Abstract
In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are designed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, the splines use either clean or noised arbitrarily-spaced samples. Formulas for the spline’s extrapolation beyond the sampling interval are established. Sharp estimations of the approximation errors are presented. The capability to adapt the grid to the structure of an object and to have minimal requirements to the operating memory are of great advantages for offline processing of signals and multidimensional data arrays. The designed splines serve as a source for generating real-time wavelet transforms to apply to signals in scenarios where the signal’s samples subsequently arrive one after the other at random times. The wavelet transforms are executed by six-tap weighted moving averages of the signal’s samples without delay. On arrival of new samples, only a couple of adjacent transform coefficients are updated in a way that no boundary effects arise.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York (1972)
Aldroubi, A., Cabrellib, C., Molterb, U.M.: Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for \(l^{2}(\mathbb {R}^{d})\) . Applied and Comp Harm. Analysis 17(2), 119–140 (2004)
Averbuch, A.Z., Neittaanmäki, P., Zheludev, V. A.: Spline and spline wavelet methods with applications to signal and image processing, Volume I: Periodic splines. Springer (2014)
Averbuch, A.Z., Neittaanmäki, P., Zheludev, V.A.: Spline and spline wavelet methods with applications to signal and image processing, Volume II: Non-periodic splines. Springer (2015)
Brislawn, C.M.: Classification of nonexpansive symmetric extension transforms for multirate filter banks. Appl. Comput. Harmon. Anal. 3(4), 337–357 (1996)
Cai, T., Brown, L.: Wavelet shrinkage for nonequispaced samples. Annals of Statistics 26(5), 1783–1799 (1998)
Daubechies, I., Guskov, I., Schroder, P., Sweldens, W.: Wavelets on irregular point sets. Phil. Trans. R. Soc. Lond. A 357, 2397–2413 (1999)
de Boor, C.: A practical guide to splines. Springer, New York (1978)
de Boor, C.: Divided differences. Surveys in Approximation Theory 1 (2005)
Hall, C. A.: On error bounds for spline interpolation. J. Approx. Theory 1, 209–218 (1968)
Dahmen, W., Carnicer, J.M., Pen, a, J.M. : Local decomposition of renable spaces and wavelets. Appl. Comput Harmon. Anal 3 (1996)
Liu, Z., Mi, Y., Mao, Y.: An improved real-time denoising method based on lifting wavelet transform. Measurement Science Review 14(3), 152–159 (2014)
Lyche, T., Schumaker, L.L.: Local Spline Approximation Methods. MRC Technical Summary Mathematics Research Center University of Wisconsin (1974)
Schoenberg, I.J.: Contributions to the problem of approximation of equidistant data by analytic functions, vol. 4:45–99 (1946). Parts A and B
Schumaker, L.L.: Spline functions: Basic theory. John Wiley & Sons, New York (1981)
Stoer, J., Bulirsch, R.: Introduction to numerical analysis, Second edition. Springer–Verlag, New York (1993)
Suturin, M.G., Zheludev, V.: On the approximation on finite intervals and local spline extrapolation . Russian J. Numer. Anal. Math. Modelling 9(1), 75–89 (1994)
Sweldens, W.: The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3(2), 186–200 (1996)
Sweldens, W.: The lifting scheme: A construction of second generation wavelets. SIAM. J. Math. Anal. 29(2), 511–546 (1997)
Vanraesa, E., Jansenb, M., Bultheela, A.: Stabilised wavelet transforms for non-equispaced data smoothing. Signal Processing 82(12), 1979–1990 (2002)
Xia, R., Meng, K., Qian, F., Wang, Z.L.: Online wavelet denoising via a moving window. Acta Automatica Sinica 33(9), 897–901 (2007)
Zav’yalov, Y.S., Kvasov, B.I., Miroshnichenko, V.L.: Methods of spline-functions. Nauka, Moscow, 1980. (In Russian)
Zheludev, V.: Local spline approximation on arbitrary meshes. Sov. Math. 8, 16–21 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Arieh Iserles
Rights and permissions
About this article
Cite this article
Averbuch, A., Neittaanmäki, P., Shefi, E. et al. Local cubic splines on non-uniform grids and real-time computation of wavelet transform. Adv Comput Math 43, 733–758 (2017). https://doi.org/10.1007/s10444-016-9504-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-016-9504-x