The wavelet transform: A surfing guide

Wavelets in Medicine and Biology

Volumn , Issue , 2017, Pages 3-36

  Aldroubi, Akram ^a  

^a NIH BEIP   (United States)

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EID: 85130295828     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (94)

1. P. Abry and A. Aldroubi. Designing multiresolution analysis-type wavelets and their fast algorithms. J. Fourier Anal. Appl., to appear.
2. A. Aldroubi. Oblique projections in atomic spaces. Proc. Am. Math. Soc., to appear.
3. A. Aldroubi. Portraits of frames: overcomplete representations with applications to image processing. In [60], pages 322-329.
4. A. Aldroubi. Portraits of frames. Proc. Am. Math. Soc., 123(6): 1661-1668, 1995.
5. A. Aldroubi, M. Eden, and M. Unser. Discrete spline filters for multiresolutions and wavelets of ι2· SIAM J. Math. Anal., 25(5): 1412-1432, 1994.
6. A. Aldroubi and M. Unser. Families of wavelet transforms in connection with Shannon's sampling theory and the Gabor transform. In [18], pages 509-528.
7. A. Aldroubi and M. Unser. Families of multiresolution and wavelet spaces with optimal properties. Numer. Fund. Anal. Optimiz., 14(5): 417-446, 1993.
8. A. Aldroubi and M. Unser. Oblique projections in discrete signal subspaces of £2 and the wavelet transform. In [62], pages 36-45.
9. A. Aldroubi and M. Unser. Sampling procedure in function spaces and asymptotic equivalence with Shannon's sampling theory. Numer. Funct. Anal. Optimiz., 15(1): 1-21, 1994.
10. M. Antoníni, M. Barlaud, P. Mathieu, and I. Daubechies. Image coding using the wavelet transform. IEEE Trans. Image Process., 1(2): 205-220, 1992.
11. A. Arneodo, E. Bacry, and J. F. Muzy. The multifractal formalism revisited with wavelets. Int. J. Bifurcation Chaos, 42: 245-302, 1994.
12. P. Auscher. Wavelet bases for L(TZ) with rational dilation factor. In [78], pages 439-452.
13. J. J. Benedetto. Irregular sampling and frames. In [18], pages 445-507.
14. J. J. Benedetto and M.W. Frazier, editors. Wavelets-Mathematics and Applications. CRC Press, Boca Raton, FL, 1993.
15. J. J. Benedetto and S. Li. Multiresolution analysis frames with applications. IEEE-ICASSP-93, 1993. Proc. IEEE-ICASSP-93, Minneapolis, MN, 1993, 304-307.
16. O. Christensen. A Paley-Wiener theorem on frames. Proc. Am. Math. Soc., 123: 2199-2202, 1994.
17. C. Chui and E. Quak. Wavelets on a bounded interval. In Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker, editors, Birkhäuser-Verlag, Basel, 1992, pages 1-24.
18. C. K. Chui, editor. Wavelets: A Tutorial in Theory and Applications. Academic Press, San Diego, CA, 1992.
19. C. K. Chui and J. Z. Wang. A cardinal spline approach to wavelets. Proc. Am. Math. Soc., 113: 785-793, 1991.
20. A. Cohen, I. Daubechies, and J. C. Fauveau. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45: 485-560, 1992.
21. A. Cohen, I. Daubechies, B. Jawerth, and P. Vial. Multiresolution analysis, wavelets and fast algorithms on an interval. C. R. Acad. Sei. Paris Sér. I Math., 1(316): 417-421, 1993.
22. A. Cohen, I. Daubechies, and P. Vial. Multiresolution analysis, wavelets and fast algorithms on an interval. Appl. Comput. Harmon. Anal., 1(1): 54-81, 1993.
23. R. R. Coifman. A real variable characterization of Hp. Studia Math., 51: 269-274, 1974.
24. R. R. Coifman and Y. Meyer. Orthonormal wave packet bases. Preprint.
25. R. R. Coifman, Y. Meyer, S. Quake, and M. V. Wickerhauser. Signal processing and compression with wave packets. In Proc. Int. Conf. Wavelets, Marseille, 1989, Y. Meyer, editor, Masson, Paris, 1992.
26. R. R. Coifman, Y. Meyer, and M. V. Wickerhauser. Size properties of wavelet packets. In [78], pages 453-470.
27. S. Dahlke, W. Dahmen, E. Schmitt, and I. Weinreich. Multiresolution analysis and wavelets on S2 and S3. Numer. Fund. Anal. Optimiz., 16(1&2): 19-41, 1993.
28. W. Dahmen, S. Prössdorf, and R. Schneider. Wavelet approximation methods for pseudodifferential equations ii: Matrix compression and fast solution. Adv. Comp. Math., 1: 259-335, 1993.
29. I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math., 41: 909-996, 1988.
30. I. Daubechies. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory, 36: 961-1005, 1990.
31. I. Daubechies. Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia, 1992.
32. I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5): 1271-1283, 1986.
33. C. de Boor, R. A. DeVore, and A. Ron. On the construction of multivariate pre-wavelets. Constr. Approx., 9(2): 123-166, 1993.
34. B. Deng, B. Jawerth, G. Peters, and W. Sweldens. Wavelet probing for compression based segmentation. In [60], pages 266-276.
35. G. Deslauriers and S. Dubuc. Interpolation dyadique. In Fractals, dimensions non entières et applications, Masson, Paris, 1987, pages 44-55.
36. R. A. DeVore, B. Jawerth, and B. J. Lucier. Image compression through wavelet transform coding. IEEE Trans. Inform. Theory, 38(2): 719-746, 1992.
37. R. A. DeVore and B. J. Lucier. Fast wavelet techniques for near-optimal image processing. IEEE Military Communications Conference, New York, NY, 1992, pages 48.3.1-48.3.7.
38. D. L. Donoho and I. M. Johnstone. Ideal spatial adaptation via wavelet shrinkage. Biometrika, to appear.
39. G. Donovan, D. P. Hardin, and J. S. Geronimo. Fractal functions, splines, intertwining multiresolution analysis, and wavelets. In [60], pages 238-243.
40. G. C. Donovan, D. P. Hardin, and P. R. Massopust. Fractal functions and wavelet expansions based on several scaling function. J. Approx. Theory, 1994.
41. R. J. Duffin and A. C. Schaeffer. A class of nonharmonic Fourier series. Trans. Am. Math. Soc., 72(2): 341-366, 1952.
42. H. G. Feichtinger and K. Grôchenig. Non-orthogonal wavelet and Gabor expansions, and group representations. In [78], pages 353-375.
43. H. G. Feichtinger and K. Grôchenig. Banach spaces related to integrable group representations and their atomic decompositions. Int. J. Fund. Anal., 86: 307-340, 1989.
44. P. Flandrin. Some aspects of non-stationary signal processing with emphasis on time-frequency and time-scale methods. In Wavelets, Time-Frequency Methods and Phase Space, Marseille, 1987.
45. P. Flandrin. Temps-fréquence. Hermes, 1993.
46. M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J., 34: 777-799, 1985.
47. T. N. T. Goodman, S. L. Tang, and W. S. Lee. Wavelet wandering subspaces. Trans. Am. Math. Soc., 338(2): 639-654, 1993.
48. K. Grôchenig. Orthogonality criteria for compactly supported scaling functions. Appl. Comput. Harmon. Anal., 1: 242-245, 1994.
49. A. Grossmann and J. Morlet. Decompostion of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal., 15(4): 723-736, 1984.
50. A. Grossmann, J. Morlet, and T. Paul. Transforms associated to square integrable group representations. I. General results. J. Math. Phys., 26(10): 2473-2479, 1985.
51. C. E. Heil and D. F. Walnut. Continuous and discrete wavelet transforms. SIAM Rev., 31(4): 628-666, 1989.
52. C. Herley and M. Vetterli. Orthogonal time-varying filter banks and wavelet packets. IEEE Trans. Signal Process., 42(10): 2650-2663, 1994.
53. M. Holschneider and P. Tchamitchian. Régularité locale de la fonction non-différentiable de Riemann. In [65], pages 102-124.
54. C. Houdré. Wavelets, probability, and statistics: some bridges. In [14], pages 365-398.
55. S. Jaffard. Exposants de Hôlder en des points donnés et coefficients d'ondelettes. C. R. Acad. Sei. Paris, 308(10): 79-81, 1989.
56. B. Jawerth and W. Sweldens. An overview of wavelet based multiresolution analyses. SIAM Rev., 36(3): 377-412, 1994.
57. R. Jia and C. A. Micchelli. Using the refinement equation for the construction of pre-wavelets II: Powers of two. In Curves and Surfaces, P. J. Laurent, A. le Méhauté, and L. L. Schumaker, editors. Academic Press, New York, 1991, 209-249.
58. P. T. Jorgensen and S. Pederson. Harmonic analysis of fractal measures induced by representations of a certain C-algebra. Bull. Am. Math. Soc., 29: 228-234, 1993.
59. J. Kovačević and M. Vetterli. Perfect reconstruction filter banks with arbitrary rational sampling rates. IEEE Trans. Signal Process., 41(6): 2047-2066, 1993.
60. A. F. Laine, editor. Mathematical Imaging: Wavelet Applications in Signal and Image Processing, Proc. SPIE, Vol. 2034. SPIE-The International Society for Optical Engineering, Bellingham, WA, 1993.
61. A. F. Laine, S. Schuler, and J. Fan. Mammographic feature enhancement by multiscale analysis. IEEE Trans. Medical Imaging, 13(14), 1994.
62. A. F. Laine and M. Unser, editors. Mathematical Imaging: Wavelet Applications in Signal and Image Processing, Proc. SPIE, Vol. 2303. SPIE-The International Society for Optical Engineering, Bellingham, WA, 1994.
63. R. Latter. A decomposition of Hp(R2) in terms of atoms. Studia Math., 62: 92-101, 1978.
64. P. G. Lemarié. Ondelettes a localisation exponentielles. J. Math. Pures Appl., 67: 227-236, 1988.
65. P. G. Lemarié, editor. Les Ondelettes en 1989. Number 1438 in Lecture Notes in Math. Springer-Verlag, 1990.
66. P. G. Lemarié and Y. Meyer. Ondelettes et bases hilbertiennes. Rev. Math. Iberoam., 2, 1986.
67. S. Li and D. Healy. General frame decompositions, pseudo-duals and its application to Weyl-Heisenberg frames. Numer. Fund. Anal. Op-timiz., to appear.
68. S. Mallat. Multiresolution approximations and wavelet orthonormal bases of L2(R). Trans. Am. Math. Soc., 315(1): 69-97, 1989.
69. S. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Signal Proc., II (7): 674-693, 1989.
70. S. Mallat and W. L. Hwang. Singularity detection and processing with wavelets. IEEE Trans. Inform. Theory, 38(2): 617-643, 1992.
71. S. Mallat and S. Zhong. Characterization of signals from multiscale edges. IEEE Trans. Patt. Anal. Mach. Intell., 14: 710-732, 1992.
72. Y. Meyer. Principe d'incertitude, bases hilbertiennes et algèbres d'opérateurs. In Sèmin. Bourbaki, Vol. 662, 1985.
73. Y. Meyer. Ondelettes et opérateurs. Hermann, Paris, 1990.
74. Y. Meyer. Ondelettes sur l'intervalle. Rev. Math. Iberoam., 7: 115-133, 1992.
75. C. A. Micchelli. Using the refinement equation for the construction of pre-wavelet. Numerical Algorithms, 1: 75-116, 1991.
76. T. Olson and J. DeStefano. Wavelet localization of the Radon transform. IEEE Trans. Signal Process., 42(10): 2055-2067, 1994.
77. O. Rioul. A discrete-time multiresolution theory. IEEE Trans. Signal Process., 41(8): 2591-2606, 1993.
78. M. B. Ruskai, G. Beylkin, R. R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, and L. Raphael, editors. Wavelets and their Applications. Jones and Bartlett, Boston, 1992.
79. P. Schröder and W. Sweldens. Spherical wavelets: Efficiently representing functions on the sphere. Computer Graphics, (SIGGRAPH '95 Proceedings), to appear.
80. P. Schröder and W. Sweldens. Spherical wavelets: Texture processing. Tech. Rep. 1995:4, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, Columbia, 1995.
81. V. Strela and G. Strang. Finite element multiwavelets. In [60], pages 202-213.
82. J. O. Strömberg. A modified Franklin system and higher order spline systems on Rn as unconditional bases for Hardy spaces. In Conf. Harmonic Analysis in Honor of Antoni Zygmund, Beckner, et al., editors, volume II, University of Chicago Press, Chicago, 1981, 475-494.
83. W. Sweldens. The lifting scheme: A construction of second generation wavelets. Tech. Rep. 1994:6, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, Columbia, 1994.
84. W. Sweldens. The lifting scheme: A custom-design construction of biorthogonal wavelets. Tech. Rep. 1994:7, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina, Columbia, 1994.
85. W. Sweldens and R. Piessens. Quadrature formulae and asymptotic error expansions for wavelet x approximations of smooth functions. SIAM J. Num. Anal., 31(4): 1240-1264, 1994.
86. M. Unser and A. Aldroubi. Polynomial splines and wavelets-a signal processing perspective. In [18], pages 543-601.
87. M. Unser and A. Aldroubi. Multiresolution image registration procedure using spline pyramids. In [60], pages 160-170.
88. M. Unser, A. Aldroubi, and M. Eden. On the asymptotic convergence of B-spline wavelets to Gabor functions. IEEE Trans. Inf. Theory, 38: 864-872, 1992.
89. M. Unser, A. Aldroubi, and M. Eden. A family of polynomial spline wavelet transforms. Signal Process., 30:141-162, 1993.
90. M. Vetterli and C. Herley. Wavelets and filter banks. IEEE Trans. Signal Process., 40: 2207-2231, 1993.
91. M. V. Wickerhauser. Acoustic signal compression with wavelet packets. In [18], pages 679-700.
92. J. M. Wilson. On the atomic decomposition for hardy spaces. Pacific J. Math., 116: 201-207, 1985.
93. H. Woerdeman and A. Aldroubi. Extrapolation in multiresolutions. In [60], pages 120-128.
94. X. G. Xia and Z. Z. Zhang. On sampling theorem, wavelets, and wavelet transforms. IEEE Trans. Signal Process., 41(12): 3524-3535, 1993.