On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem

The problem of restoring the orthonormality of a noisy rotation matrix by finding its nearest correct rotation matrix arises in many areas of robotics, computer graphics, and computer vision. When the Frobenius norm is taken as the measure of closeness, the solution is usually computed using the sin...

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Authors: Sarabandi, Soheil, Shabani, Arya, Porta, Josep M., Thomas, Federico
Format: article
Status:Versión aceptada para publicación
Publication Date:2020
Country:España
Institution:Consejo Superior de Investigaciones Científicas (CSIC)
Repository:DIGITAL.CSIC. Repositorio Institucional del CSIC
OAI Identifier:oai:digital.csic.es:10261/227704
Online Access:http://hdl.handle.net/10261/227704
Access Level:Open access
Keyword:Rotation matrices
Quaternions
Singular value decomposition
Third degree polynomials.
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spelling On Closed-Form Formulas for the 3-D Nearest Rotation Matrix ProblemSarabandi, SoheilShabani, AryaPorta, Josep M.Thomas, FedericoRotation matricesQuaternionsSingular value decompositionThird degree polynomials.The problem of restoring the orthonormality of a noisy rotation matrix by finding its nearest correct rotation matrix arises in many areas of robotics, computer graphics, and computer vision. When the Frobenius norm is taken as the measure of closeness, the solution is usually computed using the singular value decomposition (SVD). A closed-form formula exists but, as it involves the roots of a polynomial of third degree, it is assumed to be too complicated and numerically ill-conditioned. In this article, we show how, by carefully using some algebraic recipes scattered in the literature, it is possible to derive a simple and yet numerically stable formula for most practical applications. Moreover, by relying on a result that permits obtaining the quaternion corresponding to the sought optimal rotation matrix, we present another closed-form formula that provides a good approximation to the optimal one using only the elementary algebraic operations of addition, subtraction, multiplication, and division. These two closed-form formulas are compared with respect to the SVD in terms of accuracy and computational cost.This work was partially supported by the Spanish Ministry of Economy and Competitiveness through the projects DPI2017-88282-P and MDM-2016-0656.Institute of Electrical and Electronics EngineersMinisterio de Economía y Competitividad (España)Ministerio de Ciencia, Innovación y Universidades (España)Agencia Estatal de Investigación (España)Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]2021202120202021info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Postprintinfo:eu-repo/semantics/acceptedVersionhttp://hdl.handle.net/10261/227704reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)Inglés#PLACEHOLDER_PARENT_METADATA_VALUE##PLACEHOLDER_PARENT_METADATA_VALUE##PLACEHOLDER_PARENT_METADATA_VALUE#info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/DPI2017-88282-PDPI2017-88282-P/AEI/10.13039/501100011033info:eu-repo/grantAgreement/MINECO/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MDM-2016-0656http://dx.doi.org/10.1109/TRO.2020.2973072Síinfo:eu-repo/semantics/openAccessoai:digital.csic.es:10261/2277042026-05-22T06:33:51Z
dc.title.none.fl_str_mv On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem
title On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem
spellingShingle On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem
Sarabandi, Soheil
Rotation matrices
Quaternions
Singular value decomposition
Third degree polynomials.
title_short On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem
title_full On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem
title_fullStr On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem
title_full_unstemmed On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem
title_sort On Closed-Form Formulas for the 3-D Nearest Rotation Matrix Problem
dc.creator.none.fl_str_mv Sarabandi, Soheil
Shabani, Arya
Porta, Josep M.
Thomas, Federico
author Sarabandi, Soheil
author_facet Sarabandi, Soheil
Shabani, Arya
Porta, Josep M.
Thomas, Federico
author_role author
author2 Shabani, Arya
Porta, Josep M.
Thomas, Federico
author2_role author
author
author
dc.contributor.none.fl_str_mv Ministerio de Economía y Competitividad (España)
Ministerio de Ciencia, Innovación y Universidades (España)
Agencia Estatal de Investigación (España)
Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]
dc.subject.none.fl_str_mv Rotation matrices
Quaternions
Singular value decomposition
Third degree polynomials.
topic Rotation matrices
Quaternions
Singular value decomposition
Third degree polynomials.
description The problem of restoring the orthonormality of a noisy rotation matrix by finding its nearest correct rotation matrix arises in many areas of robotics, computer graphics, and computer vision. When the Frobenius norm is taken as the measure of closeness, the solution is usually computed using the singular value decomposition (SVD). A closed-form formula exists but, as it involves the roots of a polynomial of third degree, it is assumed to be too complicated and numerically ill-conditioned. In this article, we show how, by carefully using some algebraic recipes scattered in the literature, it is possible to derive a simple and yet numerically stable formula for most practical applications. Moreover, by relying on a result that permits obtaining the quaternion corresponding to the sought optimal rotation matrix, we present another closed-form formula that provides a good approximation to the optimal one using only the elementary algebraic operations of addition, subtraction, multiplication, and division. These two closed-form formulas are compared with respect to the SVD in terms of accuracy and computational cost.
publishDate 2020
dc.date.none.fl_str_mv 2020
2021
2021
2021
dc.type.none.fl_str_mv info:eu-repo/semantics/article
http://purl.org/coar/resource_type/c_6501
Postprint
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10261/227704
url http://hdl.handle.net/10261/227704
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv #PLACEHOLDER_PARENT_METADATA_VALUE#
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#PLACEHOLDER_PARENT_METADATA_VALUE#
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/DPI2017-88282-P
DPI2017-88282-P/AEI/10.13039/501100011033
info:eu-repo/grantAgreement/MINECO/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MDM-2016-0656
http://dx.doi.org/10.1109/TRO.2020.2973072
Sí
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Institute of Electrical and Electronics Engineers
publisher.none.fl_str_mv Institute of Electrical and Electronics Engineers
dc.source.none.fl_str_mv reponame:DIGITAL.CSIC. Repositorio Institucional del CSIC
instname:Consejo Superior de Investigaciones Científicas (CSIC)
instname_str Consejo Superior de Investigaciones Científicas (CSIC)
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