Geodesic
Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 1, pp. 77-82 
E. B. Postnikov. Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 1, pp. 77-82. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_1_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that the problem of evaluating the continuous Morlet wavelet transform can be stated as the Cauchy problem for a system of two partial differential equations. The initial conditions for the desired functions, i.e., for the real and imaginary parts of the wavelet transform, are the analyzed function and a vanishing function, respectively. Numerical examples are given.

[1] Koronovskii A. A., Khramov A. E., Nepreryvnyi veivlet-analiz i ego prilozheniya, Fizmatlit, M., 2003

[2] A. Poularikas (ed.), The transforms and applications handbook, CRC Press; IEEE Press, 2000 | MR

[3] Barnsley M., Fractals everywere, Acad. Press, Boston, 1993 | MR | Zbl

[4] Alvarez L., Guichard F., Lions P.-L., Morel J.-M., “Axioms and fundamental equations of image processing”, Arch. Ration. Mech. and Analys., 123 (1993), 199–257 | DOI | MR | Zbl

[5] “Spacial issue on partial differential equations and geometry-driven diffusion in image processing and analysis”, IEEE Trans. Image Proc., 7:3 (1998), 269–473 | DOI

[6] Mallat S., Zhong S., “Characterization of signals from multiscale edges”, IEEE Trans. Patt. Analys and Mach. Internat., 14:7 (1992), 710–732 | DOI | MR

[7] Mallat S., Hwang W. L., “Singularity detection and processing with wavelets”, IEEE Trans. Inform. Theory, 38:2 (1992), 617–643 | DOI | MR | Zbl

[8] Kestener P., Arneodo A., “Three-dimensional wavelet-based multifractal method: The need for revising the multifractal description of turbulence dissipation data”, Phys. Rev. Letts, 91 (2003), 494–501 | DOI

[9] Dobeshi I., Desyat lektsii o veivletakh, Regulyarnaya i khaoticheskaya dinamika, M., Izhevsk, 2001

[10] Haase M., Widjajakusuma J., Bader R., “Scaling laws and frequency decomposition from wavelet transform maxima lines and ridges”, Emergent Nature, World Scient., Singapore, 2001, 365–374

[11] Zeldovich Ya. B., Myshkis A. D., Elementy matematicheskoi fiziki, Nauka, M., 1973 | MR

[12] Skeel R. D., Berzins M., “A method for the spatial discretization of parabolic equations in one space variable”, SIAM J. Scand. and Stat. Comput., 11 (1990), 1–32 | DOI | MR | Zbl