A weighted H¹ seminorm regularization method for Fredholm integral equations of the first kind

References

• R. Acar, C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Probl. 10 (1994), pp. 1217–1229 doi: 10.1088/0266-5611/10/6/003
   Web of Science ®Google Scholar
• P. Blomgren, T. Chan, P. Mulet, Extensions to total variation denoising, SPIE, Proceedings of the International Society of Photo-Optical Instrumentation Engineers, Advanced Signal Processing: Algorithms, Architectures and Implementation VII, Vol. 3162, F. T. Luk, SPIE, San Diego, CA, (1997), pp. 367–375
   Google Scholar
• K. Bredies, K. Kunisch, T. Pock, Total generalized variation, SIAM J. Imag. Sci. 3 (2010), pp. 492–526 doi: 10.1137/090769521
   Web of Science ®Google Scholar
• C. Brito-Loeza, K. Chen, Multigrid algorithm for high order denoising, SIAM J. Imag. Sci. 3 (2010), pp. 363–389 doi: 10.1137/080737903
   Web of Science ®Google Scholar
• R. Chan, K. Chen, Multilevel algorithm for a Poisson noise removal model with total-variation regularization, Int. J. Comput. Math. 84 (2007), pp. 1183–1198 doi: 10.1080/00207160701450390
   Web of Science ®Google Scholar
• R. Chan, M. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38 (1996), pp. 427–482 doi: 10.1137/S0036144594276474
   Web of Science ®Google Scholar
• R. Chan, H. X. Liang, J. Ma, Positively constrained total variation penalized image restoration, Adv. Adapt. Data Anal. 3 (2011), pp. 1–15 doi: 10.1142/S1793536911000817
   Google Scholar
• Q. S. Chang, X. C. Tai, L. Xing, A compound algorithm of denoising using second-order and fourth-order partial differential equations, Numer. Math. Theor. Meth. Appl. 2 (2009), pp. 353–376
   Web of Science ®Google Scholar
• Q. Chen, P. Montesinos, Q. S. Sun, P. A. Heng, D. S. Xia, Adaptive total variation denoising based on difference curvature, Image Vis. Comput. 28 (2010), pp. 298–306 doi: 10.1016/j.imavis.2009.04.012
   Web of Science ®Google Scholar
• A. Chopra, H. Lian, Total variation, adaptive total variation and nonconvex smoothly clipped absolute deviation penalty for denoising blocky images, Pattern Recognit. 43 (2010), pp. 2609–2619 doi: 10.1016/j.patcog.2010.03.022
   Web of Science ®Google Scholar
• M. Defrise, C. Vanhove, X. Liu, An algorithm for total variation regularization in high-dimensional linear problems, Inverse Probl. 27 (2011), pp. 065002 doi: 10.1088/0266-5611/27/6/065002
   Web of Science ®Google Scholar
• A. El Hamidi, M. Ménard, M. Lugiez, C. Ghannam, Weighted and extended total variation for image restoration and decomposition, Pattern Recognit. 43 (2010), pp. 1564–1576 doi: 10.1016/j.patcog.2009.10.011
   Web of Science ®Google Scholar
• J. M. Fadili, G. Peyré, Total variation projection with first order schemes, IEEE Trans. Image Process. 20 (2011), pp. 657–669 doi: 10.1109/TIP.2010.2072512
   PubMed Web of Science ®Google Scholar
• P. C. Hansen, The truncated SVD as a method for regularization, BIT 27 (1987), pp. 534–553 doi: 10.1007/BF01937276
   Web of Science ®Google Scholar
• P. C. HansenDiscrete Inverse Problems: Insight and AlgorithmsSIAM, Philadelphia, 2010
   Google Scholar
• P. C. Hansen, K. Mosegaard, Piecewise polynomial solutions to linear inverse problems, Lect. Notes Earth Sci. 63 (1996), pp. 284–294 doi: 10.1007/BFb0011787
   Google Scholar
• P. C. Hansen, T. Sekii, H. Shibahashi, The modified truncated SVD method for regularization in general form, SIAM J. Sci. Stat. Comput. 13 (1992), pp. 1142–1150 doi: 10.1137/0913066
   Web of Science ®Google Scholar
• Y. M. Huang, M. K. Ng, Y. W. Wen, A fast total variation minimization method for image restoration, Multiscale Model. Simul. 7 (2008), pp. 774–795 doi: 10.1137/070703533
   Web of Science ®Google Scholar
• Y. M. Huang, M. K. Ng, Y. W. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imag. Sci. 2 (2009), pp. 20–40 doi: 10.1137/080712593
   Web of Science ®Google Scholar
• X. Q. JinDevelopments and Applications of Block Toeplitz Iterative SolversScience Press/Kluwer Academic Publishers, Beijing/Dordrecht, 2002
   Google Scholar
• Y. F. Lou, X. Q. Zhang, S. Osher, A. Bertozzi, Image recovery via nonlocal operators, J. Sci. Comput. 42(2) (2010), pp. 185–197 doi: 10.1007/s10915-009-9320-2
   Web of Science ®Google Scholar
• K. Maleknejad, S. Sohrabi, Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets, Appl. Math. Comput. 186 (2006), pp. 836–843 doi: 10.1016/j.amc.2006.08.023
   Web of Science ®Google Scholar
• K. Maleknejad, N. Aghazadeh, R. Mollapourasl, Numerical solution of Fredholm integral equation of the first kind with collocation method and estimation of error bound, Appl. Math. Comput. 179 (2006), pp. 352–359 doi: 10.1016/j.amc.2005.11.159
   Web of Science ®Google Scholar
• C. A. Micchelli, L. X. Shen, Y. S. Xu, Proximity algorithms for image models: Denoising, Inverse Probl. 27 (2011), pp. 045009 doi: 10.1088/0266-5611/27/4/045009
   Web of Science ®Google Scholar
• K. Papafitsoros, C.-B. Schonlieb, A combined first and second order variational approach for image reconstruction(2012), Available at arXiv:1202.6341.
   Google Scholar
• A. D. Polynin, A. V. Manzhirov, Handbook of Integral Equations, CRC Press, New York, 1998
   Google Scholar
• M. T. Rashed, Numerical solutions of the integral equations of the first kind, Appl. Math. Comput. 145 (2003), pp. 413–420 doi: 10.1016/S0096-3003(02)00497-6
   Web of Science ®Google Scholar
• J. Savage, K. Chen, An improved and accelerated non-linear multigrid method for total-variation denoising, Int. J. Comput. Math. 82 (2005), pp. 1001–1015 doi: 10.1080/00207160500069904
   Web of Science ®Google Scholar
• J. Savage, K. Chen, On multigrids for solving a class of improved total variation based staircasing reduction models in Image Processing Based on Partial Differential Equations Mathematics and Visualization, X. C. Tai, K. A. Lie, T. F. Chan, S. Osher, eds., Springer, Berlin, Heidelberg (2007), pp. 69–94
   Google Scholar
• D. M. Strong, P. Blomgren, T. F. Chan, Spatially adaptive local feature-driven total variation minimizing image restoration, in Proceedings of the International Society of Photo-Optical Instrumentation Engineers, Statistical and Stochastic Methods in Image Processing II, SPIE, SPIE, Vol. 3167, F. Pretnus, J. L. Davidson, E. R. Dougherty, eds., San Diego, CA (1997), pp. 222–233
   Google Scholar
• A. N. Tikhonov, Regularization of incorrectly posed problems, Soviet Math. Doklady 4 (1964), pp. 1624–1627
   Google Scholar
• C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, 2002
   Google Scholar
• Y. Wang, J. Yang, W. Yin, Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imag. Sci. 1(3) (2008), pp. 248–272 doi: 10.1137/080724265
   Web of Science ®Google Scholar
• W. Zhu, T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imag. Sci. 5 (2011), pp. 1–32 doi: 10.1137/110822268
   Web of Science ®Google Scholar