### Abstract

This paper addresses the topic of fitting reduced data represented by the sequence of interpolation points (Formula Presented) in arbitrary Euclidean space E^{m}. The parametric curve γ together with its knots (Formula Presented) (for which (Formula Presented)) are both assumed to be unknown. We look at some recipes to estimate T in the context of dense versus sparse M for various choices of interpolation schemes (Formula Presented). For M dense, the convergence rate to approximate γ with (Formula Presented) is considered as a possible criterion to force a proper choice of new knots (Formula Presented). The latter incorporates the so-called exponential parameterization “retrieving” the missing knots T from the geometrical spread of M. We examine the convergence rate in approximating γ by commonly used interpolants (Formula Presented) based here on M and exponential parameterization. In contrast, for M sparse, a possible optional strategy is to select (Formula Presented) which optimizes a certain cost function depending on the family of admissible knots (Formula Presented). This paper focuses on minimizing “an average acceleration” within the family of natural splines (Formula Presented) fitting M with (Formula Presented) admitted freely in the ascending order. Illustrative examples and some applications listed supplement theoretical component of this work.

Original language | English |
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Title of host publication | Advances in Soft and Hard Computing |

Editors | Janusz Kacprzyk, Jerzy Pejaś, Imed El Fray, Tomasz Hyla |

Publisher | Springer-Verlag Berlin |

Pages | 3-17 |

Number of pages | 15 |

ISBN (Print) | 9783030033132 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

Event | 21st International Multi-Conference on Advanced Computer Systems, ACS 2018 - Międzyzdroje, Poland Duration: 24 Sep 2018 → 26 Sep 2018 http://acs.zut.edu.pl/ |

### Publication series

Name | Advances in Intelligent Systems and Computing |
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Volume | 889 |

ISSN (Print) | 2194-5357 |

### Conference

Conference | 21st International Multi-Conference on Advanced Computer Systems, ACS 2018 |
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Abbreviated title | ACS2018 |

Country | Poland |

City | Międzyzdroje |

Period | 24/09/18 → 26/09/18 |

Other | Artificial intelligence, software technologies, biometrics and IT security are established fields of computer science of both theoretical and practical significance. The aim of ACS 2018 Multi-Conference is to bring artificial intelligence, software technologies, biometrics, IT security and open and distance learning researchers in contact with the ACS community, and to give ACS attendees an opportunity to exchange some significant knowledge according to this areas of interest. Industrial and systems presentations bearing new ideas and solution paradigms are welcome as well. |

Internet address |

### Fingerprint

### Cite this

*Advances in Soft and Hard Computing*(pp. 3-17). (Advances in Intelligent Systems and Computing; Vol. 889). Springer-Verlag Berlin. https://doi.org/10.1007/978-3-030-03314-9_1

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*Advances in Soft and Hard Computing.*Advances in Intelligent Systems and Computing, vol. 889, Springer-Verlag Berlin, pp. 3-17, 21st International Multi-Conference on Advanced Computer Systems, ACS 2018, Międzyzdroje, Poland, 24/09/18. https://doi.org/10.1007/978-3-030-03314-9_1

**Fitting Dense and Sparse Reduced Data.** / Kozera, Ryszard; Wiliński, Artur.

Research output: Chapter in Book/Conference paper › Conference paper

TY - GEN

T1 - Fitting Dense and Sparse Reduced Data

AU - Kozera, Ryszard

AU - Wiliński, Artur

PY - 2019/1/1

Y1 - 2019/1/1

N2 - This paper addresses the topic of fitting reduced data represented by the sequence of interpolation points (Formula Presented) in arbitrary Euclidean space Em. The parametric curve γ together with its knots (Formula Presented) (for which (Formula Presented)) are both assumed to be unknown. We look at some recipes to estimate T in the context of dense versus sparse M for various choices of interpolation schemes (Formula Presented). For M dense, the convergence rate to approximate γ with (Formula Presented) is considered as a possible criterion to force a proper choice of new knots (Formula Presented). The latter incorporates the so-called exponential parameterization “retrieving” the missing knots T from the geometrical spread of M. We examine the convergence rate in approximating γ by commonly used interpolants (Formula Presented) based here on M and exponential parameterization. In contrast, for M sparse, a possible optional strategy is to select (Formula Presented) which optimizes a certain cost function depending on the family of admissible knots (Formula Presented). This paper focuses on minimizing “an average acceleration” within the family of natural splines (Formula Presented) fitting M with (Formula Presented) admitted freely in the ascending order. Illustrative examples and some applications listed supplement theoretical component of this work.

AB - This paper addresses the topic of fitting reduced data represented by the sequence of interpolation points (Formula Presented) in arbitrary Euclidean space Em. The parametric curve γ together with its knots (Formula Presented) (for which (Formula Presented)) are both assumed to be unknown. We look at some recipes to estimate T in the context of dense versus sparse M for various choices of interpolation schemes (Formula Presented). For M dense, the convergence rate to approximate γ with (Formula Presented) is considered as a possible criterion to force a proper choice of new knots (Formula Presented). The latter incorporates the so-called exponential parameterization “retrieving” the missing knots T from the geometrical spread of M. We examine the convergence rate in approximating γ by commonly used interpolants (Formula Presented) based here on M and exponential parameterization. In contrast, for M sparse, a possible optional strategy is to select (Formula Presented) which optimizes a certain cost function depending on the family of admissible knots (Formula Presented). This paper focuses on minimizing “an average acceleration” within the family of natural splines (Formula Presented) fitting M with (Formula Presented) admitted freely in the ascending order. Illustrative examples and some applications listed supplement theoretical component of this work.

KW - Computer vision and graphics

KW - Interpolation

KW - Reduced data

UR - http://www.scopus.com/inward/record.url?scp=85059038390&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-03314-9_1

DO - 10.1007/978-3-030-03314-9_1

M3 - Conference paper

SN - 9783030033132

T3 - Advances in Intelligent Systems and Computing

SP - 3

EP - 17

BT - Advances in Soft and Hard Computing

A2 - Kacprzyk, Janusz

A2 - Pejaś, Jerzy

A2 - El Fray, Imed

A2 - Hyla, Tomasz

PB - Springer-Verlag Berlin

ER -