TOPICS
Search

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)=|a|^(-1/2)psi((x-b)/a).
(1)

Then

W_psi(f)(a,b)=1/(sqrt(a))int_(-infty)^inftyf(t)psi((t-b)/a)dt,
(2)

and Calderón's formula gives

f(x)=C_psiint_(-infty)^inftyint_(-infty)^infty<f,psi^(a,b)>psi^(a,b)(x)a^(-2)dadb.
(3)

A common type of wavelet is defined using Haar functions.

The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features wavelets.

See also

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

Explore with Wolfram|Alpha

References

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, 2nd rev. upd. ed. New York: A K Peters, 1998.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.Resnikoff, H. L. and Wells, R. O. J. Wavelet Analysis: The Scalable Structure of Information. New York: Springer-Verlag, 1998.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.Vidakovic, B. Statistical Modeling by Wavelets. New York: Wiley, 1999.Walker, J. S. A Primer on Wavelets and their Scientific Applications. Boca Raton, FL: CRC Press, 1999.Walter, G. G. Wavelets and Other Orthogonal Systems with Applications. Boca Raton, FL: CRC Press, 1994.Weisstein, E. W. "Books about Wavelets." http://www.ericweisstein.com/encyclopedias/books/Wavelets.html.Wickerhauser, M. V. Adapted Wavelet Analysis from Theory to Software. Wellesley, MA: Peters, 1994.

Referenced on Wolfram|Alpha

Wavelet

Cite this as:

Weisstein, Eric W. "Wavelet." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Wavelet.html

Subject classifications