The exponential map for the group of similarity transformations and applications to motion interpolation

Spyridon Leonardos; Christine Allen-Blanchette; Jean Gallier

doi:10.1109/ICRA.2015.7139026

The exponential map for the group of similarity transformations and applications to motion interpolation

Spyridon Leonardos, Christine Allen-Blanchette, Jean Gallier

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

3 Scopus citations

Abstract

In this paper, we explore the exponential map and its inverse, the logarithm map, for the group SIM(n) of similarity transformations in ⁿ which are the composition of a rotation, a translation and a uniform scaling. We give a formula for the exponential map and we prove that it is surjective. We give an explicit formula for the case of n = 3 and show how to efficiently compute the logarithmic map. As an application, we use these algorithms to perform motion interpolation. Given a sequence of similarity transformations, we compute a sequence of logarithms, then fit a cubic spline that interpolates the logarithms and finally, we compute the interpolating curve in SIM(3).

Original language	English (US)
Title of host publication	2015 IEEE International Conference on Robotics and Automation, ICRA 2015
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	377-382
Number of pages	6
Edition	June
ISBN (Electronic)	9781479969234
DOIs	https://doi.org/10.1109/ICRA.2015.7139026
State	Published - Jun 29 2015
Externally published	Yes
Event	2015 IEEE International Conference on Robotics and Automation, ICRA 2015 - Seattle, United States Duration: May 26 2015 → May 30 2015

Publication series

Name	Proceedings - IEEE International Conference on Robotics and Automation
Number	June
Volume	2015-June
ISSN (Print)	1050-4729

Conference

Conference	2015 IEEE International Conference on Robotics and Automation, ICRA 2015
Country/Territory	United States
City	Seattle
Period	5/26/15 → 5/30/15

All Science Journal Classification (ASJC) codes

Software
Artificial Intelligence
Electrical and Electronic Engineering
Control and Systems Engineering

Access to Document

10.1109/ICRA.2015.7139026

Cite this

Leonardos, S., Allen-Blanchette, C., & Gallier, J. (2015). The exponential map for the group of similarity transformations and applications to motion interpolation. In 2015 IEEE International Conference on Robotics and Automation, ICRA 2015 (June ed., pp. 377-382). Article 7139026 (Proceedings - IEEE International Conference on Robotics and Automation; Vol. 2015-June, No. June). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ICRA.2015.7139026

Leonardos, Spyridon ; Allen-Blanchette, Christine ; Gallier, Jean. / The exponential map for the group of similarity transformations and applications to motion interpolation. 2015 IEEE International Conference on Robotics and Automation, ICRA 2015. June. ed. Institute of Electrical and Electronics Engineers Inc., 2015. pp. 377-382 (Proceedings - IEEE International Conference on Robotics and Automation; June).

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abstract = "In this paper, we explore the exponential map and its inverse, the logarithm map, for the group SIM(n) of similarity transformations in n which are the composition of a rotation, a translation and a uniform scaling. We give a formula for the exponential map and we prove that it is surjective. We give an explicit formula for the case of n = 3 and show how to efficiently compute the logarithmic map. As an application, we use these algorithms to perform motion interpolation. Given a sequence of similarity transformations, we compute a sequence of logarithms, then fit a cubic spline that interpolates the logarithms and finally, we compute the interpolating curve in SIM(3).",

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Leonardos, S, Allen-Blanchette, C & Gallier, J 2015, The exponential map for the group of similarity transformations and applications to motion interpolation. in 2015 IEEE International Conference on Robotics and Automation, ICRA 2015. June edn, 7139026, Proceedings - IEEE International Conference on Robotics and Automation, no. June, vol. 2015-June, Institute of Electrical and Electronics Engineers Inc., pp. 377-382, 2015 IEEE International Conference on Robotics and Automation, ICRA 2015, Seattle, United States, 5/26/15. https://doi.org/10.1109/ICRA.2015.7139026

The exponential map for the group of similarity transformations and applications to motion interpolation. / Leonardos, Spyridon; Allen-Blanchette, Christine; Gallier, Jean.
2015 IEEE International Conference on Robotics and Automation, ICRA 2015. June. ed. Institute of Electrical and Electronics Engineers Inc., 2015. p. 377-382 7139026 (Proceedings - IEEE International Conference on Robotics and Automation; Vol. 2015-June, No. June).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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T1 - The exponential map for the group of similarity transformations and applications to motion interpolation

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N2 - In this paper, we explore the exponential map and its inverse, the logarithm map, for the group SIM(n) of similarity transformations in n which are the composition of a rotation, a translation and a uniform scaling. We give a formula for the exponential map and we prove that it is surjective. We give an explicit formula for the case of n = 3 and show how to efficiently compute the logarithmic map. As an application, we use these algorithms to perform motion interpolation. Given a sequence of similarity transformations, we compute a sequence of logarithms, then fit a cubic spline that interpolates the logarithms and finally, we compute the interpolating curve in SIM(3).

AB - In this paper, we explore the exponential map and its inverse, the logarithm map, for the group SIM(n) of similarity transformations in n which are the composition of a rotation, a translation and a uniform scaling. We give a formula for the exponential map and we prove that it is surjective. We give an explicit formula for the case of n = 3 and show how to efficiently compute the logarithmic map. As an application, we use these algorithms to perform motion interpolation. Given a sequence of similarity transformations, we compute a sequence of logarithms, then fit a cubic spline that interpolates the logarithms and finally, we compute the interpolating curve in SIM(3).

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Leonardos S, Allen-Blanchette C, Gallier J. The exponential map for the group of similarity transformations and applications to motion interpolation. In 2015 IEEE International Conference on Robotics and Automation, ICRA 2015. June ed. Institute of Electrical and Electronics Engineers Inc. 2015. p. 377-382. 7139026. (Proceedings - IEEE International Conference on Robotics and Automation; June). doi: 10.1109/ICRA.2015.7139026

The exponential map for the group of similarity transformations and applications to motion interpolation

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