Wavelet

Wavelets are a class of a functions used to localize a given function in both space and scaling. A family of wavelets can be constructed from a function $\psi(x)$, sometimes known as a ``mother wavelet,'' which is confined in a finite interval. ``Daughter wavelets'' $\psi^{a,b}(x)$ are then formed by translation ($b$) and contraction ($a$). Wavelets are especially useful for compressing image data, since a Wavelet Transform has properties which are in some ways superior to a conventional Fourier Transform.

An individual wavelet can be defined by

$$\psi^{a,b}(x)=\vert a\vert^{-1/2}\psi\left({x-b\over a}\right).$$ (1)

Then

$$W_\psi(f)(a,b)={1\over\sqrt{a}} \int_{-\infty}^\infty f(t)\psi\left({t-b\over a}\right)\,dt,$$ (2)

and Calderón's Formula gives

$$f(x)=C_\psi \int_{-\infty}^\infty \int_{-\infty}^\infty \left\langle{f,\psi^{a,b}}\right\rangle{}\psi^{a,b}(x)a^{-2}\,da\,db.$$ (3)

A common type of wavelet is defined using Haar Functions.

See also Fourier Transform, Haar Function, Lemarié's Wavelet, Wavelet Transform

References

Wavelets

Benedetto, J. J. and Frazier, M. (Eds.). Wavelets: Mathematics and Applications. Boca Raton, FL: CRC Press, 1994.

Chui, C. K. An Introduction to Wavelets. San Diego, CA: Academic Press, 1992.

Chui, C. K. (Ed.). Wavelets: A Tutorial in Theory and Applications. San Diego, CA: Academic Press, 1992.

Chui, C. K.; Montefusco, L.; and Puccio, L. (Eds.). Wavelets: Theory, Algorithms, and Applications. San Diego, CA: Academic Press, 1994.

Daubechies, I. Ten Lectures on Wavelets. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992.

Erlebacher, G. H.; Hussaini, M. Y.; and Jameson, L. M. (Eds.). Wavelets: Theory and Applications. New York: Oxford University Press, 1996.

Foufoula-Georgiou, E. and Kumar, P. (Eds.). Wavelets in Geophysics. San Diego, CA: Academic Press, 1994.

Hernández, E. and Weiss, G. A First Course on Wavelets. Boca Raton, FL: CRC Press, 1996.

Hubbard, B. B. The World According to Wavelets: The Story of a Mathematical Technique in the Making. New York: A. K. Peters, 1995.

Jawerth, B. and Sweldens, W. ``An Overview of Wavelet Based Multiresolution Analysis.'' SIAM Rev. 36, 377-412, 1994.

Kaiser, G. A Friendly Guide to Wavelets. Cambridge, MA: Birkhäuser, 1994.

Massopust, P. R. Fractal Functions, Fractal Surfaces, and Wavelets. San Diego, CA: Academic Press, 1994.

Meyer, Y. Wavelets: Algorithms and Applications. Philadelphia, PA: SIAM Press, 1993.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Wavelet Transforms.'' §13.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 584-599, 1992.

Schumaker, L. L. and Webb, G. (Eds.). Recent Advances in Wavelet Analysis. San Diego, CA: Academic Press, 1993.

Stollnitz, E. J.; DeRose, T. D.; and Salesin, D. H. ``Wavelets for Computer Graphics: A Primer, Part 1.'' IEEE Computer Graphics and Appl. 15, No. 3, 76-84, 1995.

Stollnitz, E. J.; DeRose, T. D.; and Salesin, D. H. ``Wavelets for Computer Graphics: A Primer, Part 2.'' IEEE Computer Graphics and Appl. 15, No. 4, 75-85, 1995.

Strang, G. ``Wavelets and Dilation Equations: A Brief Introduction.'' SIAM Rev. 31, 614-627, 1989.

Strang, G. ``Wavelets.'' Amer. Sci. 82, 250-255, 1994.

Taswell, C. Handbook of Wavelet Transform Algorithms. Boston, MA: Birkhäuser, 1996.

Teolis, A. Computational Signal Processing with Wavelets. Boston, MA: Birkhäuser, 1997.

Walter, G. G. Wavelets and Other Orthogonal Systems with Applications. Boca Raton, FL: CRC Press, 1994.

``Wavelet Digest.'' http://www.math.sc.edu/~wavelet/.

Wickerhauser, M. V. Adapted Wavelet Analysis from Theory to Software. Wellesley, MA: Peters, 1994.