Framelets: MRA-based constructions of wavelet frames

Applied and Computational Harmonic Analysis

Volumn 14, Issue 1, 2003, Pages 1-46

  Daubechies, Ingrid ^a   Han, Bin ^b   Ron, Amos ^c   Shen, Zuowei ^d  

^a PRINCETON UNIVERSITY   (United States)

^b UNIVERSITY OF ALBERTA   (Canada)

^c CS Department   (United States)

^d NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Fast frame transform; Framelets; Frames; Multiresolution analysis; Oblique extension principle; Pseudo splines; Tight frames; Unitary extension principle; Wavelets

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; BESSEL FUNCTIONS; IMAGE COMPRESSION; MATHEMATICAL OPERATORS; REDUNDANCY;

MULTIRESOLUTION ANALYSIS;

WAVELET TRANSFORMS;

EID: 0242573570     PISSN: 10635203     EISSN: None     Source Type: Journal    
DOI: 10.1016/S1063-5203(02)00511-0     Document Type: Article

References (44)

1. G. Battle, A block spin construction of ondelettes, Part I: Lemarie functions, Comm. Math. Phys. 110 (1987) 601-615.
2. J. Benedetto, S.D. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (4) (1998) 389-427.
3. J. Benedetto, O. Treiber, Wavelet frames: Multiresolution analysis and extension principles, in: I. Debnath (Ed.), Wavelet Transforms and Time-Frequency Signal Analysis, Birkhäuser, Boston, 2001, pp. 3-36, Chap. 1.
4. C. de Boor, A. Ron, The exponentials in the span of the integer translates of a compactly supported function, J. London Math. Soc. 45 (1992) 519-535.
5. C. de Boor, R. DeVore, A. Ron, On the construction of multivariate (pre)wavelets, Constr. Approx. 9 (1993) 123-166.
6. d), Trans. Amer. Math. Soc. 341 (1994) 787-806.
7. d), J. Funct. Anal. 119 (1) (1994) 37-78.
8. n), Appl. Comput. Harmon. Anal. 8 (2) (2000) 203-221.
9. A. Cohen, I. Daubechies, J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. XLV (1992) 485-560.
10. C.K. Chui, W. He, Compactly supported tight frames associated with refinable functions, Appl. Comput. Harmon. Anal. 8 (3) (2000) 293-319.
11. C.K. Chui, W. He, J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (3) (2002) 224-262.
12. C.K. Chui, X. Shi, Bessel sequences and affine frames, Appl. Comput. Harmon. Anal. 1 (1993) 29-49.
13. C.K. Chui, X. Shi, J. Stockler, Affine frames, quasi-frames and their duals, Adv. Comput. Math. 8 (1998) 1-17.
14. C.K. Chui, J. Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992) 903-915.
15. I. Daubechies, Ten Lectures on Wavelets, in: CBMS Conf. Series in Appl. Math., Vol. 61, SIAM, Philadelphia, 1992.
16. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909-996.
17. I. Daubechies. B. Han, Pairs of dual wavelet frames from any two refinable functions, 2000, preprint.
18. I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) 1271-1283.
19. G. Donovan, J.S. Geronimo, D.P. Hardin, Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets, SIAM J. Math. Anal. 27 (6) (1996) 1791-1815.
20. R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 147-158.
21. M. Frazier, G. Garrigós, X. Wang, G. Weiss, A characterization of functions that generate wavelet and related expansion, J. Fourier Anal. Appl. 3 (1997) 883-906.
22. J.S. Geronimo, D.P. Hardin, P.R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (3) (1994) 373-401.
23. G.-M. Greuel, G. Pfister, H. Schönemann, Singular version 1.2 User Manual. In Reports On Computer Algebra, number 21. Centre for Computer Algebra, University of Kaiserslautern, June 1998. http://www.mathematik.uni-kl.de/̃zca/Singular.
24. K. Gröchenig, A. Ron, Tight compactly supported wavelet frames of arbitrarily high smoothness, Proc. Amer. Math. Soc. 126 (4) (1998) 1101-1107.
25. B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (4) (1997) 380-413.
26. E. Hernándes, G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, 1996.
27. K. Jetter, D.X. Zhou, Order of linear approximation from shift invariant spaces, Constr. Approx. 11 (4) (1995) 423-438.
28. K. Jetter, D.X. Zhou, Order of linear approximation on finitely generated shift invariant spaces, 1998, preprint.
29. R.Q. Jia, C.A. Micchelli, Using the refinement equation for the construction of pre-wavelets II: Powers of two, in: P.J. Laurent, A. Le Méhauté, L.L. Schumaker (Eds.), Curves and Surfaces, Academic Press, New York, 1991, pp. 209-246.
30. W. Lawton, S.L. Lee, Z. Shen, Characterization of compactly supported refinable splines, Adv. Comput. Math. 3 (1995) 137-145.
31. P.G. Lemarié, Ondelettes à localisation exponentielle, J. Math. Pures Appl. 67 (1988) 227-236.
32. 2(ℝ), Trans. Amer. Math. Soc. 3-15 (1989) 69-87.
33. Y. Meyer, Ondelettes et Opérateurs I: Ondelettes, Hermann, Paris, 1990.
34. A. Petukhov, Explicit construction of framelets, Preprint #3, IMI, University of South Carolina, 2000. Also available at http://www.math.sc.edu/̃imip/0003.ps.
35. A. Petukhov, Symmetric framelets, preprint, 2000.
36. G. Plonka, Approximation order provided by refinable function vectors, Constr. Approx. 13 (2) (1997) 221-244.
37. G. Plonka, A. Ron, A new factorization technique of the matrix mask of univariate refinable functions, Numer. Math. 87 (3) (2001) 555-595.
38. d), Canad. J. Math. 47 (5) (1995) 1051-1094; Ftp site: ftp://ftp.cs.wisc.edu/Approx/framel.ps.
39. A. Ron, Z. Shen, Gramian analysis of affine bases and affine frames, in: C.K. Chui, L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2. Wavelets, World Scientific, Singapore, 1995, pp. 375-382.
40. d): The analysis of the analysis operator, J. Funct. Anal. 148 (1997) 408-447; Ftp site: ftp://ftp.cs.wisc.edu/Approx/affine.ps.
41. d) H: Dual systems, J. Fourier Anal. Appl. 3 (1997) 617-637; Ftp site: ftp://ftp.cs.wisc.edu/Approx/dframe.ps.
42. d). Math. Comput. 67 (1998) 191-207; Ftp site: ftp://ftp.cs.wisc.edu/Approx/tight.ps.
43. d), in: K.S. Lau (Ed.), Advances in Wavelets, Springer, New York, 1998, pp. 27-49; Ftp site: ftp://ftp.cs.wise.edu/Approx/hk.ps.
44. I. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2) (2001) 163-181.