKA), under Grants PD 108406, NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003), NK 78439, by the J{\'a}nos Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the DAAD. Orit E. Raz was supported by Grant 892/13 from the Israel Science Foundation. Micha Sharir was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Funding Information: Centers for Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski– MINERVA Center for Geometry at Tel Aviv University. Igor Tubis was supported by the Deutsch Institute. A preliminary version of this paper has appeared in [5]. Funding Information: Rinat Ben-Avraham was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Matthias Henze was supported by ESF EUROCORES programme EuroGIGA-VORONOI, (DFG): RO 2338/5-1. Rafel Jaume was supported by ?Obra Social La Caixa? and the DAAD. Bal?zs Keszegh was supported by Hungarian National Science Fund (OTKA), under Grants PD 108406, NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003), NK 78439, by the J?nos Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the DAAD. Orit E. Raz was supported by Grant 892/13 from the Israel Science Foundation. Micha Sharir was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (Center No.?4/11), and by the Hermann Minkowski?MINERVA Center for Geometry at Tel Aviv University. Igor Tubis was supported by the Deutsch Institute. A preliminary version of this paper has appeared in?[5 ]. Publisher Copyright: {\textcopyright} 2017, Springer Science+Business Media New York.",

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month = aug,

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doi = "10.1007/s00453-017-0326-0",

language = "American English",

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AU - Ben-Avraham, Rinat

AU - Henze, Matthias

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AU - Raz, Orit E.

AU - Sharir, Micha

AU - Tubis, Igor

N1 - Funding Information: Acknowledgements Rinat Ben-Avraham was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Matthias Henze was supported by ESF EUROCORES programme EuroGIGA-VORONOI, (DFG): RO 2338/5-1. Rafel Jaume was supported by “Obra Social La Caixa” and the DAAD. Balázs Keszegh was supported by Hungarian National Science Fund (OTKA), under Grants PD 108406, NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003), NK 78439, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the DAAD. Orit E. Raz was supported by Grant 892/13 from the Israel Science Foundation. Micha Sharir was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Funding Information: Centers for Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski– MINERVA Center for Geometry at Tel Aviv University. Igor Tubis was supported by the Deutsch Institute. A preliminary version of this paper has appeared in [5]. Funding Information: Rinat Ben-Avraham was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Matthias Henze was supported by ESF EUROCORES programme EuroGIGA-VORONOI, (DFG): RO 2338/5-1. Rafel Jaume was supported by ?Obra Social La Caixa? and the DAAD. Bal?zs Keszegh was supported by Hungarian National Science Fund (OTKA), under Grants PD 108406, NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003), NK 78439, by the J?nos Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the DAAD. Orit E. Raz was supported by Grant 892/13 from the Israel Science Foundation. Micha Sharir was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (Center No.?4/11), and by the Hermann Minkowski?MINERVA Center for Geometry at Tel Aviv University. Igor Tubis was supported by the Deutsch Institute. A preliminary version of this paper has appeared in?[5 ]. Publisher Copyright: © 2017, Springer Science+Business Media New York.

PY - 2018/8/1

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N2 - We consider the problem of minimizing the RMS distance (sum of squared distances between pairs of points) under translation between two point sets A and B, in the plane, with m= | B| ≪ n= | A| , in the partial-matching setup, in which each point in B is matched to a distinct point in A. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision DB , A of the plane and derive improved bounds on its complexity. Specifically, we show that this complexity is O(n2m3.5(eln m+ e) m) , so it is only quadratic in |A|. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time.

AB - We consider the problem of minimizing the RMS distance (sum of squared distances between pairs of points) under translation between two point sets A and B, in the plane, with m= | B| ≪ n= | A| , in the partial-matching setup, in which each point in B is matched to a distinct point in A. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision DB , A of the plane and derive improved bounds on its complexity. Specifically, we show that this complexity is O(n2m3.5(eln m+ e) m) , so it is only quadratic in |A|. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time.

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KW - Local minimum

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KW - RMS distance

KW - Shape matching

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ER -

Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms

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Keywords

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