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Popular Wavelet Families and Filters and Their Use

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Encyclopedia of Complexity and Systems Science

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Introduction

Wavelets are one or a few functions whose integertranslations and dilations can generate a basis fora Hilbert space. The concept of wavelets was introduced inthe 1980's and has since been generalized and extended in manydirections. The theory and applications have been continuouslydeveloped. One of its significant features is that it providesa systematical approach for designing various filters andfilter banks for signal and image processing . Another feature isthat wavelets leads to the theory of multi-resolutionapproximation (MRA). Wavelets and MRA have found manyapplications in most areas of science and technology,e.?g., astronomy, electric engineering, fuzzy logic,geoscience, medical imaging, physics, and statistics. Waveletshave become an important subject in applied mathematics,approximation theory, numerical analysis and harmonicanalysis.

In this article we present only the discrete wavelettransform, omitting the discussion of continuous wavelettransform. We shall...

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Abbreviations

B-splines:

N d is the uniform B-spline of order d based on integer knot sequence. It is a function of piecewise polynomial of degree \( { d-1 } \) and smoothness \( { d-2 } \). Trigonometric B-splines T d will also be used.

Box splines:

\( { B_{\ell,m,n} } \) is a box spline of degree \( \ell+m+n-2 \) on three direction mesh. \( { B_{k,\ell,m,n} } \) is a box spline of degree \( { k+\ell+m+n-2 } \) on four direction mesh. They are bivariate piecewise polynomial functions of certain smoothness dependent on integers \( { k, \ell, m, n } \).

Filter:

A filter is a sequence of real numbers. For example, a FIR filter is a finite sequence of real numbers. An IIR filter is a sequence of real numbers whose discrete Fourier transform is a rational function in \( { z=\text{e}^{i\omega} } \).

Filter process:

A filter process is to convolute a digital signal with a filter, converting an input digital signal to an output digital signal. A subband coding scheme is a synthetic filter process which convert an input signal to several output signals.

Image compression:

A procedure to use less bytes of information to represent the same image (within tolerance). That is, for an image of size \( { 512\times 512 } \) and standard integer gray level [0, 255], the image needs a file of \( { 512\times 512 \times 8 } \) bytes to store in a computer or to be sent over the internet. If one can use a file of some bytes less than \( { 512\times 512 \times 8 } \) to represent this image (storage or transmission), then the file is a compressed image.

Image denoise:

A procedure to remove noises from a noised image to make the image sharper and clearer.

Image edge detection:

A procedure to find features, skeletonor segmentation of images.

L 2 spaces:

A space of all square integrable functions.

Mask:

A mask is a finite sequence of real numbers. A mask polynomial is the discrete Fourier transform of a mask. Sometimes, a mask polynomial is also called the symbol of a mask.

MRA:

MRA stands for multi-resolution approximation (of a L 2 space).

Wavelet:

A function or a group of functions which can generate a basis for Hilbert space by its translations and dilations is called wavelet. Many generalized versions of wavelets will be discussed including orthonormal wavelets, biorthogonal wavelets , prewavelets, wavelet frames, multi-wavelets, q-dilated wavelets, multivariate nonseparable wavelets.

Bibliography

Primary Literature

  1. Ayache A (1999) Construction of non-separable dyadic compactly supported orthonormal wavelet bases for \( L\sp 2({R\sp 2) } \) of arbitrarily high regularity. Rev Mat Iberoamericana 15:37–58

    MathSciNet  MATH  Google Scholar 

  2. Ayache A (2001) Some methods for constructing nonseparable, orthonormal, compactly supported wavelet bases. Appl Comput Harmon Anal 10(1):99–111

    MathSciNet  MATH  Google Scholar 

  3. Bastin F, Boigelot C (1998) Biorthogonal wavelets in \( { H^m (\mathbb{R}) } \). J Fourier Anal Appl 4:749–768

    MathSciNet  MATH  Google Scholar 

  4. Bastin F, Laubin P (1997) Regular compactly supported wavelets in Sobolev spaces. Duke Math J 87:481–508

    MathSciNet  MATH  Google Scholar 

  5. Belogay E, Wang Y (1999) Arbitrarily smooth orthogonal non-separable wavelets in \( R^2 \). SIAM J Math Anal 30:678–697

    MathSciNet  MATH  Google Scholar 

  6. de Boor C (1978) A practical guide to splines. Springer, New York

    MATH  Google Scholar 

  7. de Boor C, Hölig K, Riemenschneider S (1993) Box splines. Springer, New York

    Google Scholar 

  8. Chen G, Chui CK (1992) Signal processing and system theory. Springer, New York

    Google Scholar 

  9. Cho O, Lai MJ (2006) A class of compactly supported orthonormal B-Spline wavelets. In: Chen G, Lai MJ (eds) Wavelets and Splines. Nashboro Press, Brentwood, pp 123–151

    Google Scholar 

  10. Chrina M, Stöckler J (2007) Tight wavelet frames for irregular multiresolution analysis. Appl Comput Harmonic Anal (accepted for publication)

    Google Scholar 

  11. Chui CK (1988) Multivariate splines. SIAM Publications, Philadelphia

    Google Scholar 

  12. Chui CK (1992) An introduction to wavelets. Academic Press, San Diego

    MATH  Google Scholar 

  13. Chui CK (1997) Wavelets: A Mathematical Tool for Signal Analysis. SIAM Publication, Philedalphia

    MATH  Google Scholar 

  14. Chui CK, He W (2000) Compactly supported tight frames associated with refinable functions. Appl Comp Harmonic Anal 8:293–319

    MathSciNet  MATH  Google Scholar 

  15. Chui CK, He W (2001) Construction of multivariate tight frames via Kronecker products. Appl Comp Harmonic Anal 11:305–312

    MathSciNet  MATH  Google Scholar 

  16. Chui CK, Lian JA (1996) A study of orthonormal multi-wavelets. Appl Numer Math 20:273–298

    MathSciNet  MATH  Google Scholar 

  17. Chui CK, Shi XL (1993) Bessel sequences and affine frames. Appl Comp Harmonic Anal 1:29–49

    MathSciNet  MATH  Google Scholar 

  18. Chui CK, Wang JZ (1992) On compactly supported spline-wavelets and a duality principle. Trans Amer Math Soc 330:903–915

    MathSciNet  MATH  Google Scholar 

  19. Chui CK, Stöckler J, Ward JD (1992) On compactly supported box-spline wavelets. Approx Theory Appl 8:77–100

    Google Scholar 

  20. Chui CK, He W, Stöckler J (2002) Compactly supported tight and sibling frames with maximum vanishing moments. Appl Comp Harmonic Anal 13:224–262

    Google Scholar 

  21. Cohen A, Daubechies I (1993) Nonseparable dimensional wavelet bases. Revista Math Ibereamericana 9:51–137

    MathSciNet  MATH  Google Scholar 

  22. Cohen A, Daubechies I, Feauveau J-C (1992) Biorthogonal bases of compactly supported wavelets. Commun Pure Appl Math XLV:485–560

    MathSciNet  Google Scholar 

  23. Daubechies I (1988) Orthonormal bases of compactly supported wavelets. Comm Pure Appl Math 41:909–996

    MathSciNet  MATH  Google Scholar 

  24. Daubechies I (1992) Ten lectures on wavelets. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  25. Daubechies I, Han B, Ron A, Shen ZW (2003) Framelets: MRA-based constructions of wavelet frames. Appl Comp Harmonic Anal 14:1–46

    MathSciNet  MATH  Google Scholar 

  26. Dierckx P (1986) An algorithm for fitting data over a circle using tensor product splines. J Comp Appl Math 15:161–173

    MATH  Google Scholar 

  27. Donoho D (1995) De-noising by soft-thresholding. IEEE Trans Inform Theory 41:613–627

    MathSciNet  MATH  Google Scholar 

  28. Donovan GC, Geronimo JS, Hardin DP, Massopust PR (1996) Construction of orthogonal wavelets using fractal interpolation functions. SIAM J Math Anal 27:1158–1192

    MathSciNet  MATH  Google Scholar 

  29. Donovan GC, Geronimo JS, Hardin DP (1996) Interwining multiresolution analyses and the construction of piecewise polynomial wavelets. SIAM J Math Anal 27:1791–1815

    MathSciNet  MATH  Google Scholar 

  30. Geronimo J, Lai MJ (2006) Factorization of multivariate positive Laurent polynomials. J Approx Theory 139:327–345

    MathSciNet  MATH  Google Scholar 

  31. Goodman T (2003) A class of orthonormal refinable functions and wavelets. Constr Approx 19:525–540

    MathSciNet  MATH  Google Scholar 

  32. Goodman T, Micchelli CA (1994) Orthonormal cardinal functions. In: Chui CK et al (eds) Wavelets: Theory, algorithms and applications. Academic, San Diego, pp 53–88

    Google Scholar 

  33. Han B (2007) Construction of wavelets and framelets by the projection method. Int J Math Sci, to appear

    Google Scholar 

  34. Hardin D, Hogan A, Sun Q (2004) The matrix-valued Riesz lemma and local orthonormal bases in shift-invariant spaces. Adv Comput Math 20:367–384

    MathSciNet  MATH  Google Scholar 

  35. He W, Lai MJ (1997) Examples of bivariate non-separable continuous compactly supported orthonormal wavelets. Proc SPIE 3169:303–314. (It also appears in IEEE Trans Image Proces 9:949–953 (2000)

    ADS  Google Scholar 

  36. He W, Lai MJ (1999) Construction of bivariate compactly supported biorthogonal box spline wavelets with arbitrarily high regularities. J Appl Comput Harmonic Anal 6:53–74

    MathSciNet  MATH  Google Scholar 

  37. He W, Lai MJ (2003) Construction of trivariate compactly supported biorthogonal box wavelets. J Approx Theor 120:1–19

    MathSciNet  MATH  Google Scholar 

  38. Hong J, Lai MJ (2007) New constructions of orthonormal multiwavelets. Manuscript

    Google Scholar 

  39. Jia RQ, Micchelli CA (1992) Using the refinement equation for the construction of pre-wavelets, V. Extensibility of trigonometric polynomials. Comput 48:61–72

    MathSciNet  MATH  Google Scholar 

  40. Jia RQ, Shen ZW (1994) Multiresolution and wavelets. Proc Edinburgh Math Soc 37:271–300

    MathSciNet  MATH  Google Scholar 

  41. Jia R-Q, Wang J, Zhou D-X (2003) Compactly supported wavelet bases for Sobolev spaces. Appl Comput Harmon Anal 15(3):224–241

    MathSciNet  MATH  Google Scholar 

  42. Johnstone I, Donoho D (1994) Ideal spatial adaptation via wavelet shrinkage. Biometrika 81:425–455

    MathSciNet  MATH  Google Scholar 

  43. Karoui A (2003) A note on the construction of nonseparable wavelet bases and multiwavelet matrix filters of \( { L_2(\mathbb{R}^n) } \), where \( { n\geq 2 } \). Electron Res Announc Amer Math Soc 9:32–39

    MathSciNet  MATH  Google Scholar 

  44. Kotyczka U, Oswald P (1995) Piecewise linear prewavelets of small support. In: Chui CK, Schumaker LL (eds) Approximation Theory VIII, vol 2. World Scientific, Singapore, pp 235–242

    Google Scholar 

  45. Kovacevic J, Vetterli M (1992) Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for \( { (R)^n } \). IEEE Trans Info Theory 38:533–555

    Google Scholar 

  46. Kovacevic J, Vetterli M (1995) Nonseparable two- and three- dimensional wavelets. IEEE Trans Signal Proc 43:1260–1273

    Google Scholar 

  47. Lai M-J (1992) Fortran subroutines for B-nets of box splines on three and four directional meshes. Numerical Algo 2:33–38

    ADS  MATH  Google Scholar 

  48. Lai M-J (2002) Methods for constructing nonseparable compactly supported orthonormal wavelets. In: Zhou DX (ed) Wavelet analysis: Twenty year's development. World Scientific, Singapore, pp 231–251

    Google Scholar 

  49. Lai M-J (2006) Construction of multivariate compactly supported orthonormal wavelets. Adv Comput Math 25:41–56

    MathSciNet  MATH  Google Scholar 

  50. Lai M-J, Lian J-A (2007) A private communication on CDF9/7, May 2007

    Google Scholar 

  51. Lai M-J, Lian J-A (2007) Construction of tight multi-wavelet frames (under preparation)

    Google Scholar 

  52. Lai M-J, Liu HP (2007) Prewavelet solution to Poisson equation. Manuscript

    Google Scholar 

  53. Lai M-J, Lyche T (2007) Tight wavelets frames using trigonometric B-splines. Manuscript

    Google Scholar 

  54. Lai M-J, Nam K (2006) Tight wavelet frames over bounded domains. In: Chen G, Lai M-J (eds) Wavelets and splines. Nashboro Press, Athens, pp 313–326

    Google Scholar 

  55. Lai M-J, Petukhov A (2007) Method of virtual components for constructing redundant filter banks and wavelet frames. Appl Comput Harmonic Anal 22:304–318

    MathSciNet  MATH  Google Scholar 

  56. Lai M-J, Petukhov A (2007) Method of virtual components in the multivariate setting. (submitted)

    Google Scholar 

  57. Lai M-J, Roach DW (1999) Nonseparable symmetric wavelets with short support. In: Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing VII, vol 3813, pp 132–146

    Google Scholar 

  58. Lai M-J, Roach DW (2001) Construction of bivariate symmetric orthonormal wavelets with short support. In: Kopotun K, Lyche T, Neamtu M (eds) Trends in approximation theory. Vanderbilt University Press, Nashville, pp 213–223

    Google Scholar 

  59. Lai M-J, Roach D (2002) Parameterizations of univariate orthogonal wavelets with short support. In: Chui CK, Schumaker LL, Stoeckler J (eds) Approximation theory X: Wavelets, splines, and applications. Vanderbilt University Press, Nashville, pp 369–384

    Google Scholar 

  60. Lai M-J, Schumaker LL (2007) Spline functions over triangulations. Cambridge University Press, Cambridge

    Google Scholar 

  61. Lai M-J, Stoeckler J (2006) Construction of multivariate compactly supported tight wavelet frames. Appl Comput Harmonic Anal 21:324–348

    MATH  Google Scholar 

  62. Lawton W, Lee SL, Shen ZW (1996) An algorithm for matrix extension and wavelet construction. Math Comp 65:723–737

    MathSciNet  ADS  MATH  Google Scholar 

  63. Lebrun J, Vetterli M (1998) Balanced multiwavelets: Theory and design. IEEE Trans Signal Process 46:1119–1125

    MathSciNet  ADS  Google Scholar 

  64. Lyche T (1999) Trigonometric splines: A survey with new results. In: Pena J (ed) Shape preserving representations in computer-aided geometric design. Noa Science Publishers, New York

    Google Scholar 

  65. Lyche T, Schumaker LL (1994) L-spline wavelets. In: Chui C, Montefusco L, Puccio L (eds) Wavelets: Theory, algorithms, and applications. Academic Press, New York, pp 197–212

    Google Scholar 

  66. Lyche T, Schumaker LL (2000) A multiresolution tensor spline method for fitting functions on the sphere. SIAM J Sci Comput 22:724–746

    MathSciNet  MATH  Google Scholar 

  67. Lyche T, Winther R (1979) A stable recurrence relation for trigonometric B-splines. J Approx Theory 3:266–279

    MathSciNet  Google Scholar 

  68. Lyche T, Schumaker LL, Stanley S (1998) Quasi-interpolants based on trigonometric splines. J Approx Theory 95:280–309

    MathSciNet  MATH  Google Scholar 

  69. Maass P (1997) Families of orthogonal two-dimensional wavelets. SIAM J Math Anal 27:1454–1481

    MathSciNet  Google Scholar 

  70. Mallat S (1989) Multi-resolution approximations and wavelet orthonormal bases of \( { L_2(\mathbb{R}) } \). Trans Amer Math Soc 315:69–87

    MathSciNet  MATH  Google Scholar 

  71. Nam K (2005) Box spline tight frames and their applications for image processing, Ph.D. Dissertation. University of Georgia, Athens

    Google Scholar 

  72. Petukhov A (2001) Explicit construction of framelets. Appl Comp Harmonic Anal 11:313–327

    MathSciNet  MATH  Google Scholar 

  73. Petukhov A (2003) Symmetric framelets. Constr Approx 19:309–328

    MathSciNet  MATH  Google Scholar 

  74. Riemenschneider S, Shen Z (1991) Box splines, cardinal series, and wavelets. In: Chui CK (ed) Approximation theory and functional analysis. Academic, Boston, pp 133–149

    Google Scholar 

  75. Riemenschneider S, Shen Z (1992) Wavelets and pre-wavelets in low dimensions. J Approx Theory 71:18–38

    MathSciNet  MATH  Google Scholar 

  76. Ron A, Shen ZW (1997) Affine systems in \( { L_2(\mathbb{R}^d) } \): The analysis of the analysis operator. J Func Anal 148:408–447

    MathSciNet  MATH  Google Scholar 

  77. Ron A, Shen ZW (1997) Affine system in \( { L_2(\mathbb{R}^d) } \), II. Dual systems. J Fourier Anal Appl 3:617–637

    MathSciNet  Google Scholar 

  78. Ron A, Shen ZW (1998) Compactly supported tight affine spline frames in \( { L_2(R^d) } \). Math Comp 67:191–207

    MathSciNet  ADS  MATH  Google Scholar 

  79. Ron A, Shen ZW (1998) Construction of compactly supported affine frames in \( { L_2(R^d) } \). In: Lau KS (ed) Advances in Wavelets. Springer, New York, pp 27–49

    Google Scholar 

  80. Rudin W (1963) The existence problem for positive definite functions. Illinois J Math 7:532–539

    MathSciNet  MATH  Google Scholar 

  81. Said A, Pearlman WA (1996) A new fast and efficient image codec based on set partitioning in hierarchical trees. IEEE Trans Circ Syst Video Technol 6:243–250

    Google Scholar 

  82. Schoenberg IJ (1964) On trigonometric spline interpolation. J Math Mech 13:795–825

    MathSciNet  MATH  Google Scholar 

  83. Schoenberg IJ (1973) Cardinal spline interpolation. SIAM Publication, Philadelphia

    MATH  Google Scholar 

  84. Schumaker LL (2007) Spline functions, basic theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  85. Schumaker LL, Traas C (1991) Fitting scattered data on sphere like surface using tensor products of trigonometric and polynomial splines. Num Math 60:133–144

    MathSciNet  MATH  Google Scholar 

  86. Simoncelli EP, Adelson EH (1990) Non-separable extensions of quadrature mirror filters to multiple dimensions. Proc IEEE 78:652–664

    Google Scholar 

  87. Stanhill D, Zeevi YY (1996) Two dimensional orthogonal wavelets with vanishing moments. IEEE Trans Signal Proces 46:2579–2590

    ADS  Google Scholar 

  88. Stanhill D, Zeevi YY (1998) Two dimensional orthogonal filter banks and wavelets with linear phase. IEEE Trans Signal Proces 46:183–190

    ADS  Google Scholar 

  89. Strang G, Fix G (1973) A Fourier analysis of the finite element variational method. In: Geymonat G (ed) Constructive aspects of functional analysis, C.I.M.E. II Ciclo 1971; 793–840

    Google Scholar 

  90. Vetterli M (1984) Multidimensional subband coding: some theory and algorithms. Signal Proces 6:97–112

    MathSciNet  Google Scholar 

  91. Wickerhauser MV (1994) Adapted wavelet analysis from theory to software. AK Peters Ltd, Wellesley

    MATH  Google Scholar 

  92. Zhou J (2006) Construction of orthonormal wavelets of dilation factor 3 with application in image compression. Ph.D. dissertation, University of Georgia, Athens

    Google Scholar 

Books and Reviews

  1. Antoine JP, Murenzi R, Vandergheynst P, Ali ST (2004) Two-dimensional wavelets and their relatives. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  2. Chen G, Lai M-J (2006) Wavelets and Splines: Athens, 2005. Nashboro Press, Brentwood

    MATH  Google Scholar 

  3. Cohen A (2003) Numerical analysis of wavelet methods. North-Holland Publishing Co, Amsterdam

    MATH  Google Scholar 

  4. Hernández E, Weiss G (1996) A first course on wavelets, with a foreword by Yves Meyer. CRC Press, Boca Raton

    Google Scholar 

  5. Mallat S (1998) Wavelet tour of signal processing. Academic Press, San Diego

    MATH  Google Scholar 

  6. MathWorks, Inc (2007) MATLAB wavelet toolbox

    Google Scholar 

  7. Meyer Y, Coifman R (1997) Wavelets, Calderon-Zygmund and multilinear operators. Hermann, Paris, 1990 and Cambridge University University Press, Cambridge

    MATH  Google Scholar 

  8. Strang G, Nguyen T (1996) Wavelets and filter banks. Wellesley-Cambridge Press, Wellesley

    MATH  Google Scholar 

  9. Vetterli M and Kovacevic J (1995) Wavelets and subband coding. Prentice Hall, New Jersey

    Google Scholar 

  10. Vidakovic B (1999) Statistical modeling by wavelets. A Wiley-Interscience Publication. Wiley, New York

    Google Scholar 

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  1. Department of Mathematics, The University of Georgia, Athens, USA

    Ming-Jun Lai

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  1. Ming-Jun Lai
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Lai, MJ. (2009). Popular Wavelet Families and Filters and Their Use. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_411

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