Framelet based deconvolution

Journal of Computational Mathematics

Volumn 28, Issue 3, 2010, Pages 289-308

  Cai, Jian Feng ^a   Shen, Zuowei ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Deconvolution; Framelet; Soft thresholding; Tight frame; Wavelet

Indexed keywords

EID: 77953996853     PISSN: 02549409     EISSN: None     Source Type: Journal    
DOI: 10.4208/jcm.1001-m1002     Document Type: Article

References (26)

1. G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms. I, Commun. Pur. Appl. Math., 44 (1991), 141-183.
2. L. Borup, R. Gribonval and M. Nielsen, Biframelet systems with few vanishing moments characterize Besov spaces, Appl. Comput. Harmon. A., 17 (2004), 3-28.
3. J.-F. Cai, R.H. Chan, L. Shen, and Z. Shen, Restoration of chopped and nodded images by framelets, SIAM J. Sci. Comput, 30 (2008), 1205-1227.
4. J.-F. Cai, R.H. Chan, and Z. Shen, A framelet-based image inpainting algorithm, Appl. Comput. Harmon. A., 24 (2008), 131-149.
5. J.-F. Cai, S. Osher, Z. Shen, Linearized Bregman iterations for frame-based image deblurring, SIAM J. Imaging Sci., 2 (2009), 226-252.
6. A. Chai and Z. Shen, Deconvolution: A wavelet frame approach, Numer. Math., 106 (2007), 529-587.
7. R.H. Chan, T.F. Chan, L. Shen, and Z. Shen, Wavelet algorithms for high-resolution image reconstruction, SIAM J. Sci. Comput, 24 (2003), 1408-1432.
8. R.H. Chan, S.D. Riemenschneider, L. Shen, and Z. Shen, Tight frame: an efficient way for high-resolution image reconstruction, Appl. Comput. Harmon. A., 17 (2004), 91-115.
9. R.H. Chan, S.D. Riemenschneider, L. Shen, and Z. Shen, High-resolution image reconstruction with displacement errors: A framelet approach, Int. J. Imag. Syst. Tech., (2005), 91-104.
10. R.H. Chan, L. Shen, and Z. Shen, A framelet-based approach for image inpainting, 2005. Research Report 2005-04(325), Department of Mathematics, The Chinese University of Hong Kong.
11. R.H. Chan, Z. Shen, and T. Xia, A framelet algorithm for enhancing video stills, Appl. Comput. Harmon. A., 23 (2007), 153-170.
12. A. Cohen, M. Hoffmann, and M. Reib, Adaptive wavelet Galerkin methods for linear inverse problems, SIAM J. Numer. Anal, 42 (2004), 1479-1501.
13. P.L. Combettes and V.R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Sim., 4 (2005), 1168-1200.
14. I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pur. Appl. Math., 57 (2004), 1413-1457.
15. I. Daubechies, B. Han, A. Ron, and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. A., 14 (2003), 1-46.
16. I. Daubechies, G. Teschke, and L. Vese, Iteratively solving linear inverse problems under general convex constraints, Inverse Problems and Imaging, 1 (2007), 29-46.
17. C. de Boor, R.A. DeVore, and A. Ron, On the construction of multivariate (pre)wavelets, Constr. Approx., 9 (1993), 123-166.
18. C. De Mol and M. Defrise, A note on wavelet-based inversion algorithms, in Inverse problems, image analysis, and medical imaging (New Orleans, LA, 2001), vol. 313 of Contemp. Math., Amer. Math. Soc, Providence, RI, 2002, 85-96.
19. B. Dong and Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. A., 22 (2007), 78-104.
20. D.L. Donoho, De-noising by soft-thresholding, IEEE T. Inform. Theory, 41 (1995), 613-627.
21. D.L. Donoho, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition, Appl. Comput. Harmon. A., 2 (1995), 101-126.
22. Y. Hur and A. Ron, CAPlets: wavelet representations without wavelets, preprint, (2005).
23. R. Jia and Z. Shen, Multiresolution and Wavelets, Proc. Edinburgh Math. Soc., 37 (1994), 271-300.
24. J. Kalifa and S. Mallat, Thresholding estimators for linear inverse problems and deconvolutions, Ann. Stat, 31 (2003), 58-109.
25. J. Kalifa, S. Mallat, and B. Rouge, Deconvolution by thresholding in mirror wavelet bases, IEEE T. Image Process., 12 (2003), 446-457.
26. 2(Rd): the analysis of the analysis operator, J. Funct. Anal, 148 (1997), 408-447.