The lifting scheme: A construction of second generation wavelets

SIAM Journal on Mathematical Analysis

Volumn 29, Issue 2, 1998, Pages 511-546

  Sweldens, Wim ^a  

^a LUCENT TECHNOLOGIES   (United States)

Author keywords

Lifting scheme; Multiresolution; Second generation wavelet; Wavelet

Indexed keywords

EID: 0032388955     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141095289051     Document Type: Article

References (117)

1. R. ABGRALL AND A. HARTEN, Multiresolution representation in unstructured meshes, SIAM J. Numer. Anal., submitted.
2. A. ALDROUBI, M. EDEN, AND M. UNSER, Discrete spline filters for multiresolutions and wavelets of ℓ2, SIAM J. Math. Anal., 25 (1994), pp. 1412-1432.
3. A. ALDROUBI AND M. UNSER, Families of multiresolution and wavelet spaces with optimal properties, Numer. Funct. Anal. Optim., 14 (1993), pp. 417-446.
4. 2 for the sparse representation of integral operators. SIAM J. Math. Anal., 24 (1993), pp. 246-262.
5. B. ALPERT, G. BEYLKIN, R. COIFMAN, AND V. ROKHLIN, Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput., 14 (1993), pp. 159-184.
6. B. K. ALPERT, Wavelets and other bases for fast numerical linear algebra, in [20], pp. 181-216.
7. A. AMELLAOUI AND P. G. LEMARIÉ-RIEUSSET, Ondelettes splines sur grilles irregulieres, manuscript.
8. L. ANDERSSON, N. HALL, B. JAWERTH, AND G. PETERS, Wavelets on closed subsets of the real line, in [101], pp. 1-61.
9. L. ANDERSSON, B. JAWERTH, AND M. MITREA, The Cauchy singular integral operator and Clifford wavelets, in [14], pp. 525-546.
10. P. AUSCHER, Wavelets with boundary conditions on the interval, in [20], pp. 217-236.
11. P. AUSCHER AND PH. TCHAMITCHIAN, Bases d'ondelettes sur les courbes corde-arc, noyau de, Cauchy et espaces de Hardy associés, Rev. Mat. Iberoamericana, 5 (1989), pp. 139-170.
12. P. AUSCHER AND PH. TCHAMITCHIAN, Ondelettes et conjecture de Kato, C. R. Acad. Sci. Paris Sér. I Math. I, 313 (1991), pp. 63-66.
13. G. BATTLE, A block spin construction of ondelettes, Comm. Math. Phys., 110 (1987), pp. 601-615.
14. J. BENEDETTO AND M. FRAZIER, EDS., Wavelets: Mathematics and Applications, CRC Press, Boca Raton, FL, 1993.
15. M. D. BUHMANN AND C. A. MICCHELLI, Spline prewavelets for non-uniform knots, Numer. Math., 61 (1992), pp. 455-474.
16. R. CALDERBANK, I. DAUBECHIES, W. SWELDENS, AND B.-L. YEO, Wavelet transforms that map integers to integers, Appl. Comput. Harm. Anal., to appear.
17. J. M. CARNICER, W. DAHMEN, AND J. M. PEÑA, Local decompositions of refinable spaces, J. Appl. Comput. Harmonic Anal., 3 (1996), pp. 127-153.
18. C. CHUI AND E. QUAK, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory. D. Braess and L. L. Schumaker, eds., Birkhäuser-Verlag, Basel, 1992, pp. 1-24.
19. C. K. CHUI, An Introduction to Wavelets, Academic Press, San Diego, CA, 1992.
20. C. K. CHUI, ED., Wavelets: A Tutorial in Theory and Applications, Academic Press, San Diego, CA, 1992.
21. C. K. CHUI, L. MONTEFUSCO, AND L. PUCCIO, EDS., Conference on Wavelets: Theory, Algorithms, and Applications. Academic Press, San Diego, CA, 1994.
22. C. K. CHUI, J. STÖCKLER, AND J. D. WARD, Compactly Supported Box-Spline Wavelets, Technical report CAT 230, Center for Approximation Theory, Department of Mathematics, Texas A&M University, College Station, TX, 1991.
23. C. K. CHUI AND J. Z. WANG, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113 (1991), pp. 785-793.
24. C. K. CHUI AND J. Z. WANG, A general framework of compactly supported splines and wavelets, J. Approx. Theory, 71 (1992), pp. 263-304.
25. C. K. CHUI AND J. Z. WANG, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc., 330 (1992), pp. 903-915.
26. A. COHEN, Ondelettes, analyses multiresolutions et filtres miroirs en quadrature, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), pp. 439-459.
27. A. COHEN, W. DAHMEN, AND R. DEVORE, Multiscale decompositions on bounded domains, Trans. Amer. Math. Soc., to appear.
28. A. COHEN AND I. DAUBECHIES, Non-separable bidimensional wavelet bases, Rev. Mat. Iberoamericana, 9 (1993), pp. 51-137.
29. A. COHEN, I. DAUBECHIES, AND J. FEAUVEAU, Bi-orthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45 (1992), pp. 485-560.
30. A. COHEN, I. DAUBECHIES, B. JAWERTH, AND P. VIAL, Multiresolution analysis, wavelets and fast algorithms on an interval, C. R. Acad. Sci. Paris Sér. I Math. I, 316 (1993), pp. 417-421.
31. A. COHEN, I. DAUBECHIES, AND P. VIAL, Multiresolution analysis, wavelets and fast, algorithms on an interval, J. Appl. Comput. Harmonic Anal., 1 (1993), pp. 54-81.
32. A. COHEN AND J.-M. SCHLENKER, Compactly supported bidimensional wavelet bases with hexagonal symmetry, Constr. Approx., 9 (1993), pp. 209-236.
33. 2 boundedness of Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc., 2 (1989), pp. 553-564.
34. R. R. COIFMAN, Y. MEYER, S. QUAKE, AND M. V. WICKERHAUSER, Signal processing and compression with wave packets, in Proc. International Conference on Wavelets, Y. Meyer, ed., Marseille, 1989, Masson, Paris, 1992.
35. R. R. COIFMAN, Y. MEYER, AND V. WICKERHAUSER, Size properties of wavelet packets, in [95], pp. 453-470.
36. R. R. COIFMAN AND M. L. WICKERHAUSER, Entropy based algorithms for best basis selection, IEEE Trans. Inform. Theory, 38 (1992), pp. 713-718.
37. D. COLELLA AND C. HEIL, The characterization of continuous four-coefficient scaling functions and wavelets, IEEE Trans. Inform. Theory, 38 (1992), pp. 876-881.
38. D. COLELLA AND C. HEIL, Characterizations of scaling functions: Continuous solutions, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 496-518.
39. S. DAHLKE AND R. DEVORE, Elliptic Boundary Value Problems and Besov Spaces, Preprint, Dept. of Mathematics, University of South Carolina, Columbia, SC, 1996.
40. W. DAHMEN, Decomposition of refinable spaces and applications to operator equations, Numer. Algorithms. 5 (1993), pp. 229-245.
41. W. DAHMEN, Some remarks on multiscale transformations, stability, and biorthogonality, in Wavelets, Images, and Surface Fitting, P. J. Laurent, A. Le Méhauté, and L. L. Schumaker, eds.. A. K. Peters, Wellesley, MA, 1994, pp. 157-188.
42. W. DAHMEN, Stability of Multiscale Transformations, Technical report, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 1994.
43. W. DAHMEN, B. KLEEMANN, S. PRÖSSDORF, AND R. SCHNEIDER, Multiscale methods for the double layer potential equation on a polyhedron, in Advances in Computational Mathematics, H. P. Dikshit and C. A. Micchelli, eds., World Scientific. Singapore, 1994, pp. 15-57.
44. W. DAHMEN AND A. KUNOTH, Multilevel preconditioning, Numer. Math., 63 (1992). pp. 315-344.
45. W. DAHMEN AND C. A. MICCHELLI, Banded matrices with banded inverses II: Locally finite decompositions of spline spaces, Constr. Approx., 9 (1993), pp. 263-281.
46. W. DAHMEN, S. PRÖSSDORF, AND R. SCHNEIDER, Multiscale methods for pseudo-differential equations on smooth manifolds, in [21], 1994, pp. 385-424.
47. I. DAUBECHIES, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41 (1988), pp. 909-996.
48. I. DAUBECHIES, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Series in Appl. Math. 61, SIAM, Philadelphia, PA, 1992.
49. I. DAUBECHIES, Orthonormal bases of compactly supported wavelets II: Variations on a theme, SIAM J. Math. Anal., 24 (1993), pp. 499-519.
50. I. DAUBECHIES AND J. C. LAGARIAS, Two-scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22 (1991), pp. 1388-1410.
51. I. DAUBECHIES AND J. C. LAGARIAS, Two-scale difference, equations II, Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal., 23 (1992), pp. 1031-1079.
52. I. DAUBECHIES AND W. SWELDENS, Factoring wavelet transforms into lifting steps, J. Fourier Anal. Appl., to appear.
53. B. DENG, B. JAWERTH, G. PETERS, AND W. SWELDENS, Wavelet probing for compression based segmentation, in Wavelet Applications in Signal and Image Processing, A. F. Laine, ed., Proc. SPIE 2034, International Society for Optical Engineering, 1993, pp. 266-276.
54. G. DESLAURIERS AND S. DUBUC, Interpolation dyadique, in Fractals, dimensions non entières et applications, Masson, Paris, 1987, pp. 44-55.
55. G. DESLAURIERS AND S. DUBUC, Symmetric iterative interpolation processes, Constr. Approx., 5 (1989), pp. 49-68.
56. D. L. DONOHO, Smooth wavelet decompositions with blacky coefficient kernels, in [101], pp. 259-308.
57. D. L. DONOHO, Interpolating Wavelet Transforms, Department of Statistics, Preprint, Stanford University, Stanford, CA, 1992.
58. D. L. DONOHO, Unconditional bases are optimal bases for data compression and for statistical estimation, J. Appl. Comput. Harmonic Analysis, 1 (1993), pp. 100-115.
59. D. L. DONOHO, On minimum entropy segmentation, Preprint, Department of Statistics, Stanford University, Stanford, CA, 1994.
60. G. C. DOKOVAN, J. S. GERONIMO, AND D. P. HARDIN, A class of orthogonal multiresolution analyses in 2D, in Mathematical Methods for Curves and Surfaces, Daehlen et al., eds., Vanderbilt Univ. Press, Nashville, TN, 1995, pp. 99-110.
61. G. C. DONOVAN, J. S. GERONIMO, AND D. P. HARDIN, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, SIAM J. Math. Anal., 27 (1996), pp. 1791-1815.
62. G. C. DONOVAN, J. S. GERONIMO, D. P. HARDIN, AND P. R. MASSOPUST, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal., 27 (1996), pp. 1158-1192.
63. T. EIROLA, Sobolev characterization of solutions of dilation equations, SIAM J. Math. Anal., 23 (1992), pp. 1015-1030.
64. G. FERNÁNDEZ, S. PERIASWAMY, AND W. SWELDENS, LIFTPACK: A software package for wavelet transforms using lifting, in Wavelet Applications in Signal and Image Processing IV, M. Unser, A. Aldroubi, and A. F. Laine, eels., Proc. SPIE 2825, International Society of Optical Engineering, 1996.
65. M. FRAZIER AND B. JAWERTH, Decora-position of Besov spaces, Indiana Univ. Math. J., 34 (1995), pp. 777-799.
66. M. FRAZIER AND B. JAWERTH, The φ-Transform and Applications to Distribution Spaces, in Function Spaces and Applications, M. Cwikel et al., eds., Lecture Notes in Math. 1302, Springer, New York, 1988, pp. 223-246.
67. M. FRAZIER AND B. JAWERTH, A discrete transform, and decompositions of distribution spaces, J. Funct. Anal., 93 (1990), pp. 34-170.
68. p on general measure spaces. J. Fourier Anal. Appl., 3 (1997), pp. 457-474.
69. S. J. GORTLER, P. SCHRÖDER, M. F. COHEN, AND P. HANRAHAN, Wavelet radiosity, in Computer Graphics (SIGGRAPH '93 Proceedings), ACM SIGGRAPH, New York, 1993. pp. 221-230.
70. D. P. HARDIN, B. KESSLER, AND P. R. MASSOPUST, Multiresolution analyses based on fractal functions, J. Approx. Theory, 71 (1992), pp. 104-120.
71. A. HARTEN, Multiresolution representation of data: A general framework. SIAM J. Numer. Anal., 33 (1996), pp. 1205-1256.
72. P. N. HELLER, Rank M wavelets with N vanishing moments, SIAM J. Matrix. Anal. Appl., 16 (1995), pp. 502-519.
73. C. HERLEY AND M. VETTERLI, Wavelets and recursive filter banks, IEEE Trans. Signal Process, 41 (1993), pp. 2536-2556.
74. n, manuscript.
75. B. JAWERTH AND W. SWELDENS, Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations, in Sixth Copper Mountain Conference on Multigrid Methods, N. D. Melson, T. A. Manteuffel, and S. F. McCormick, eds., NASA Conference Publication 3224, 1993, pp. 259-273.
76. R.-Q. JIA AND C. A. MICCHELLI, Using the refinement equations for the construction of prewavelets II: Powers of two, in Curves and Surfaces, P. J. Laurent, A, Le Méhauté, and L. L. Schumaker, eds., Academic Press, New York, 1991.
77. n. IEEE Trans. Inform. Theory, 38 (1992), pp. 533-555.
78. P.-G. LEMARIÉ, Ondelettes a localisation exponentielle, J. Math. Pures Appl., 67 (1988), pp. 227-236.
79. M. LOUNSBERY, Multiresolution Analysis for Surfaces of Arbitrary Topological Type, Ph.D. thesis, Department of Computer Science, University of Washington, Seattle, WA, 1994.
80. M. LOUNSBERY, T. D. DEROSE, AND J. WARREN, Multiresolution surfaces of arbitrary topological type, ACM Trans. Graphics, 16 (1997), pp. 34-73.
81. M. LUNDBERG AND G. WELLAND, Construction of compact p-wavelets, Constr. Approx., 9 (1993), pp. 347-370.
82. S. MALLAT AND S. ZHONG, Wavelet transform maxima and multiscale edges, in [95], pp. 67-104.
83. S. MALLAT AND S. ZHONG, Characterization of signals from multiscale edges, IEEE Trans. Patt. Anal. Mach. Intell., 14 (1992), pp. 710-732.
84. S. G. MALLAT, Multifreguency channel decompositions of images and wavelet models, IEEE Trans. Acoust. Speech Signal Process., 37 (1989), pp. 2091-2110.
85. 2(R), Trans. Amer. Math. Soc., 315 (1989), pp. 69-87.
86. S. G. MALLAT, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Patt. Anal. Mach. Intell., 11 (1989), pp. 674-693.
87. Y. MEYER, Omlelettes et Opérateurs, I: Ondelettes, II: Opérateurs de Calderön-Zygmund, III: (with R. Coifman), Opérateurs multilinéaires, Hermann, Paris. 1990. English translation of Vol. 1, Wavelets and Operators, is published by Cambridge University Press, 1993.
88. Y. MEYER, Omlelettes et Opérateurs, I: Ondelettes, II: Opérateurs de Calderön-Zygmund, III: (with R. Coifman), Opérateurs multilinéaires, Hermann, Paris. 1990. English translation of Vol. 1, Wavelets and Operators, is published by Cambridge University Press, 1993.
89. Y. MEYER, Ondelettes sur l'intervalle, Rev. Mat. Iberoamericana, 7 (1992), pp. 115-133.
90. C. A. MICCHELLI, Using the refinement equations for the construction of pre-wavelets, Numer. Algorithms. 1 (1991), pp. 75-116.
91. C. A. MICCHELLI, C. RABUT, AND F. I. UTRETAS, Using the refinement equations for the construction of pre-wavelets III: Elliptic splines, Numer. Algorithms. 1 (1991), pp. 331-352.
92. M. MITREA. Singular integrals, Hardy spaces and Clifford wavelets, Lecture Notes in Math 1575, Springer, New York, 1994.
93. S. D. RIEMENSCHNEIDER AND Z. SHEN, Wavelets and pre-wavelets in low dimensions, J. Approx. Theory, 71 (1992), pp. 18-38.
94. S. D. RIEMENSCHNEIDER AND Z. W. SHEN, Box splines, cardinal series and wavelets, in Approximation Theory and Functional Analysis, C. K. Chui, ed., Academic Press, New York, 1991.
95. O. RIOUL, Simple, regularity criteria for subdivision schemes, SIAM J. Math. Anal., 23 (1992), pp. 1544-1576.
96. M. B. RUSKAI, G. BEYLKIN, R. COIFMAN, I. DAUBECHIES, S. MALLAT, Y. MEYER, AND L. RAPHAEL, EDS., Wavelets and Their Applications, Jones and Bartlett, Boston, MA, 1992.
97. R. SCHNEIDER, Multiskalen- und Wavelet-Matrixkornpression: Analysisbasierte Methoden zur effizienten Losing großer vollbesetzter Gleichungssysteme, Habilitationsschrift, TH Darmstadt, 1995
98. P. SCHRÖDER, Wavelet Algorithms for Illumination Computations, Ph.D. thesis, Department of Computer Science, Princeton University, Princeton, NJ, 1994.
99. P. SCHRÖDER, S. J. GORTLER, M. F. COHKN, AND P. HANRAHAN, Wavelet, projections for radiosity, Computer Graphics Forum, 13 (1994), pp 141-152.
100. P. SCHRÖDER AND W. SWELDENS, Spherical wavelets: Efficiently representing functions on the sphere, Computer Graphics Proceedings (SIGGRAPH 95), ACM SIGGRAPH, New York, 1995, pp. 161-172.
101. P. SCHRÖDER AND W. SWELDENS, Spherical wavelets: Texture processing, in Rendering Techniques '95, P. Hanrahan and W. Purgathofer, eds., Springer-Verlag. Wien, New York, 1995.
102. L. L. SCHUMAKER AND G. WEBB, EDS., Recent Advances in Wavelet Analysis, Academic Press, New York, 1993.
103. P. STEFFEN, P. HELLER, R. A. GOPINATH, AND C. S. BURRUS, Theory of regular m-band wavelets, IEEE Trans. Signal Process., 41 (1993), pp. 3497-3511.
104. J. O. STRÖMBERG, A modified Franklin system and higher order spline systems on R" as unconditional bases for Hardy spaces, in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II, Beckner et al., ed., University of Chicago Press, Chicago, IL, 1981, pp. 475-494.
105. W. SWELDENS, Construction and Applications of Wavelets in Numerical Analysis, Ph.D. thesis, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, 1994.
106. W. SWELDENS, The lifting scheme: A custom-design construction of biorthogonal wavelets, J. Appl. Comput. Harmonic Analysis, 3 (1996), pp. 186-200.
107. W. SWELDENS AND P. SCHRÖDER, Building your own wavelets at home, in Wavelets in Computer Graphics, ACM SIGGRAPH Course notes, 1996, pp. 15-87; also available online from http://cm.bell-labs.com/who/wim/papers/papers.html#athome.
108. M. UNSER, A. ALDROUBI, AND M. EDEN, On the asymptotic convergence of B-spline wavelets to Gabor functions, IEEE Trans. Inform. Theory, 38 (1992), pp. 864-872.
109. M. UNSER, A. ALDROUEII, AND M. EDEN, A family of polynomial spline wavelet transforms, Signal Process., 30 (1993), pp. 141-162.
110. P. P. VAIDYANATHAN, T. Q. NGUYEN, Z. DOGANATA, AND T. SARAMAKI, Improved technique for design of perfect reconstruction fir QMF banks with lossless polyphase matrices, IEEE Trans. Acoust. Speech Signal Process., 37 (1989), pp. 1042-1055.
111. M. VETTERLI AND C. HERLEY, Wavelets and filter banks: Theory and design, IEEE Trans. Acoust. Speech Signal Process., 40 (1992), pp. 2207-2232.
112. M. VETTERLI AND J. KOVACEVIC, Wavelets and Subband Coding. Prentice-Hall, Englewood Cliffs, NJ, 1995.
113. L. F. VILLEMOES, Energy moments in time and frequency for two-scale difference equation solutions and wavelets, SIAM J. Math. Anal., 23 (1992), pp. 1119-1543.
114. L. F. VILLEMOES, Wavelet analysis of refinement equations, SIAM J. Math. Anal., 25 (1994), pp. 1433-1460.
115. M. V. WICKERHAUSER, Acoustic signal compression with wavelet packets, in [20], pp. 679-700.
116. M. V. WICKERHAUSER, Adapted Wavelet Analysis from Theory to Software, A. K. Peters, Wellesley, MA, 1994.
117. H. YSERENTANT, On the multi-level splitting of finite element spaces, Numer. Math., 49 (1986), pp. 379-412.