Wavelet Transform: Theory and Aplucations

Theory

• Part A: Introduction to method
• Part B: Theory of Wavelet Transform
• Part C: Wavelets in the INTERNET

Astrophysical applications

• Part D: Line profile variations
• Part E: Analysis of Time Series
• Part F: Images Analysis
• Part G: Using the Wavelet for analysis of internal Structure of the astrophysical objects + N-Dimensions

Part A: INTRODUCTION TO METHOD: Books and Review papers

1. Daubechies, Ingrid, Ten lectures on wavelets, University of Lowell, Philadelphia: Society for Industrial and Applied Mathematics (SIAM) (1992)

Comment: the best and rather simple description of method (There is a russian translatio, M-Izhevsk, 2001)

2. Chui, Charles K., An Introduction to wavelets, Academic Press, New York, Boston, Philadelphia: Society for Industrial and Applied Mathematics (SIAM), (1992)

3. Meyer Y., Wavelets and Operators, Cambridge, Cambridge University press (1992)

4. Meyer Y., Wavelets: Algorithms and Applications, Philadelphia, SIAM (1993)

5. Farge M., Hunt J.R.C., Vassilivos J.C., Wavelets, Fractals and Fourier Transform, Oxford, Oxford University press (1995)

6. Combes, Jean-Michel; Grossmann, Alexander, Tchamitchian, Philippe, Wavelets. Time-Frequency Methods and Phase Space, Proceedings of the International Conference, Marseille, France, December 14-18, 1987, IX, 315 pp. 88 figs.. Springer-Verlag Berlin Heidelberg New York.

7. Grossman A., Kroland-Martinet R., Morlet J., 1989, Wavelets, (eds) J.M.Combes, A.Grossman, P.Thamithian, pp. 2-20, Springer

8. Chui, Charles K., Wavelets: A tutorial in theory and applications, Wavelet Analysis and its Applications, San Diego, CA: Academic Press, ed. by Chui, Charles K. (1992)

9. Ruskai, Mary B.; Beylkin, Gregory; Coifman, Ronald, Wavelets and their applications, Jones and Bartlett Books in Mathematics, Boston: Jones and Bartlett, 1992, edited by Ruskai, Mary B.; Beylkin, Gregory; Coifman, Ronald

10. Erlebacher G., Hussaini M.Y., Jameson, L.M., Wavelets, Theory and Applications, Ed. by Gordon Erlebacher, M. Yousuff Hussaini, and Leland M. Jameson. Oxford University Press (New York), (1996)

PART B: Theory of Wavelet Transform

Comment: Most papers of the list are rather hard for understanding for astronomers!

1. Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41, 1988, pp. 909-996.

2. Meyer Y., Wavelets and Operators, in Analysis at Urbana, Vol. 1, E. Berkson, N. T. Peck, and J. Uhl, eds., London Math. Soc., Lecture Notes Series 137, 1989, pp. 256-365.

3. Beylkin, G., Coifman, R., Rokhlin, V., Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44, 1991, pp. 141-183.

4. David G., Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Math., V. 1465, 1991.

Part C: Wavelets in the INTERNET

1. J. Lewalle, Tutorial on continuous Wavelet Analysis of Experimental data

2. R. Polikar, The Wavelet Tutorial

3. Amara Graps, An Introduction to Wavelets

4. Swiss Federal Institute of Technology, Everything you ever wanted to know about Wavelets

5. Swiss Federal Institute of Technology, Links to the most essential resources related to wavelets

6. Christopher Torrence, Gilbert P. Compo, A Practical Guide to Wavelet Analysis

7. Wikipedia, the free encyclopedia Wavelet

8. Andreas Uhl, Page on the book The World According to Wavelets

9. Edward Aboufadel and Steven Schlicker (Grand valley State University) Discovering Wavelets: Home Page

10. C. Valens, A Really Friendly Guide To Wavelets

Part D: Line profile variations

1. R.H.D.Townsend, MN, 310, 851-862 (1999), Spatial wavelet analysis of line-profile variations

e-Mail: rhdt@star.ucl.ac.uk

2. Kiss, L. L., SzatmАry K., Szabe G., Mattei J.A., Multiperiodicity in semiregular variables. II. Systematic amplitude variations, A&AS, 145, 283, (2000)

3. Meiksin A., Spectral analysis of the Lyalpha; forest using wavelets MNRAS, 314, 566 (2000)

4. Gill, A. G.; Henriksen, R. N., A first use of wavelet analysis for molecular clouds, Astrophysical Journal, vol. 365, Dec. 10, p. L27-L30. (1990)

Comment: Using WL for investigation of fractal structure

5. Lepine, Sebastien, Wavelet analysis of Wolf-Rayet emission line variability: Evidence for clumping, Astrophysics and Space Science vol. 221, no. 1-2, p. 371-382 (1994)

6. Moffat, A. F. J.; Lepine, S.; Henriksen, R. N.; Robert, C., First wavelet analysis of emission line variations in Wolf-Rayet stars, Astrophysics and Space Science, vol. 216, no. 1-2, p. 55-65 (1994)

Comment (3-4): Analysis of line profiles. Extracting of details from line profiles

7. Lepine, S., Moffat, A.F.J., Henriksen, R.N., Wind Inhomogeneities in Wolf-Rayet Stars. I. Search for Scaling Laws Using Wavelet Transforms, ApJ, 466, p.392 (1996)

8. Lepine, S., Moffat, A.F.J., Wind Inhomogeneities in Wolf-Rayet Stars. II. Investigation of Emission-Line Profile Variations, The Astrophysical Journal, V.514, pp. 909-931. (1999)

Comment (5-6): Analysis of line profiles. Using of Wavelet Power Spectra to study the global features of the profiles.

9. Townsend, R. H. D., Spatial wavelet analysis of line-profile variations, Monthly Notices R.A.S., V. 310, 851-862. (1999)

Comment: The technique of wavelet analysis in the context of line-profile variations in rapidly rotating stars undergoing non-radial pulsation.

Part E: Analysis of Time Series

1. Goupil, M. J.; Auvergne, M.; Baglin, A., Wavelet analysis of pulsating white dwarfs, Astr.Astroph. v. 250, 89-98 (1991)

Comment: Parts of light curves of variable white dwarfs ate investigated by means of a wavelet analysis. This time-frequency analysis decomposes the light curves into their different oscillating components. In addition to an oscillation of large amplitude, small amplitude oscillations are thereby clearly emphasized for both stars.

2. Bocchialini, K.; Baudin, F., Wavelet analysis of chromospheric solar oscillations, Astr. Astroph., v.299, p.893 (1995)

3. Frick, P.; Baliunas, S. L.; Galyagin, D., Sokoloff, D.; Soon, W., Wavelet Analysis of Stellar Chromospheric Activity Variations, ApJ, 483, p.426 (1997)

Comment: The quasi-periodic nature of chromospheric activity variations is not completely compatible with the standard Fourier analysis, so a wavelet analysis is applied to study the nature of regularities in the data. A modified wavelet technique that is suitable for analysis of data with gaps is pesented.

4. Soon W., Frick P., Baliunas S., Lifetime of Surface Features and Stellar Rotation: A Wavelet Time-Frequency Approach, The Astroph. J., 510, L135-L138 (1999)

5. Fligge M., Solanki S. K., Noise reduction in astronomical spectra using wavelet packets, A & A Suppl., 124, 579-587 (1997)

6. Starck, Jean-Luc; Siebenmorgen, Ralf; Gredel, Roland, Spectral Analysis Using the Wavelet Transform, ApJ, 482, p.1011 (1997)

Comment: A new signal processing technique to analyze noisy spectra. The method is based on the wavelet transform and employs the a grave trous algorithm. Noise determination and detection criteria are discussed, together with pitfalls related to the use of wavelets in the analysis of spectra.

Part F: IMAGES ANALYSIS

1. Press, William H., Wavelet-Based Compression Software for FITS Images, Astronomical Data Analysis Software and Systems I, A.S.P. Conference Series, 25, 1992, Diana M. Worrall, Chris Biemesderfer, and Jeannette Barnes, eds., p. 3. (1992)

2. Nunez, J.; Otazu, X., Multiresolution image reconstruction using wavelets, Vistas in Astronomy, 40, 555-562 (1996)

3. 4. Pantin, E.; Starck, J.-L., Deconvolution of astronomical images using the multiscale maximum entropy method, A&A. Suppl.Ser., 118, p.575-585 (1996)

PART G: Using the Wavelet for analysis of internal Structure of the astrophysical objects