Abstract
Periodic discrete splines with different periods and spans are introduced in Sect. 3.4 of Volume I (Averbuch, Neittaanmäki and Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Springer, Berlin, 2014) [2]. In this chapter, we regard periodic discrete splines as a base for the design of periodic discrete-time wavelets, wavelet packets and wavelet frames. Therefore, only the discrete splines whose spans are 2 are outlined. These discrete splines are linear combinations of the discrete B-splines. So also, the so-called discrete-time splines are discussed in the chapter that are linear combinations of the discrete-time B-splines. The discrete-time B-splines are defined by the polynomial B-splines sampling. Utilization of the Fast Fourier transform (FFT) provides a fast explicit way to implement all the computations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Aldroubi, M. Eden, M. Unser, Discrete spline filters for multiresolutions and wavelets of \(l_2\). SIAM J. Math. Anal. 25(5), 1412–1432 (1994)
A.Z. Averbuch, P. Neittaanmäki, V.A. Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Periodic Splines, vol I (Springer, Berlin, 2014)
A.Z. Averbuch, P. Neittaanmäki, V.A. Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Non-periodic Splines, vol II (Springer, Berlin, 2015)
G. Battle, A block spin construction of ondelettes. I. lemarié Funct. Comm. Math. Phys. 110(4), 601–615 (1987)
V.N. Malozemov, A.B. Pevnyi, Discrete periodic splines and their numerical applications. Comput. Math. Math. Phys. 38(8), 1181–1192 (1998)
L.L. Schumaker, Spline Functions: Basic Theory (Wiley, New York, 1981)
M. Unser, A. Aldroubi, M. Eden, Fast B-spline transforms for continuous image representation and interpolation. IEEE Trans. Pattern Anal. Mach. Intell. 13(3), 277–285 (1991)
M. Unser, A. Aldroubi, M. Eden, B-spline signal processing. I. Theory. IEEE Trans. Signal Process. 41(2), 821–833 (1993). IEEE Signal Processing Society’s 1995 best paper award
K.F. Üstüner, L.A. Ferrari, Discrete splines and spline filters. IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process. 39(7), 417–422 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Averbuch, A.Z., Neittaanmäki, P., Zheludev, V.A. (2019). Periodic Discrete and Discrete-Time Splines. In: Spline and Spline Wavelet Methods with Applications to Signal and Image Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-92123-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-92123-5_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92122-8
Online ISBN: 978-3-319-92123-5
eBook Packages: EngineeringEngineering (R0)