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小波

小波是一類函式，用於在空間和尺度上定位給定函式。一族小波可以從一個函式 psi(x) 構建，有時稱為“母小波”，它被限制在有限區間內。然後透過平移 (b) 和伸縮 (a) 形成“子小波” psi^(a,b)(x)。小波特別適用於壓縮影像資料，因為小波變換在某些方面優於傳統的傅立葉變換。

一個單獨的小波可以定義為

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

那麼

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

且 Calderón 公式給出

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

一種常見的小波型別是使用 Haar 函式定義的。

電視劇犯罪劇集 數字追兇 第一季劇集 “Counterfeit Reality” (2005) 以小波為特色。

另請參閱

傅立葉變換, Haar 函式, Lemarié 小波, 小波變換

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參考文獻

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.

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小波

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韋斯坦因，埃裡克·W. "小波。" 來自 Web 資源。 https://mathworld.tw/Wavelet.html

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