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Divergence free polar wavelets for the analysis and representation of fluid flows. (English) Zbl 1411.42010
Summary: We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the ideal, analytic sense, have a multi-resolution structure and fast transforms, and an intuitive correspondence to common flow phenomena. Our construction also allows for well defined directional selectivity, e.g. to model the behavior of divergence free vector fields in the vicinity of boundaries or to represent highly directional features like in a von Kármán vortex street. We demonstrate the practicality and efficiency of our construction by analyzing the representation of different divergence free vector fields in our wavelets.
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
76B99 Incompressible inviscid fluids
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[1] Abry, P.; Flandrin, P., On the initialization of the discrete wavelet transform algorithm, IEEE Signal Process. Lett., 1, 32-34, (1994)
[2] Battle, G.; Federbush, P., Divergence-free vector wavelets, Mich. Math. J., 40, 181-195, (1993) · Zbl 0798.42022
[3] Battle, G.; Federbush, P.; Uhlig, P., Wavelets for quantum gravity and divergence-free wavelets, Appl. Comput. Harmon. Anal., 1, 295-297, (1994) · Zbl 0798.42024
[4] Bostan, E.; Unser, M.; Ward, JP, Divergence-free wavelet frames, IEEE Signal Process. Lett., 22, 1142-1146, (2015)
[5] Candès, E.; Donoho, DL, New tight frames of curvelets and optimal representations of objects with piecewise \(\text{C}^2\) singularities, Commun. Pure Appl. Math., 57, 219-266, (2004) · Zbl 1038.94502
[6] Candès, EJ; Donoho, DL, Ridgelets: a key to higher-dimensional intermittency?, R. Soc. Lond. Philos. Trans. Ser. A, 357, 2495, (1999) · Zbl 1082.42503
[7] Candès, EJ; Donoho, DL, Continuous curvelet transform: I. Resolution of the wavefront set, Appl. Comput. Harmon. Anal., 19, 162-197, (2005) · Zbl 1086.42022
[8] Chenouard, N.; Unser, M., 3D steerable wavelets in practice, IEEE Trans. Image Process., 21, 4522-4533, (2012) · Zbl 1373.42043
[9] Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992) · Zbl 0776.42018
[10] Do, MN; Vetterli, M., The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process., 14, 2091-2106, (2005)
[11] Federbush, P., A phase cell approach to Yang-Mills theory. I. Modes, lattice-continuum duality, Commun. Math. Phys., 107, 319-329, (1986)
[12] Federbush, P., A phase cell approach to Yang-Mills theory. III. Local stability, modified renormalization group transformation, Commun. Math. Phys., 110, 293-309, (1987)
[13] Federbush, P., A phase cell approach to Yang-Mills theory. IV. The choice of variables, Commun. Math. Phys., 114, 317-343, (1988)
[14] Freeman, D.; Poore, D.; Wei, AR; Wyse, M., Moving Parseval frames for vector bundles, Houst. J. Math., 40, 817-832, (2014) · Zbl 1306.42046
[15] Freeman, WT; Adelson, EH, The design and use of steerable filters, IEEE Trans. Pattern Anal. Mach. Intell., 13, 891-906, (1991)
[16] Hörmander, L.: The Analysis of Linear Partial Differential Operators. Springer, Berlin (2004) · Zbl 1083.32033
[17] Kovacevic, J.; Chebira, A., Life beyond bases: the advent of frames (part I), IEEE Signal Process. Mag., 24, 86-104, (2007)
[18] Kovacevic, J.; Chebira, A., Life beyond bases: the advent of frames (part II), IEEE Signal Process. Mag., 24, 115-125, (2007)
[19] Labate, D., Lim, W.-Q., Kutyniok, G., Weiss, G.: Sparse multidimensional representation using shearlets. In: Papadakis, M., Laine, A.F., Unser, M.A. (eds.) Wavelets XI, pp. 254-262. International Society for Optics and Photonics, Aug 2005
[20] Lemarie-Rieusset, P-G; Singh, SP (ed.), Wavelets, splines and divergence-free vector functions, 381-390, (1992), Dordrecht · Zbl 0747.41007
[21] Lessig, C.: Polar wavelets in space. IEEE Signal Process. Lett. (submitted) (2018). https://arxiv.org/abs/1805.02061
[22] Mallat, S.G.: A Wavelet Tour of Signal Processing: The Sparse Way, 3rd edn. Academic Press, London (2009) · Zbl 1170.94003
[23] McEwen, JD; Durastanti, C.; Wiaux, Y., Localisation of directional scale-discretised wavelets on the sphere, Appl. Comput. Harmon. Anal., 44, 59-88, (2016) · Zbl 1376.42052
[24] Perona, P.: Deformable kernels for early vision. In: Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 222-227. IEEE Computer Society Press (1991)
[25] Portilla, J.; Simoncelli, EP, A parametric texture model based on joint statistics of complex wavelet coefficients, Int. J. Comput. Vis., 40, 49-70, (2000) · Zbl 1012.68698
[26] Simoncelli, E.P., Freeman, W.T.: The steerable pyramid: a flexible architecture for multi-scale derivative computation. In: Proceedings, International Conference on Image Processing, vol. 3, pp. 444-447. IEEE Computer Society Press (1995)
[27] Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series. Princeton University Press, Princeton (1993)
[28] Suter, D.: Divergence-free wavelets made easy. Technical report, Monash University, Clayton, Australia (1994)
[29] Unser, M.; Chenouard, N., A unifying parametric framework for 2D steerable wavelet transforms, SIAM J. Imaging Sci., 6, 102-135, (2013) · Zbl 1279.68340
[30] Urban, K., On divergence-free wavelets, Adv. Comput. Math., 4, 51-81, (1995) · Zbl 0822.42020
[31] Walter, GG; Cai, L., Periodic wavelets from scratch, J. Comput. Anal. Appl., 1, 25-41, (1999)
[32] Ward, JP; Unser, M., Harmonic singular integrals and steerable wavelets in L2(Rd), Appl. Comput. Harmon. Anal., 36, 183-197, (2014) · Zbl 1347.42059
[33] Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1922) · JFM 48.0412.02
[34] Zhang, X-P; Tian, L-S; Peng, Y-N, From the wavelet series to the discrete wavelet transform—the initialization, IEEE Trans. Signal Process., 44, 129-133, (1996)
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