Skip to main content
SpringerLink
Log in

Scattered data approximation by regular grid weighted smoothing

  • Published:
Sādhanā Aims and scope Submit manuscript

Abstract

Scattered data approximation refers to the computation of a multi-dimensional function from measurements obtained from scattered spatial locations. For this problem, the class of methods that adopt a roughness minimization are the best performing ones. These methods are called variational methods and they are capable of handling contrasting levels of sample density. These methods express the required solution as a continuous model containing a weighted sum of thin-plate spline or radial basis functions with centres aligned to the measurement locations, and the weights are specified by a linear system of equations. The main hurdle in this type of method is that the linear system is ill-conditioned. Further, getting the weights that are parameters of the continuous model representing the solution is only a part of the effort. Getting a regular grid image requires re-sampling of the continuous model, which is typically expensive. We develop a computationally efficient and numerically stable method based on roughness minimization. The method leads to an algorithm that uses standard regular grid array operations only, which makes it attractive for parallelization. We demonstrate experimentally that we get these computational advantages only with a little compromise in performance when compared with thin-plate spline methods.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13

Similar content being viewed by others

References

  1. Vio R, Strohmer T and Wamsteker W 2000 On the reconstruction of irregularly sampled time series. Publ. Astron. Soc. Pac. 112(767): 74–90

    Article  Google Scholar 

  2. Arigovindan M, Suhling M, Jansen C, Hunziker P and Unser M 2007 Full motion and flow field recovery from echo Doppler data. IEEE Trans. Med. Imag. 26(1): 31–45

    Article  Google Scholar 

  3. Park S C, Park M K and Kang M G 2003 Super-resolution image reconstruction: a technical overview. IEEE Sig. Process. Mag. 20(3): 21–36

    Article  Google Scholar 

  4. Rauth M and Strohmer T 1998 Smooth approximation of potential fields from noisy scattered data. Geophysics 63(1): 85–94

    Article  Google Scholar 

  5. Faugeras O and Keriven R 2002 Variational principles, surface evolution, PDEs, level set methods and the stereo problem. In: 5th IEEE EMBS International Summer School on Biomedical Imaging, 83 pp

  6. Dijndam A, Sehonewille M and Hindriks C 1999 Reconstruction of band-limited signals, irregularly sampled along one spatial direction. Geophysics 64(2): 524–538

  7. Rashidi M 2010 Non-uniform sampling and reconstruction of multi-band signals and its application in wideband spectrum sensing of cognitive radio. arXiv preprint arXiv:1010.2158

  8. Fenn M and Steidl G 2006 Robust local approximation of scatterred data. In: Geometric properties for incomplete data Springer, Netherlands 317–334

  9. Schall O, Belyaev A and Seidel H P 2005 Robust filtering of noisy scattered point data. In: Point-Based Graphics, 2005. Eurographics/IEEE VGTC Symposium Proceedings IEEE 71–144

  10. Stein M L 2012 Interpolation of spatial data: some theory for kriging. Springer Science & Business Media

  11. Lai M J 1996 Scattered data interpolation and approximation using bivariate C1 piecewise cubic polynomials. Comput. Aided Geom. Des. 13(1): 81–88

    Article  MathSciNet  MATH  Google Scholar 

  12. Arigovindan M 2005 Variational reconstruction of vector and scalar images from non-uniform samples. These de doctorat, École Polytechnique Fédérale de Lausanne (EPFL) 94: 96

  13. Cazals F and Giesen J 2006 Effective computational geometry for curves and surfaces. In: Delaunay triangulation based surface reconstruction. Springer 231–276

  14. Rauth M 1995 Application of 2D methods for scattered data approximation to geophysical data sets. In: Sampling Theory and Applications, Proceedings of the SAMPTA’95 Workshop, Jurmala (Riga), Latvia 38–43

  15. Shepard D 1968 A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference 517–524

  16. Strohmer T and Tanner J 2006 Fast reconstruction methods for bandlimited functions from periodic nonuniform sampling. SIAM J. Num. Anal. 44(3): 1073–1094

    Article  MathSciNet  MATH  Google Scholar 

  17. Duchon J 1977 Splines minimizing rotation-invariant semi-norms in Sobolev spaces. 85–100

  18. Franke R and Nielson G 1980 Smooth interpolation of large sets of scattered data. Int. J. Num. Methods Eng. 15(11): 1691–1704

    Article  MathSciNet  MATH  Google Scholar 

  19. Strang G and Nguyen T 1996 Wavelets and filter banks. SIAM. Philadelphia, USA

  20. Micchelli C A 1984 Interpolation of scattered data: distance matrices and conditionally positive definite functions. In: Approximation theory and spline functions Springer 143–145. Berlin, Germany

  21. Arigovindan M, Suhling M and Unser M 2005 Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution. IEEE Trans. Image Process. 14(4): 450–460

  22. Beatson R K, Cherrie J B and Mouat C T 1999 Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11(2): 253–270

    Article  MathSciNet  MATH  Google Scholar 

  23. Carr J C, Fright W R and Beatson R K 1997 Surface interpolation with radial basis functions for medical imaging. IEEE Trans. Med. Imag. 16(1): 96–107

    Article  Google Scholar 

  24. Fasshauer G E 2007 Meshfree approximation methods with Matlab. vol. 6 World Scientific Publishing Co Inc, Singapore

  25. Buhmann M D 2003 Radial basis functions: theory and implementations. Cambridge University Press, Cambridge, England

  26. Baxter B 1992 The interpolation theory of radial basis functions. Cambridge, England

  27. Mongillo M 2011 Choosing basis functions and shape parameters for radial basis function methods. SIAM Undergrad. Res. Online 4: 190–209

    Article  Google Scholar 

  28. Fasshauer G E and McCourt M J 2012 Stable evaluation of Gaussian radial basis function interpolants. SIAM J. Sci. Comput. 34(2): A737–A762

    Article  MathSciNet  MATH  Google Scholar 

  29. Wendland H 2006 Computational aspects of radial basis function approximation. Stud. Comput. Math. 12: 231–256

    Article  MathSciNet  MATH  Google Scholar 

  30. Cavoretto R 2013 Partition of unity algorithm for two-dimensional interpolation using compactly supported radial basis functions. Commun. Appl. Commun. Appl. Ind. Math. 3(2)

  31. Cavoretto R, De Rossi A and Perracchione E 2015 Partition of unity interpolation on multivariate convex domains. Int. J. Model. Simul. Sci. Comput. 6(04): 1550034

    Article  Google Scholar 

  32. Morozov O V, Unser M and Hunziker P 2011 Reconstruction of large, irregularly sampled multidimensional images. A Tensor-Based Approach. IEEE Trans. Med. Imag. 30(2): 366–374

  33. Shewchuk J R et al. 1994 An introduction to the conjugate gradient method without the agonizing pain

  34. Salomon D 2004 Data compression: the complete reference. Springer Science & Business Media, New York, USA

  35. Harten A and Yad-Shalom I 1994 Fast multiresolution algorithms for matrix-vector multiplication. SIAM J. Num. Anal. 31(4): 1191–1218

    Article  MathSciNet  MATH  Google Scholar 

  36. Unser M 1995 Multigrid adaptive image processing. In: Proceedings of the International Conference on Image Processing IEEE vol. 1 49–52

    Google Scholar 

  37. Saad Y and Schultz M H 1986 GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3): 856–869

    Article  MathSciNet  MATH  Google Scholar 

  38. Brandner D and Withers G The Cell Image Library, CIL: 248, 1585, 7437 and 10016. Available at http://www.cellimagelibrary.org

Download references

Acknowledgement

This work was supported by the Young Scientist Scheme of Department of Science and Technology, Government of India, under Grant SB/FTP/ETA-240/2013.

Author information

Authors and Affiliations

  1. Imaging Systems Laboratory (ISL), Department of Electrical Engineering, Indian Institute of Science, Bangalore, 560012, India

    Bibin Francis, Sanjay Viswanath & Muthuvel Arigovindan

Authors
  1. Bibin Francis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sanjay Viswanath
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muthuvel Arigovindan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muthuvel Arigovindan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Francis, B., Viswanath, S. & Arigovindan, M. Scattered data approximation by regular grid weighted smoothing. Sādhanā 43, 5 (2018). https://doi.org/10.1007/s12046-017-0765-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12046-017-0765-y

Keywords

Search

Navigation