Abstract
This chapter presents a design scheme to generate tight and so-called semi-tight frames in the space of discrete-time periodic signals. The frames originate from oversampled perfect reconstruction periodic filter banks. The filter banks are derived from discrete-time and discrete periodic splines. Each filter bank comprises one linear phase low-pass filter (in most cases interpolating) and one high-pass filter, whose magnitude response mirrors that of a low-pass filter. In addition, these filter banks comprise a number of band-pass filters. In this chapter, frames generated by four-channel filter banks are briefly outlined (see Chap. 17 in [2] for details) and tight frames generated by six- and eight-channel filter banks are introduced. These latter frames provide an additional redundancy to the frame representations of signals. The design scheme enables us to design framelets with any number of local discrete vanishing moments (LDVMs). The computational complexity of the framelet transforms practically does not depend on the number of LDVMs and on the size of the impulse response of filters.
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. Averbuch, P. Neittaanmäki, V. Zheludev, Periodic spline-based frames: design and applications for image restoration. Inverse Probl. Imaging 9(3), 661–707 (2015)
A.Z. Averbuch, P. Neittaanmäki, V.A. Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Volume I: Periodic Splines (Springer, 2014)
A.Z. Averbuch, P. Neittaanmäki, V.A. Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Volume II: Non-periodic Splines (Springer, 2015)
L.M. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7(3), 200–217 (1967)
J. Cai, S. Osher, Z. Shen, Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2009/2010)
Z. Cvetković, M. Vetterli, Oversampled filter banks. IEEE Trans. Signal Process. 46(5), 1245–1255 (1998)
B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon. Anal. 22(1), 78–104 (2007)
G. Polya, G. Szegö, Aufgaben and Lehrsätze aus der Analysis, vol. II (Springer, Berlin, 1971)
W. Yin, S. Osher, D. Goldfarb, J. Darbon, Bregman iterative algorithms for \(l_1\)-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1(1), 143–168 (2008)
V. Zheludev, V.N. Malozemov, A.B. Pevnyi, Filter banks and frames in the discrete periodic case, in Proceedings of the St. Petersburg Mathematical Society, vol. XIV, ed, by N. Uraltseva. American Mathematical Society Transl., Ser. 2 (2009), pp. 1–11
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). Wavelet Frames Generated by Spline-Based p-Filter Banks. 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_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-92123-5_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92122-8
Online ISBN: 978-3-319-92123-5
eBook Packages: EngineeringEngineering (R0)