Effect of Spatial Decomposition on the Efficiency of k Nearest Neighbors Search in Spatial Interpolation

  • Naijie Fan
  • Gang MeiEmail author
  • Zengyu Ding
  • Salvatore Cuomo
  • Nengxiong Xu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11339)


Spatial interpolations are commonly used in geometric modeling for life science applications. In large-scale spatial interpolations, it is always needed to find a local set of data points for each interpolated point using the k Nearest Neighbor (kNN) search procedure. To improve the computational efficiency of kNN search, spatial decomposition structures such as grids and trees are employed to fastly locate the nearest neighbors. Among those spatial decomposition structures, the uniform grid is the simplest one, and the size of the grid cell could strongly affect the efficiency of kNN search. In this paper, we evaluate the effect of the size of uniform grid cell on the efficiency of kNN search. Our objective is to find the relatively optimal size of grid cell by considering the distribution of scattered points (i.e., the data points and the interpolated points). We employ the Standard Deviation of points’ coordinates to measure the spatial distribution of scattered points. For the irregularly distributed scattered points, we perform several series of kNN search procedures in two dimensions. Benchmark results indicate that: in two dimensions, with the increase of the Standard Deviation of points’ coordinates, the relatively optimal size of the grid cell decreases and eventually converges. The relationships between the Standard Deviation of scattered points’ coordinates and the relatively optimal size of grid cell are also fitted. The fitted relationships could be applied to determine the relatively optimal grid cell in kNN search, and further, improve the computational efficiency of spatial interpolations.


Spatial interpolation k nearest neighbors (kNN) search Uniform grid Spatial distribution Standard error 



This work was supported by the Natural Science Foundation of China (Grant Numbers 11602235 and 41772326), the China Postdoctoral Science Foundation (2015M571081), the Fundamental Research Funds for the Central Universities (2652017086).


  1. 1.
    Al Aghbari, Z., Al-Hamadi, A.: Efficient KNN search by linear projection of image clusters. Int. J. Intell. Syst. 26(9), 844–865 (2011)CrossRefGoogle Scholar
  2. 2.
    Allen, G., Gandevia, S., Mckenzie, D.: Reliability of measurements of muscle strength and voluntary activation using twitch interpolation. Muscle Nerve 18(6), 593–600 (1995)CrossRefGoogle Scholar
  3. 3.
    Beliakov, G., Li, G.: Improving the speed and stability of the k-nearest neighbors method. Pattern Recognit. Lett. 33(10), 1296–1301 (2012)CrossRefGoogle Scholar
  4. 4.
    Cavoretto, R., Rossi, A.D., Dell’Accio, F., Tommaso, F.D.: Fast computation of triangular shepard interpolants. J. Comput. Appl. Math. (2018).
  5. 5.
    Cuomo, S., Galletti, A., Giunta, G., Marcellino, L.: Reconstruction of implicit curves and surfaces via RBF interpolation. Appl. Numer. Math. 116(1), 157–171 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cuomo, S., Galletti, A., Giunta, G., Starace, A.: Surface reconstruction from scattered point via RBF interpolation on GPU. In: Ganzha, M., Maciaszek, L., Paprzycki, M (eds.) 2013 Federated Conference on Computer Science and Information Systems (Fedcsis), pp. 433–440 (2013)Google Scholar
  7. 7.
    Ding, Z., Mei, G., Cuomo, S., Xu, N., Tian, H.: Performance evaluation of GPU-accelerated spatial interpolation using radial basis functions for building explicit surfaces. Int. J. Parallel Program. 46(5), 963–991 (2018)CrossRefGoogle Scholar
  8. 8.
    Dong, W., Zhang, L., Lukac, R., Shi, G.: Sparse representation based image interpolation with nonlocal autoregressive modeling. IEEE Trans. Image Process. 22(4), 1382–1394 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Huang, F., Bu, S., Tao, J., Tan, X.: OpenCL implementation of a parallel universal Kriging algorithm for massive spatial data interpolation on heterogeneous systems. ISPRS Int. J. Geo Inf. 5(6), 96 (2016)CrossRefGoogle Scholar
  10. 10.
    Huang, F., Liu, D., Tan, X., Wang, J., Chen, Y., He, B.: Explorations of the implementation of a parallel IDW interpolation algorithm in a Linux cluster-based parallel GIS. Comput. Geosci. 37(4), 426–434 (2011)CrossRefGoogle Scholar
  11. 11.
    Lehmann, T., Gonner, C., Spitzer, K.: Survey: interpolation methods in medical image processing. IEEE Trans. Med. Image 18(11), 1049–1075 (1999)CrossRefGoogle Scholar
  12. 12.
    Li, S., Harner, E.J., Adjeroh, D.A.: Random KNN. In: Zhou, Z.H., et al. (eds.) 2014 IEEE International Conference on Data Mining Workshop (ICDMW), pp. 629–636 (2014)Google Scholar
  13. 13.
    Liu, J., Nowinski, W.L.: A hybrid approach to shape-based interpolation of stereotactic atlases of the human brain. Neuroinformatics 4(2), 177–198 (2006)CrossRefGoogle Scholar
  14. 14.
    Liu, S.g., Wei, Y.w.: Fast nearest neighbor searching based on improved VP-tree. Pattern Recognit. Lett. 60–61, 8–15 (2015)CrossRefGoogle Scholar
  15. 15.
    Mei, G.: Evaluating the power of GPU acceleration for IDW interpolation algorithm. Sci. World J. 2014, 8 (2014)Google Scholar
  16. 16.
    Mei, G., Xu, L., Xu, N.: Accelerating adaptive inverse distance weighting interpolation algorithm on a graphics processing unit. R. Soc. Open Sci. 4(9), 170436 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mei, G., Xu, N., Xu, L.: Improving GPU-accelerated adaptive IDW interpolation algorithm using fast kNN search. SpringerPlus 5, 1389 (2016)CrossRefGoogle Scholar
  18. 18.
    Meijering, E.: A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proc. IEEE 90(3), 319–342 (2002)CrossRefGoogle Scholar
  19. 19.
    Mirzaei, D.: Analysis of moving least squares approximation revisited. J. Comput. Appl. Math. 282, 237–250 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nutanong, S., Zhang, R., Tanin, E., Kulik, L.: V*-kNN: an efficient algorithm for moving k nearest neighbor queries. In: ICDE: 2009 IEEE International Conference on Data Engineering, pp. 1519–1522 (2009)Google Scholar
  21. 21.
    Pan, J., Manocha, D.: Bi-level locality sensitive hashing for k-nearest neighbor computation. In: 2012 IEEE 28th IEEE International Conference on Data Engineering, pp. 378–389 (2012)Google Scholar
  22. 22.
    Pan, M.s., Yang, X.l., Tang, J.t.: Research on interpolation methods in medical image processing. J. Med. Syst. 36(2), 777–807 (2012)CrossRefGoogle Scholar
  23. 23.
    Parrott, R., Stytz, M., Amburn, P., Robinson, D.: Towards statistically optimal interpolation for 3-D medical imaging. IEEE Eng. Med. Biol. Mag. 12(3), 49–59 (1993)CrossRefGoogle Scholar
  24. 24.
    Pesquer, L., Cortes, A., Pons, X.: Parallel ordinary Kriging interpolation incorporating automatic variogram fitting. Comput. Geosci. 37(4), 464–473 (2011)CrossRefGoogle Scholar
  25. 25.
    Shankar, V., Wright, G.B., Kirby, R.M., Fogelson, A.L.: A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction-diffusion equations on surfaces. J. Sci. Comput. 63(3), 745–768 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Volkau, I., Aziz, A., Nowinski, W.: Indirect interpolation of subcortical structures in the Talairach-Tournoux atlas, vol. 5367, pp. 533–537 (2004)Google Scholar
  27. 27.
    Wang, J., Liu, G.: A point interpolation meshless method based on radial basis functions. Int. J. Numer. Methods Eng. 54(11), 1623–1648 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Naijie Fan
    • 1
  • Gang Mei
    • 1
    Email author
  • Zengyu Ding
    • 1
  • Salvatore Cuomo
    • 2
  • Nengxiong Xu
    • 1
  1. 1.School of Engineering and Technology, China University of Geosciences (Beijing)BeijingChina
  2. 2.Department of Mathematics and Applications “R. Caccioppoli”University of Naples Federico IINaplesItaly

Personalised recommendations