A new detail-preserving regularization scheme

SIAM Journal on Imaging Sciences

Volumn 7, Issue 2, 2014, Pages 1309-1334

  Guo, Weihong ^a   Qin, Jing ^a   Yin, Wotao ^b  

^a CASE WESTERN RESERVE UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Compressive sensing; Dct; Dwt; Features; Mri; Shearlet transform; Total generalized variation

Indexed keywords

COMPUTERIZED TOMOGRAPHY; MAGNETIC RESONANCE; MAGNETIC RESONANCE IMAGING; SIGNAL RECONSTRUCTION;

COMPRESSIVE SENSING; DCT; DWT; FEATURES; GENERALIZED VARIATION; SHEARLET TRANSFORMS;

MEDICAL IMAGING;

EID: 84903386069     PISSN: None     EISSN: 19364954     Source Type: Journal    
DOI: 10.1137/120904263     Document Type: Article

References (38)

1. M. Lustig, D. Donoho, and J. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magn. Reson. Med., 58 (2007), pp. 1182-1195.
2. R. Compton, S. Osher, and L. Bouchard, Hybrid regularization for MRI reconstruction with static field inhomogeneity correction, in Proceedings of the 2012 9th IEEE International Symposium on Biomedical Imaging (ISBI), IEEE, Piscataway, NJ, 2012, pp. 650-655.
3. G. Steidl and J. Weickert, Relations between soft wavelet shrinkage and total variation denoising, in Pattern Recognition, Lecture Notes in Comput. Sci. 2449, Springer, Berlin, 2002, pp. 198-205.
4. T. F. Chan, J. Shen, and H. M. Zhou, Total variation wavelet inpainting, J. Math. Imaging Vision, 25 (2006), pp. 107-125.
5. X. Zhang and T. F. Chan, Wavelet inpainting by nonlocal total variation, Inverse Problems Imaging, 4 (2010), pp. 191-210.
6. A. C. Yau, X. C. Tai, and M. K. Ng, L0-norm and total variation for wavelet inpainting, in Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. 5567, Springer, Berlin, 2009, pp. 539-551.
7. S. Durand and J. Froment, Reconstruction of wavelet coefficients using total variation minimization, SIAM J. Sci. Comput., 24 (2003), pp. 1754-1767.
8. G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data, Springer, Dordrecht, 2012.
9. G. Kutyniok, K. Guo, and D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators, in Wavelets and Splines (Athens, GA, 2005), G. Chen and M. J. Lai, eds., Nashboro Press, Nashville, TN, 2006, pp. 189-201.
10. G. Easley, D. Labate, and W. Q. Lim, Sparse directional image representations using the discrete shearlet transform, Appl. Comput. Harmon. Anal., 25 (2008), pp. 25-46.
11. J. Cai, B. Dong, S. Osher, and Z. Shen, Image restoration: Total variation, wavelet frames, and beyond, J. Amer. Math. Soc., 25 (2012), pp. 1033-1089.
12. K. Guo and D. Labate, The construction of smooth Parseval frames of shearlets, Math. Model. Natur. Phenom., 8 (2013), pp. 82-105.
13. G. Kutyniok and W.-Q. Lim, Compactly supported shearlets are optimally sparse, J. Approx. Theory, 163 (2011), pp. 1564-1589.
14. D. Donoho, Sparse components of images and optimal atomic decomposition, Constr. Approx., 17 (2001), pp. 353-382.
15. S. G. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, Academic Press, San Diego, 1999.
16. E. Candes and D. Donoho, Ridgelets: A key to higher-dimensional intermittency?, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), pp. 2495-2509.
17. M. N. Do and M. Vetterli, The contourlet transform: An efficient directional multiresolution image representation, IEEE. Trans. Image Process., 14 (2005), pp. 2091-2106.
18. E. Candes, L. Demanet, D. Donoho, and L. Ying, Fast discrete curvelet transforms, Multiscale Model. Simul., 5 (2006), pp. 861-899.
19. S. Hauser, Fast Finite Shearlet Transform, preprint, arXiv:1202.1773, 2012.
20. G. Kutyniok, M. Shahram, and X. Zhuang, Shearlab: A rational design of a digital parabolic scaling algorithm, SIAM J. Imag. Sci., 5 (2012), pp. 1291-1332.
21. L. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44 (2001), pp. 131-161.
22. M. Lysaker, A. Lundervold, and X. C. Tai, Noise removal using fourth-order partial differential equations with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), pp. 1579-1590.
23. W. Hinterberger and O. Scherzer, Variational methods on the space of functions of bounded Hessian for convexification and denoising, Computing, 76 (2006), pp. 109-133.
24. K. Papafitsoros and C. B. Schonlieb, A combined first and second order variational approach for image reconstruction, J. Math. Imaging Vision, 48 (2014), pp. 308-338.
25. K. Bredies, K. Kunisch, and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), pp. 492-526.
26. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
27. D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Math. Appl., 2 (1976), pp. 17-40.
28. R. Glowinski and A. Marrocco, Sur l'approximation par ́eĺements finis etlaŕesolution parṕenalisation-dualit́e d'uneclasse de problemes de Dirichletnonlińeaires, RAIRO Oper. Res. Ser. Rouge, 9 (1975), pp. 41-76.
29. D. Bertsekas and J. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Athena Scientific, Belmont, MA, 1997.
30. B. He, L. Z. Liao, D. Han, and H. Yang, A new inexact alternating directions method for monotone variational inequalities, Math. Program., 92 (2002), pp. 103-118.
31. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), pp. 1-122.
32. W. Deng and W. Yin, On the Global and Linear Convergence of the Generalized Alternating Direction Method of Multipliers, Rice University CAAM Technical report TR12-14, Houston, TX, 2012.
33. R. H. Chan, J. Yang, and X. Yuan, Alternating direction method for image inpainting in wavelet domains, SIAM J. Imaging Sci., 4 (2011), pp. 807-826.
34. J. Yang, Y. Zhang, and W. Yin, A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE J. Sel. Topics Signal Process., 4 (2010), pp. 288-297.
35. W. Guo and W. Yin, Edge guided reconstruction for compressive imaging, SIAM J. Imaging Sci., 5 (2012), pp. 809-834.
36. F. Knoll, K Bredies, T. Pock, and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magnet. Resonance Med., 65 (2011), pp. 480-491.
37. I. W. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal., 10 (2001), pp. 163-181.
38. H. N. Al-Taai, A novel fast computing method for framelet coefficients, Amer. J. Appl. Sci., 5 (2008), pp. 1522-1527.