Thinplate Splines on the Sphere
In this paper, we give explicit closed forms for the semi-reproducing kernels associated with thinplate spline interpolation on the sphere. Polyharmonic or thinplate splines for Rᵈ were introduced by Duchon and have become a widely used tool in myriad applications. The analogues for Sᵈ⁻¹ are the thi...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209767 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Thinplate Splines on the Sphere / R.K. Beatson, W. Zu Castell // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862534404551737344 |
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| author | Beatson, R.K. Zu Castell, W. |
| author_facet | Beatson, R.K. Zu Castell, W. |
| citation_txt | Thinplate Splines on the Sphere / R.K. Beatson, W. Zu Castell // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we give explicit closed forms for the semi-reproducing kernels associated with thinplate spline interpolation on the sphere. Polyharmonic or thinplate splines for Rᵈ were introduced by Duchon and have become a widely used tool in myriad applications. The analogues for Sᵈ⁻¹ are the thin plate splines for the sphere. The topic was first discussed by Wahba in the early 1980s, for the S² case. Wahba presented the associated semi-reproducing kernels as infinite series. These semi-reproducing kernels play a central role in expressions for the solution of the associated spline interpolation and smoothing problems. The main aims of the current paper are to give a recurrence for the semi-reproducing kernels and also to use the recurrence to obtain explicit closed-form expressions for many of these kernels. The closed-form expressions will, in many cases, be significantly faster to evaluate than the series expansions. This will enhance the practicality of using these thinplate splines for the sphere in computations.
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| first_indexed | 2025-12-03T02:39:26Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209767 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-03T02:39:26Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Beatson, R.K. Zu Castell, W. 2025-11-26T11:25:47Z 2018 Thinplate Splines on the Sphere / R.K. Beatson, W. Zu Castell // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42A82; 33C45; 42C10; 62M30 arXiv: 1801.01313 https://nasplib.isofts.kiev.ua/handle/123456789/209767 https://doi.org/10.3842/SIGMA.2018.083 In this paper, we give explicit closed forms for the semi-reproducing kernels associated with thinplate spline interpolation on the sphere. Polyharmonic or thinplate splines for Rᵈ were introduced by Duchon and have become a widely used tool in myriad applications. The analogues for Sᵈ⁻¹ are the thin plate splines for the sphere. The topic was first discussed by Wahba in the early 1980s, for the S² case. Wahba presented the associated semi-reproducing kernels as infinite series. These semi-reproducing kernels play a central role in expressions for the solution of the associated spline interpolation and smoothing problems. The main aims of the current paper are to give a recurrence for the semi-reproducing kernels and also to use the recurrence to obtain explicit closed-form expressions for many of these kernels. The closed-form expressions will, in many cases, be significantly faster to evaluate than the series expansions. This will enhance the practicality of using these thinplate splines for the sphere in computations. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Thinplate Splines on the Sphere Article published earlier |
| spellingShingle | Thinplate Splines on the Sphere Beatson, R.K. Zu Castell, W. |
| title | Thinplate Splines on the Sphere |
| title_full | Thinplate Splines on the Sphere |
| title_fullStr | Thinplate Splines on the Sphere |
| title_full_unstemmed | Thinplate Splines on the Sphere |
| title_short | Thinplate Splines on the Sphere |
| title_sort | thinplate splines on the sphere |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209767 |
| work_keys_str_mv | AT beatsonrk thinplatesplinesonthesphere AT zucastellw thinplatesplinesonthesphere |