The undecimated wavelet decomposition and its reconstruction

IEEE Transactions on Image Processing

Volumn 16, Issue 2, 2007, Pages 297-309

  Starck, Jean Luc ^a   Fadili, Jalal ^b   Murtagh, Fionn ^c  

^a UNIVERSITÉ PARIS DIDEROT   (France)

^b Centre National de la Recherche Scientifique   (France)

^c UNIVERSITY OF LONDON   (United Kingdom)

Author keywords

Denoising; Multiresolution; Multiscale decomposition; Restoration; Wavelet

Indexed keywords

FILTER BANKS; MULTIRESOLUTION ANALYSIS; WAVELET DECOMPOSITION;

DENOISING; MULTISCALE DECOMPOSITION;

IMAGE RECONSTRUCTION;

ALGORITHM; ARTICLE; COMPUTER ASSISTED DIAGNOSIS; EVALUATION; IMAGE ENHANCEMENT; INFORMATION RETRIEVAL; MATHEMATICAL COMPUTING; METHODOLOGY; SIGNAL PROCESSING;

ALGORITHMS; IMAGE ENHANCEMENT; IMAGE INTERPRETATION, COMPUTER-ASSISTED; INFORMATION STORAGE AND RETRIEVAL; NUMERICAL ANALYSIS, COMPUTER-ASSISTED; SIGNAL PROCESSING, COMPUTER-ASSISTED;

EID: 33847702157     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIP.2006.887733     Document Type: Article

References (40)

1. I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992.
2. G. Strang and T. Nguyen, Wavelet and Filter Banks. Cambridge, U.K.: Wellesley-Cambridge, 1996.
3. S. Mallat, A Wavelet Tour of Signal Processing. New York: Academic, 1998.
4. J.-L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach. Cambridge, U.K.: Cambridge Univ. Press, 1998.
5. A. Cohen, Numerical Analysis of Wavelet Methods. New York: Elsevier, 2003.
6. J. P. Antoine and R. Murenzi, "Two dimensional wavelet analysis in image processing," Phys. Mag., vol. 16, pp. 105-134, 1994.
7. A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, and J. F. Muzy, Ondelettes, Multifractales et Turbulences. Paris, France: Diderot, Arts et Sciences, 1995.
8. P. Dutilleux, "An implementation of the "algorithme à trous" to compute the wavelet transform," in Wavelets: Time-Frequency Methods and Phase-Space, J.M. Combes, A. Grossmann, and P. Tchamitchian, Eds. New York: Springer, 1989.
9. M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, "A real-time algorithm for signal analysis with the help of the wavelet transform," in Wavelets: Time-Frequency Methods and Phase-Space. New York: Springer-Verlag, 1989, pp. 286-297.
10. J.-L. Starck, M. Elad, and D. L. Donoho, "Redundant multiscale transforms and their application for morphological component analysis," Adv. Imag. Electron Phys., vol. 132, 2004.
11. J.-L. Starck and F. Murtagh, Astronomical Image and Data Analysis. New York: Springer-Verlag, 2002.
12. S. Durand and J. Froment, "Reconstruction of wavelet coefficients using total variation minimization," SIAM J. Sci. Comput., vol. 24, no. 5, pp. 1754-1767, 2003.
13. F. Malgouyres, "Minimizing the total variation under a general convex constraint for image restoration," IEEE Trans. Image Process., vol. 11, no. 12, pp. 1450-1456, Dec. 2002.
14. E. J. Candès and F. Guo, "New multiscale transforms, minimum total variation synthesis: Applications to edge-preserving image reconstruction," Signal Process., vol. 82, no. 11, pp. 1519-1543, 2002.
15. J.-L. Starck, D. L. Donoho, and E. J. Candès, "Very high quality image restoration by combining wavelets and curvelets," Proc. SPIE, vol. 4478, pp. 9-19, 2001.
16. Z. Cvetkovic and M. Vetterli, "Oversampled filter banks," IEEE Trans Signal Process., vol. 46, no. 5, pp. 1245-1255, May 1998.
17. H. Bolcskei, F. Hlawatsch, and H. G. Feichtinger, "Frame-theoretic analysis of oversampled filter banks," IEEE Trans Signal Process., vol. 46, no. 12, pp. 3256-3268, Dec. 1998.
18. n: Analysis, synthesis, and algorithms," IEEE Trans. Inf. Theory, vol. 44, no. 1, pp. 16-31, Jan. 1998.
19. Z. Cvetkovic and M. Vetterli, "Discrete-time wavelet extrema representation: Design and consistent reconstruction," IEEE Trans. Image Process., vol. 43, no. 3, pp. 1245-1255, Mar. 1995.
20. M. J. Shensa, "Discrete wavelet transforms: Wedding the à trous and Mallat algorithms," IEEE Trans. Signal Process., vol. 40, no. 10, pp. 2464-2482, Oct. 1992.
21. A. Genovesio and J.-C. Olivo-Marin, "Tracking fluorescent spots in biological video microscopy," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing X, J.-A. Conchello, C. J. Cogswell, and T. Wilson, Eds. Bellingham, WA: SPIE, 2003, vol. 4964, pp. 98-105.
22. P. J. Burt and A. E. Adelson, "The Laplacian pyramid as a compact image code," IEEE Trans. Commun., vol. COM-31, no. 4, pp. 532-540, Apr. 1983.
23. P. Fryźlewicz and G. P. Nason, "A Haar-Fisz algorithm for Poisson intensity estimation," J. Comput. Graph. Statist., vol. 13, pp. 621-638, 2004.
24. E. D. Kolaczyk, "Nonparametric estimation of intensity maps using Haar wavelets and Poisson noise characteristics," Astrophys. J., vol. 534, pp. 490-505, 2000.
25. E. D. Kolaczyk, "Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds," Statist. Sinica, vol. 9, pp. 119-135, 1999.
26. R. D. Nowak and R. G. Baraniuk, "Wavelet-domain filtering for photon imaging systems," IEEE Trans. Image Process., vol. 8, no. 5, pp. 666-678, May 1999.
27. A. Antoniadis and T. Sapatinas, "Wavelet shrinkage for natural exponential families with quadratic variance functions," Biometrika, vol. 88, pp. 805-820, 2001.
28. K. E. Timmermann and R. D. Nowak, "Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging," IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 846-862, Apr. 1999.
29. G. Jammal and A. Bijaoui, "Dequant: A flexible multiresolution restoration framework," Signal Process., vol. 84, no. 7, pp. 1049-1069, 2004.
30. G. Steidl, J. Weickert, T. Brox, P. Mrázek, and M. Welk, "On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and Sides," Tech. Rep. 26, Dept. Math., Univ. Bremen, Bremen, Germany, 2003.
31. M. Unser, "Texture classification and segmentation using wavelet frames," IEEE Trans. Image Process., vol. 4, no. 11, pp. 1549-1560, Nov. 1995.
32. J.-L. Starck, A. Bijaoui, and F. Murtagh, "Multiresolution support applied to image filtering and deconvolution," CVGIP: Graph. Models Image Process., vol. 57, pp. 420-431, 1995.
33. I. Halperin, "The product of projection operators," Acta Sci. Math., vol. 23, pp. 96-99, Mar. 1962.
34. J. A. Tropp, I. Dhillon, R. W. Heath, and T. Strohmer, "Designing structured tight frames via an alternating projection method," IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 188-209, Jan. 2005.
35. D. C. Youla, "Mathematical theory of image restoration by the method of convex projections," in Image Recovery: Theory and Application, H. Starck, Ed. New York: Academic, 1987.
36. M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, "Image coding using wavelet transform," IEEE Trans. Image Process., vol. 1, no. 2, pp. 205-220, Apr. 1992.
37. M. Elad, "Why Simple Shrinkage is Still Relevant for Redundant Representations," to be published, 2006.
38. G. Battle, "A block spin construction of ondelettes. Part I: Lemarié functions," Comm. Math. Phys., vol. 110, pp. 601-615, 1987.
39. J.-L. Starck, M. Elad, and D. Donoho, "Image decomposition via the combination of sparse representation and a variational approach," IEEE Trans. Image Process., vol. 14, no. 10, pp. 1570-1582, Oct. 2005.
40. M. Elad, J.-L. Starck, D. Donoho, and P. Querre, "Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)," J. Appl. Comput. Harmon. Anal., vol. 19, pp. 340-358, 2006.