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On the design of multi-dimensional compactly supported Parseval framelets with directional characteristics. (English) Zbl 1426.42024
Summary: In this paper, we propose a new method for the construction of multi-dimensional, wavelet-like families of affine frames, commonly referred to as framelets, with specific directional characteristics, small and compact support in space, directional vanishing moments (DVM), and axial symmetries or anti-symmetries. The framelets we construct arise from readily available refinable functions. The filters defining these framelets have few non-zero coefficients, custom-selected orientations and can act as finite-difference operators.
42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text: DOI
[1] Daubechies, I., Ten Lectures on Wavelets, CBMS, vol. 61 (1992), SIAM: Society for Industrial and Applied Mathematics · Zbl 0776.42018
[2] Vetterli, M.; Kovacevic, J., Wavelets and Subband Coding (1995), Prentice Hall PTR: Prentice Hall PTR Englewood Cliffs, NJ · Zbl 0885.94002
[3] Kovacevic, J.; Vetterli, M., Nonseparable multidimensional perfect reconstruction filter-banks, IEEE Trans. Inform. Theory, 38, 533-555 (1992)
[4] Ayache, A., Some methods for constructing nonseparable, orthonormal, compactly supported wavelet bases, Appl. Comput. Harmon. Anal., 10, 99-111 (2001) · Zbl 0977.42017
[5] Belogay, Eugene; Wang, Yang, Arbitrarily smooth orthogonal nonseparable wavelets in \(R^2\), SIAM J. Math. Anal., 30, 3, 678-697 (1999) · Zbl 0946.42025
[6] Candes, Emmanuel; Demanet, Laurent; Donoho, David; Ying, Lexing, Fast discrete curvelet transforms, Multiscale Model. Simul., 5, 3, 861-899 (2006) · Zbl 1122.65134
[7] Candes, Emmanuel J.; Demanet, Laurent, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math., 58, 11, 1472-1528 (2005) · Zbl 1078.35007
[8] Demanet, L.; Vandergheynst, P., Gabor wavelets on the sphere, (Proc. SPIE Int. Soc. Opt. Eng., vol. 5207 (2003)), Article 5207 pp.
[9] Candes, E. J., Harmonic analysis of neural networks, Appl. Comput. Harmon. Anal., 6, 197-218 (1999) · Zbl 0931.68104
[10] Candès, Emmanuel J.; Donoho, David L., New tight frames of curvelets and optimal representations of objects with piecewise \(C^2\) singularities, Comm. Pure Appl. Math., 57, 2, 219-266 (2004) · Zbl 1038.94502
[11] Labate, D.; Lim, W.; Kutyniok, G.; Weiss, G., Sparse multidimensional representation using shearlets, (SPIE Proc., vol. 5914 (2005), SPIE: SPIE Bellingham), 254-262
[12] Ron, A.; Shen, Z., Affine system in \(L^2(R^d)\): the analysis of the analysis operator, J. Funct. Anal., 148, 408-447 (1997) · Zbl 0891.42018
[13] Ron, A.; Shen, Z., Affine systems in \(L_2(R^d)\) II: dual systems, J. Fourier Anal. Appl., 3, 617-637 (1997) · Zbl 0904.42025
[14] Grohs, Philipp; Kutyniok, Gitta, Parabolic molecules, Found. Comput. Math., 14, 2, 299-337 (Apr 2014)
[15] Daubechies, Ingrid; Han, Bin; Ron, Amos; Shen, Zuowei, Framelets: Mra-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., 14, 1, 1-46 (2003) · Zbl 1035.42031
[16] Han, Bin, Nonhomogeneous wavelet systems in high dimensions, Appl. Comput. Harmon. Anal., 32, 2, 169-196 (2012) · Zbl 1241.42028
[17] Chui, Charles K.; He, Wenjie; Stöckler, Joachim, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal., 13, 3, 224-262 (2002) · Zbl 1016.42023
[18] Atreas, Nikolaos; Melas, Antonios; Stavropoulos, Theodoros, Affine dual frames and extension principles, Appl. Comput. Harmon. Anal., 36, 1, 51-62 (2014) · Zbl 1294.42004
[19] Atreas, Nikolaos D.; Papadakis, Manos; Stavropoulos, Theodoros, Extension principles for dual multiwavelet frames of \(L^2(R^s)\) constructed from multirefinable generators, J. Fourier Anal. Appl., 1-24 (2016) · Zbl 1348.42030
[20] Kutyniok, Gitta; Labate, Demetrio, Resolution of the wavefront set using continuous shearlets, Trans. Amer. Math. Soc., 361, 5, 2719-2754 (2009) · Zbl 1169.42012
[21] Kittipoom, Pisamai; Kutyniok, Gitta; Lim, Wang-Q., Construction of compactly supported shearlet frames, Constr. Approx., 35, 1, 21-72 (Feb 2012)
[22] Ayache, A., Construction de bases orthonormés d’ondelettes de \(L^2(R^2)\) non séparables, à support compact et de régularité arbitrairement grande, C. R. Acad. Sci. Paris, 325, 17-20 (1997) · Zbl 0896.46005
[24] San Antolín, A.; Zalik, R. A., A family of nonseparable scaling functions and compactly supported tight framelets, J. Math. Anal. Appl., 404, 2, 201-211 (2013) · Zbl 1306.42051
[25] Han, Bin, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, Approximation Theory, Wavelets, and Numerical Analysis. Approximation Theory, Wavelets, and Numerical Analysis, J. Comput. Appl. Math., 155, 1, 43-67 (2003) · Zbl 1021.42020
[26] Han, Bin; Jiang, Qingtang; Shen, Zuowei; Zhuang, Xiaosheng, Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness, Math. Comp., 87, 347-379 (2018) · Zbl 1375.42051
[27] Han, Bin, On dual wavelet tight frames, Appl. Comput. Harmon. Anal., 4, 4, 380-413 (1997) · Zbl 0880.42017
[28] Kingsbury, N., Image processing with complex wavelets, Philos. Trans. R. Soc. Lond. Ser. A, 357, 2543-2560 (1999) · Zbl 0976.68527
[29] Selesnick, I. W.; Sendur, L., Iterated oversampled filter banks and wavelet frames, (Unser, M.; Aldroubi, A.; Laine, A., Proc. Wavelet Applications in Signal and Image Processing VIII. Proc. Wavelet Applications in Signal and Image Processing VIII, Proceedings of SPIE, vol. 4119 (2000))
[30] Han, B.; Zhao, Z., Tensor product complex tight framelets with increasing directionality, SIAM J. Imaging Sci., 7, 2, 997-1034 (2014) · Zbl 1295.42023
[31] Han, B.; Mo, Q.; Zhao, Z., Compactly supported tensor product complex tight framelets with directionality, SIAM J. Math. Anal., 47, 3, 2464-2494 (2015) · Zbl 1319.42031
[32] Cabrelli, C. A.; Gordillo, M-L., Existence of multiwavelets in \(R^n\), Proc. Amer. Math. Soc., 130, 5, 1413-1424 (2000) · Zbl 0988.42025
[33] Adelson, E. H.; Simoncelli, E.; Hingoranp, R., Orthogonal pyramid transforms for image coding, (Visual Communications and Image Processing II, vol. 845 (1987)), 50-58
[34] Simoncelli, E. P.; Freeman, W. T., The steerable pyramid: a flexible architecture for multi-scale derivative computation, (Proc. IEEE International Conference on Image Processing (1995))
[35] Candes, E. J.; Donoho, D. L., Ridgelets: a key to higher dimensional intermittency?, Philos. Trans. R. Soc. Lond. Ser. A, 2495-2509 (1999) · Zbl 1082.42503
[36] Guo, Kanghui; Labate, Demetrio, Optimally sparse multidimensional representation using shearlets, SIAM J. Math. Anal., 39, 298-318 (2007) · Zbl 1197.42017
[37] Papadakis, M.; Bodmann, B. G.; Alexander, S. K.; Vela, D.; Baid, S.; Gittens, A. A.; Kouri, D. J.; Gertz, S. D.; Jain, S.; Romero, J. R.; Li, X.; Cherukuri, P.; Cody, D. D.; Gladish, G. W.; Aboshady, I.; Conyers, J. L.; Casscells, S. W., Texture-based tissue characterization for high-resolution CT-scans of coronary arteries, Commun. Numer. Methods Eng., 25, 6, 597-613 (2009) · Zbl 1165.92316
[38] Han, Bin; Li, Tao; Zhuang, Xiaosheng, Directional compactly supported box spline tight framelets with simple geometric structure, Appl. Math. Lett., 91, 213-219 (2019) · Zbl 1407.42025
[39] Lu, Yue; Do, M. N., The finer directional wavelet transform, (Proc. IEEE Int. Conf. Acoust. Speech Signal Process, vol. 4 (March 2005)), pp. iv/573-iv/576
[40] Lu, Y. M.; Do, M. N., Multidimensional directional filter banks and surfacelets, IEEE Trans. Image Process., 16, 4, 918-931 (April 2007)
[41] da Cunha, A. L.; Do, M. N., On two-channel filter banks with directional vanishing moments, IEEE Trans. Image Process., 16, 5, 1207-1219 (May 2007)
[42] da Cunha, A. L.; Do, M. N., Bi-orthogonal filter banks with directional vanishing moments, (Proc. IEEE Int. Conf. Acoust. Speech Signal Process, vol. 4 (March 2005)), pp. iv/553-iv/556
[43] Diao, Chenzhe; Han, Bin, Quasi-tight framelets with high vanishing moments derived from arbitrary refinable functions, Appl. Comput. Harmon. Anal. (2018)
[44] Hernandez, E.; Weiss, G., A First Course on Wavelets (1996), CRC Press: CRC Press Boca Raton, FL · Zbl 0885.42018
[45] Hast, Anders, Simple filter design for first and second order derivatives by a double filtering approach, Pattern Recogn. Lett., 42, 65-71 (2014)
[46] Song Goh, Say; Teo, K. M., Extension principles for tight wavelet frames of periodic functions, Appl. Comput. Harmon. Anal., 25, 2, 168-186 (2008) · Zbl 1258.42031
[47] Atreas, Nikolaos; Karantzas, Nikolaos; Papadakis, Manos; Stavropoulos, Theodoros, Exploring neuronal synapses with directional and symmetric frame filters with small support, (Proc. SPIE, vol. 10394 (2017)), Article 10394 pp.
[48] Daubechies, Ingrid, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41, 7, 909-996 (1988) · Zbl 0644.42026
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