Skip to main content
SpringerLink
Log in

Multidimensional Scattered Time-varying Scattered Data Meshless Interpolation for Sensor Networks

  • Conference paper
  • First Online:
Computational Science and Its Applications – ICCSA 2023 (ICCSA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13956 ))

Included in the following conference series:

  • 566 Accesses

Abstract

Interpolation and approximation of scattered scalar and vector data is a part of a solution of many engineering problems. The methods are based mostly on some triangulation of the data domain, usually limited to 2D or 3D data, followed by an interpolation or an approximation to obtain a smooth result. This contribution presents a meshless approach based on the Radial Basis Functions (RBF). It is nearly dimensionless and it enables interpolation of time varying data, i.e. interpolation of scattered spatio-temporal varying data, i.e. interpolation in space-time domain without “time-frames”. The meshless methods for scattered spatio-temporal data can be used for interpolation, approximation and evaluation of data acquired from buoys, sensor networks, sensors for tsunami, chemical and radiation detectors, ships and submarines detection, weather forecast, 3D vector fields compression and visualization, etc.

Research partially supported by the University of West Bohemia - Institutional research support.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For non DT tessellation, see Smolik [45].

  2. 2.

    In the case of high N using block matrices might help in solving large systems of linear equations, see Majdisova [28] - a system of \(6.7\ 10^6\) points was solved.

  3. 3.

    The distance defined as \(\Vert .\Vert _1\) is used in fuzzy approach, see Perfilieva [32].

  4. 4.

    It should be noted that \(r^2 \log (r)= \frac{1}{2} r^2 \log (r^2)\), i.e. actually no \(\sqrt{r^2}\) computation is needed, only the values of the weights \(\lambda _j\) are doubled.

  5. 5.

    In the actual implementation \(\varphi (r)=r^2 \ln {r^2}\) should be used as \(r^2 \ln {r^2}=2~r^2 \ln {r}\). The \(\lambda \) weights are doubled and \(\sqrt{.}\) operation is not needed.

  6. 6.

    The SN-RBF with the Gauss function and bilinear polynomial was used; parameters \(\alpha =0.255\) and \(\beta =1\) in Eq. 10.

  7. 7.

    There are several suboptimal shape parameters, see Afiatdoust [1], Skala [43], Karageorghis [22] and Wang [49].

References

  1. Afiatdoust, F., Esmaeilbeigi, M.: Optimal variable shape parameters using genetic algorithm for radial basis function approximation. Ain Shams Eng. J. 6, 639–647 (2015)

    Article  Google Scholar 

  2. Ali, F.A.M., Karim, S.A.A., Dass, S.C., Skala, V., Hasan, M.K., Hashim, I.: Efficient visualization of scattered energy distribution data by using cubic timmer triangular patches. In: Sulaiman, S.A. (ed.) Energy Efficiency in Mobility Systems, pp. 145–180. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-0102-9_8

    Chapter  Google Scholar 

  3. Ali, F., et al.: New cubic timmer triangular patches with c1 and g1 continuity. Jurnal Teknologi 81(6), 1–11 (2019)

    Google Scholar 

  4. Ali, F., et al.: Construction of cubic timmer triangular patches and its application in scattered data interpolation. Mathematics 8, 159 (2020)

    Article  Google Scholar 

  5. Ali, F.A.M., et al.: Visualizing the energy of scattered data by using cubic timmer triangular patches. J. Phys. 1366(1), 012098 (2019)

    MathSciNet  Google Scholar 

  6. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139(1–4), 3–47 (1996)

    Article  MATH  Google Scholar 

  7. Biancolini, M.E.: Fast Radial Basis Functions for Engineering Applications, 1st edn. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-319-75011-8

    Book  MATH  Google Scholar 

  8. Buhmann, M.: On quasi-interpolation with radial basis functions. J. Approx. Theory 72(1), 103–130 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carr, J., et al.: Reconstruction and representation of 3d objects with radial basis functions. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 2001, pp. 67–76 (2001)

    Google Scholar 

  10. Červenka, M., Skala, V.: Conditionality analysis of the radial basis function matrix. In: Gervasi, O., Murgante, B., Misra, S., Garau, C., Blečić, I., Taniar, D., Apduhan, B.O., Rocha, A.M.A.C., Tarantino, E., Torre, C.M., Karaca, Y. (eds.) ICCSA 2020. LNCS, vol. 12250, pp. 30–43. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58802-1_3

    Chapter  Google Scholar 

  11. Cuomo, S., Galletti, A., Giunta, G., Marcellino, L.: Reconstruction of implicit curves and surfaces via rbf interpolation. Appl. Numer. Math. 116, 157–171 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cuomo, S., Galletti, A., Giunta, G., Starace, A.: Surface reconstruction from scattered point via rbf interpolation on gpu. In: 2013 Federated Conference on Computer Science and Information Systems, FedCSIS 2013, pp. 433–440 (2013)

    Google Scholar 

  13. Drake, K.P., Fuselier, E.J., Wright, G.B.: Implicit surface reconstruction with a curl-free radial basis function partition of unity method (2021)

    Google Scholar 

  14. Fasshauer, G.: Meshfree Approximation Methods with Matlab, 1st edn. World Scientific, Singapore (2007)

    Book  MATH  Google Scholar 

  15. Floater, M.S., Iske, A.: Multistep scattered data interpolation using compactly supported radial basis functions. J. Comput. Appl. Math. 73(1–2), 65–78 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Franke, R.: A critical comparison of some methods for interpolation of scattered data. Technical report. Naval Postgraduate School Monterey, CA (1979)

    Google Scholar 

  17. Goodman, T., Said, H., Chang, L.: Local derivative estimation for scattered data interpolation. Appl. Math. Comput. 68(1), 41–50 (1995)

    MathSciNet  MATH  Google Scholar 

  18. Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76, 1905–1915 (1971)

    Article  Google Scholar 

  19. Hardy, R.L.: Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988. Comput. Math. Appl. 19, 163–208 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Iske, A.: Multiresolution Methods in Scattered Data Modelling, vol. 37. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-642-18754-4

    Book  MATH  Google Scholar 

  21. Jäger, J.: Advances in radial and spherical basis function interpolation. PhD thesis, Justus-Liebig-Universität, Otto-Behaghel-Str. 8, 35394 Gießen. PhD Thesis (2018)

    Google Scholar 

  22. Karageorghis, A., Tryfonos, P.: Shape parameter estimation in RBF function approximation. Int. J. Comput. Methods Exp. Meas. 7, 246–259 (2019)

    MATH  Google Scholar 

  23. Ku, C.-Y., Hong, L.-D., Liu, C.-Y., Xiao, J.-E.: Space-time polyharmonic radial polynomial basis functions for modeling saturated and unsaturated flows. Eng. Comput. (2021)

    Google Scholar 

  24. Macedo, I., Gois, J., Velho, L.: Hermite interpolation of implicit surfaces with radial basis functions. In: Proceedings of SIBGRAPI 2009–22nd Brazilian Symposium on Computer Graphics and Image Processing, pp. 1–8 (2009)

    Google Scholar 

  25. Macedo, I., Gois, J., Velho, L.: Hermite radial basis functions implicits. Comput. Graph. Forum 30(1), 27–42 (2011)

    Article  Google Scholar 

  26. Majdisova, Z., Skala, V.: A new radial basis function approximation with reproduction. In: CGVCVIP 2016, pp. 215–222 (2016)

    Google Scholar 

  27. Majdisova, Z., Skala, V.: A radial basis function approximation for large datasets. In: SIGRAD 2016, pp. 9–14 (2016)

    Google Scholar 

  28. Majdisova, Z., Skala, V.: Big geo data surface approximation using radial basis functions: a comparative study. Comput. Geosci. 109, 51–58 (2017)

    Article  MATH  Google Scholar 

  29. Majdisova, Z., Skala, V.: Radial basis function approximations: comparison and applications. Appl. Math. Model. 51, 728–743 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Majdisova, Z., Skala, V., Smolik, M.: Algorithm for placement of reference points and choice of an appropriate variable shape parameter for the RBF approximation. Integr. Comput. Aided Eng. 27, 1–15 (2020)

    Article  Google Scholar 

  31. Menandro, F.: Two new classes of compactly supported radial basis functions for approximation of discrete and continuous data. Eng. Rep. 1, e12028:1–30 (2019)

    Google Scholar 

  32. Perfilieva, I., Vlasanek, P., Wrublova, M.: Fuzzy Transform for Image Reconstruction, pp. 615–620. World Scientific, Singapore (2012)

    Google Scholar 

  33. Petrik, S., Skala, V.: Iso-contouring in time-varying meshes. In: SCCG 2007: 23rd Spring Conference on Computer Graphics, pp. 175–182 (2007)

    Google Scholar 

  34. Petrik, S., Skala, V.: Z-diamonds: a fast iso-surface extraction algorithm for dynamic meshes. MCCSIS 2007(3), 67–74 (2007)

    Google Scholar 

  35. Petrik, S., Skala, V.: Space and time efficient isosurface extraction. Comput. Graph. (Pergamon) 32(6), 704–710 (2008)

    Article  Google Scholar 

  36. Sarra, S.A., Sturgill, D.: A random variable shape parameter strategy for radial basis function approximation methods. Eng. Anal. Bound. Elements 33, 1239–1245 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Schaback, R.: Optimal geometric hermite interpolation of curves, mathematical methods for curves and surfaces. ii. Innov. Appl. Math., 417–428 (1998)

    Google Scholar 

  38. Skala, V.: High dimensional and large span data least square error: numerical stability and conditionality. Int. J. Appl. Phys. Math. 7(3), 148–156 (2017)

    Article  MathSciNet  Google Scholar 

  39. Skala, V.: RBF interpolation with CSRBF of large data sets. Proceedia Comput. Sci. 108, 2433–2437 (2017)

    Article  Google Scholar 

  40. Skala, V.: RBF interpolation and approximation of large span data sets. In: MCSI 2017 - Corfu, pp. 212–218. IEEE (2018)

    Google Scholar 

  41. Skala, V.: Conditionality of linear systems of equation and matrices using projective geometric algebra. In: Gervasi, O., Murgante, B., Misra, S., Garau, C., Blečić, I., Taniar, D., Apduhan, B.O., Rocha, A.M.A.C., Tarantino, E., Torre, C.M., Karaca, Y. (eds.) ICCSA 2020. LNCS, vol. 12250, pp. 3–17. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58802-1_1

    Chapter  Google Scholar 

  42. Skala, V., Kansa, E.: Why is the least square error method dangerous? Computacion y Sistemas 25(1), 149–151 (2021)

    Google Scholar 

  43. Skala, V., Karim, S.A.A., Zabran, M.: Radial basis function approximation optimal shape parameters estimation. In: Krzhizhanovskaya, V.V., et al. (eds.) ICCS 2020. LNCS, vol. 12142, pp. 309–317. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-50433-5_24

    Chapter  Google Scholar 

  44. Smolik, M., Skala, V.: Fast parallel triangulation algorithm of large data sets in E2 and E3 for in-core and out-core memory processing. In: Murgante, B., et al. (eds.) ICCSA 2014. LNCS, vol. 8580, pp. 301–314. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09129-7_23

    Chapter  Google Scholar 

  45. Smolik, M., Skala, V., In-core and out-core memory fast parallel triangulation algorithm for large data sets in E2 and E3. In: ACM SIGGRAPH,: Posters. SIGGRAPH 2014 (2014)

    Google Scholar 

  46. Smolik, M., Skala, V.: Large scattered data interpolation with radial basis functions and space subdivision. Integr. Comput. Aided Eng. 25, 49–62 (2018)

    Article  Google Scholar 

  47. Smolik, M., Skala, V.: Efficient simple large scattered 3D vector fields radial basis function approximation using space subdivision. In: Computational Science and Its Application, ICSSA 2019 proceedings, pp. 337–350 (2019)

    Google Scholar 

  48. Smolik, M., Skala, V.: Efficient speed-up of radial basis functions approximation and interpolation formula evaluation. In: Gervasi, O., et al. (eds.) ICCSA 2020. LNCS, vol. 12249, pp. 165–176. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58799-4_12

    Chapter  Google Scholar 

  49. Wang, J., Liu, G.: On the optimal shape parameters of radial basis functions used for 2-d meshless methods. Comput. Methods Appl. Mech. Eng. 191, 2611–2630 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4(1), 389–396 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  52. Wikipedia contributors. Delaunay triangulation – Wikipedia, the free encyclopedia (2022). Accessed 9 June 2022

    Google Scholar 

  53. Wikipedia contributors. Radial basis function network – Wikipedia, the free encyclopedia (2022). Accessed 22 June 2022

    Google Scholar 

  54. Wu, Z.: Compactly supported positive definite radial functions. Adv. Comput. Math. 4(1), 283–292 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhang, X., Song, K.Z., Lu, M.W., Liu, X.: Meshless methods based on collocation with radial basis functions. Comput. Mech. 26, 333–343 (2000)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The author thanks to colleagues at the Shandong University(Jinan) and Zhejiang University(Hangzhou) China, University of West Bohemia, Pilsen for their critical comments, discussions, especially to colleagues Michal Smolik, Zuzana Majdisova, Martin Cervenka, Mariia Martynova and Jan Kasak for producing some images and for numerical verification.

Author information

Authors and Affiliations

  1. Faculty of Applied Sciences, Department of Computer Science, University of West Bohemia, 301 00, Pilsen, Czech Republic

    Vaclav Skala

Authors
  1. Vaclav Skala
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vaclav Skala .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  3. Monash University, Clayton, VIC, Australia

    David Taniar

  4. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  5. University of Minho, Braga, Portugal

    Ana Cristina Braga

  6. University of Cagliari, Cagliari, Italy

    Chiara Garau

  7. National Technical University of Athens, Athens, Greece

    Anastasia Stratigea

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Skala, V. (2023). Multidimensional Scattered Time-varying Scattered Data Meshless Interpolation for Sensor Networks. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2023. ICCSA 2023. Lecture Notes in Computer Science, vol 13956 . Springer, Cham. https://doi.org/10.1007/978-3-031-36805-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-36805-9_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-36804-2

  • Online ISBN: 978-3-031-36805-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation