Abstract
Signal processing is an enabling technology that helps us to denote any operation which modifies or analyzes the information contained in a signal. In this paper, we first decompose the original signal by a wavelet packet frame and analyze the coefficients. Then, by using dual wavelet frames, we reconstruct the original signal. In this reconstruction, the standard choice for duals which plays a key role is the canonical dual. Our aim is to develop new duals to obtain more accurate results. To this end, we consider wavelet frames which Fourier transform of generators form a partition of unity. Then we introduce several explicit duals for them and compare the advantage of these duals in signal processing. This indicates that we may obtain more reliable estimates by alternate duals.
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Allabakash, S., Yasodha, P., Venkatramana Reddy, S., Srinivasulu, P.: Wavelet transform-based methods for removal of ground clutter and denoising the radar wind profiler data. Signal Process. IET 9, 440–448 (2015)
Arefijamaal, A., Zekaee, E.: Signal processing by alternate dual Gabor frames. Appl. Comput. Harmon. Anal. 35, 535–540 (2013)
Arefijamaal, A., Zekaee, E.: Image processing by alternate dual Gabor frames. Bull. Iran. Math. Soc. 42(6), 1305–1314 (2016)
Casazza, P.G., Kutyniok, G., Lammers, M.C.: Duality principles in frame theory. Fourier Anal. Appl. 10(4), 383–408 (2004)
Chen, Y., Zhang, Y., Hu, H., Ling, H.: A novel gray image watermarking scheme. J. Softw. 6(5), 849–856 (2011)
Christensen, O.: Frames and Bases. An Introductory Course. Birkhäuser, Boston (2008)
Christensen, O.: Pairs of dual Gabor frames with compact support and desired frequency localization. Appl. Comput. Harmon. Anal. 20, 403–410 (2006)
Christensen, O.: Time-frequency analysis and its applications in denoising. Department of informatics, University of Bergan, Thesis (2002)
Christensen, O., Kim, R.Y.: On dual Gabor frame pairs generated by polynomials. Fourier Anal. Appl. 16, 11–16 (2010)
Chui, C.K., Shi, X.: Orthonormal wavelets and tight frames with arbitrary real dilations. Appl. Comput. Harmon. Anal. 9(3), 243–264 (2000)
Daubechies, I.: The wavelet transform, time frequency localization and signal analysis. IEEE Trans. Inf. Theory 36(5), 961–1005 (1990)
Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)
Deng, L., Yu, C.L., Chakrabarti, C., Kim, J., Narayanan, V.: Efficient image reconstruction using partial 2D Fourier transform. In: Proceedings of the IEEE Workshop on Signal Processing Systems, pp. 49–54. SiPS, Washington, DC Metro Area, USA (2008)
Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation via wavelet shrinkage. Biometrika 81(3), 425–455 (1994)
Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. Am. Stat. Assoc. 90(432), 1200–1224 (1995)
Feichtinger, H.G., Strohmer, T.: Gabor Analysis and Algorithms: Theory and Applications. Birkhäuser, Boston (1997)
Gautier, M., Lienard, J.: Efficient wavelet packet modulation for wireless communication. In: Proceedings of the 3rd Advanced International Conference, pp. 1–19. France Telecom R and D, France (2007)
Găianu, M., Onchis, D.M.: Face and marker detection using Gabor frames on GPUs. Signal Process. 96, 90–93 (2014)
He, J., Li, Z., Qian, H.: Cryptography based on spatiotemporal chaos system and multiple maps. J. Softw. 5(4), 421–428 (2010)
Heil, C.E., Walnut, D.F.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)
Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projection, 2nd edn. Springer, New York (2009)
Huang, F., Qu, X.: Design of image encryption algorithm based on compound two-dimensional maps. J. Softw. 6(10), 1953–1960 (2011)
Janssen, A.J.E.M.: Duality and biorthogonality for Weyl–Heisenberg frames. Fourier Anal. Appl. 1(4), 403–436 (1995)
Lemvig, J.: Constructing pairs of dual bandlimited framelets with desired time localization. Adv. Comput. Math. 30, 231–248 (2009)
Mallat, S.G.: A wavelet tour of signal processing. Academic Press, San Diego (1999)
Porter, R., Canagarajah, N.: Robust rotation-invariant texture classification: wavelet, Gabor filter and GMRF based schemes. IEE Proc. Vis. Image Signal Process. 144(3), 180–188 (1997)
Ron, A., Shen, Z.: Weyl–Heisenberg frames and Riesz bases in \(L^2(\mathbb{R}^d )\). Duke Math. J. 89, 237–282 (1997)
Tong, C.S., Leung, K.T.: Super-resolution reconstruction based on linear interpolation of wavelet coefficients. Multidimens. Syst. Signal Process. 18(2–3), 153–171 (2007)
Thamarai, P., Adalarasu, K.: Denoising of EEG, ECG and PPG signals using wavelet transform. J. Pharm. Sci. and Res. 10(1), 156–161 (2018)
Verma, R., Mahrishi, R., Srivastava, G. K., Siddavatam, R.: A novel image reconstruction using second generation wavelets. In: IEEE International Conference on Advances in Recent Technologies in Communication and Computing, pp. 509–513. IEEE Press, Kerala (2009)
Wexler, J., Raz, S.: Discrete Gabor expansions. Signal Process. 21, 207–221 (1990)
Yang, X., Min, J., Shi, Y.: Parametrisation construction frame of lifting scheme. IET Signal Process. 5, 1–15 (2011)
Zhang, Y., Yuanyuan, W., Chen, Z.: Efficient discrete cosine transform model-based algorithm for photoacoustic image reconstruction. J. Biomed. Opt. 18(6), 066008(1)–066008(9) (2013)
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Arefijamaal, A.A., Arabyani Neyshaburi, F. & Matindoost, S. Construction of dual wavelet frame pairs and signal recovery. SeMA 76, 27–36 (2019). https://doi.org/10.1007/s40324-018-0156-2
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DOI: https://doi.org/10.1007/s40324-018-0156-2