A restarted iterative homotopy analysis method for two nonlinear models from image processing

References

• S. Abbasbandy, Solitary wave solutions to the Kuramoto–Sivashinsky equation by means of the homotopy analysis method, Nonlinear Dyn. 52 (2008), pp. 35–40. doi: 10.1007/s11071-007-9255-9
   Web of Science ®Google Scholar
• S. Abbasbandy and F. Zakaria, Soliton solutions for the fifth-order kdv equation with the homotopy analysis method, Nonlinear Dyn. 51(1) (2007), pp. 83–87. doi: 10.1007/s11071-006-9193-y
   Web of Science ®Google Scholar
• G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Comput. Math. Appl. 21 (1991), pp. 101–127. doi: 10.1016/0898-1221(91)90220-X
   Web of Science ®Google Scholar
• I.V. Andrianov and V.V. Danishevky, Asymptotic approach for non-linear periodical vibrations of continuous structures, J. Sound Vibration 249(3) (2002), pp. 465–481. doi: 10.1006/jsvi.2001.3878
   Web of Science ®Google Scholar
• N. Badshah and K. Chen, Multigrid method for the Chan–Vese model in variational segmentation, CiCP 4(2) (2008), pp. 294–316.
   Google Scholar
• C. Brito-Loeza and K. Chen, Multigrid algorithm for high order denoising, SIAM J. Imaging Sci. 3(3) (2010), pp. 363–389. doi: 10.1137/080737903
   Web of Science ®Google Scholar
• C. Brito-Loeza and K. Chen, On high-order denoising models and fast algorithms for vector-valued images, IEEE Trans. Image Proc. 19 (2010), pp. 1518–1527. doi: 10.1109/TIP.2010.2042655
   PubMed Web of Science ®Google Scholar
• V. Caselles, R. Kimmel, and G. Sapiro, Geodesic active contours, Proceedings of the 5th International Conference on Computer Vision, Boston, MA, 1995, pp. 694–699.
   Google Scholar
• V. Caselles, R. Kimmel, and G. Sapiro, Geodesic active contours, Int. J. Comput. Vis. 22(1) (1997), pp. 61–79. doi: 10.1023/A:1007979827043
   Web of Science ®Google Scholar
• A. Chambolle and P.L. Lions, Image recovery via total variation minimization and related problems, Numer. Math. 76 (1997), pp. 167–188. doi: 10.1007/s002110050258
   Web of Science ®Google Scholar
• R.H. Chan and 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
• T. Chan and L. Vese. An active contour model without edges, International Conference on Scale-Space Theories in Computer Vision, Greece, 1999, pp. 141–151.
   Google Scholar
• T.F. Chan and L.A. Vese, Active contours without edges, IEEE Trans. Image Process. 10(2) (2001), pp. 266–277. doi: 10.1109/83.902291
   PubMed Web of Science ®Google Scholar
• T.F. Chan, G.H. Golub, and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput. 20 (1999), pp. 1964–1977. doi: 10.1137/S1064827596299767
   Web of Science ®Google Scholar
• T.F. Chan, B.Y. Sandberg, and L.A. Vese, Active contours without edges for vector-valued images, J. Visual Commun. Image Represent. 11 (2000), pp. 130–141. doi: 10.1006/jvci.1999.0442
   Web of Science ®Google Scholar
• N. Chumchob, K. Chen, and C. Brito-Loeza, A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation, Int. J. Comput. Math. 90(1) (2013), pp. 140–161. doi: 10.1080/00207160.2012.709625
   Web of Science ®Google Scholar
• M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Equations 26(2) (2010), pp. 448–479.
   Web of Science ®Google Scholar
• T. Hayat, N. Ahmed, M. Sajid, and S. Asghar, On the MHD flow of a second grade fluid in a porous channel, Comput. Math. Appl. 54(3) (2007), pp. 407–414. doi: 10.1016/j.camwa.2006.12.036
   Web of Science ®Google Scholar
• M. Hintermuller and W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford–Shah functional, J. Math. Imaging Vis. 20 (2004), pp. 19–42. doi: 10.1023/B:JMIV.0000011317.13643.3a
   Web of Science ®Google Scholar
• H. Jafari and S. Seifi. Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. 14(5) (2009), pp. 1962–1969. doi: 10.1016/j.cnsns.2008.06.019
   Web of Science ®Google Scholar
• M. Kass, A. Witkin, and D. Terzopoulos, Snakes: Active contour models, Int. J. Comput. Vis. 1(4) (1988), pp. 321–331. doi: 10.1007/BF00133570
   Web of Science ®Google Scholar
• S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
   Google Scholar
• S.J. Liao, A challenging nonlinear problem for numerical techniques, J. Comput. Appl. Math. 181 (2005), pp. 467–472. doi: 10.1016/j.cam.2004.11.039
   Web of Science ®Google Scholar
• S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, FL, 2003.
   Google Scholar
• S.J. Liao. On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147(2) (2004), pp. 499–513. doi: 10.1016/S0096-3003(02)00790-7
   Web of Science ®Google Scholar
• S.J. Liao, A new branch of solutions of boundary-layer flows over a permeable stretching plate, Int. J. Non-Linear Mech. 42(6) (2007), pp. 819–830. doi: 10.1016/j.ijnonlinmec.2007.03.007
   Web of Science ®Google Scholar
• S.J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul. 14(4) (2009), pp. 983–997. doi: 10.1016/j.cnsns.2008.04.013
   Web of Science ®Google Scholar
• S.J. Liao, On the relationship between the homotopy analysis method and Euler transform, Commun. Nonlinear Sci. Numer. Simul. 15(6) (2010), pp. 1421–1431. doi: 10.1016/j.cnsns.2009.06.008
   Web of Science ®Google Scholar
• S.J. Liao and Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math. 119 (2007), pp. 297–355. doi: 10.1111/j.1467-9590.2007.00387.x
   Web of Science ®Google Scholar
• M. Lysaker and X.C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vis. 66 (2006), pp. 5–18. doi: 10.1007/s11263-005-3219-7
   Web of Science ®Google Scholar
• M. Lysaker, S. Osher, and X.C. Tai, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process. 13(10) (2004), pp. 1345–1357. doi: 10.1109/TIP.2004.834662
   PubMed Web of Science ®Google Scholar
• A. Marquina and S. Osher, Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J. Sci. Comput. 22 (2000), pp. 387–405. doi: 10.1137/S1064827599351751
   Web of Science ®Google Scholar
• D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math. 42 (1989), pp. 577–685. doi: 10.1002/cpa.3160420503
   Web of Science ®Google Scholar
• J.A. Murdock, Perturbations: Theory and Methods, John Wiley and Sons, New York, 1991.
   Google Scholar
• A.H. Nayfeh, Perturbations Methods, John Wiley and Sons, New York, 2000.
   Google Scholar
• L. Rada and K. Chen, A new variational model with dual level set functions for selective segmentation, Commun. Comput. Phys. 12(1) (2012), pp. 261–283.
   Web of Science ®Google Scholar
• L.I. Rudin and S. Osher, Total variation based image restoration with free local constraints, Proceedings ICIP, IEEE International Conference on Image Processing, Austin, TX, Vol. 1, 1994, pp. 31–35.
   Google Scholar
• L.I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), pp. 259–268. doi: 10.1016/0167-2789(92)90242-F
   Web of Science ®Google Scholar
• M. Sajid, T. Hayat, and S. Asghar, Non-similar solution for the axisymmetric flow of a third-grade fluid over a radially stretching sheet, Acta Mech. 189 (2007), pp. 193–205. doi: 10.1007/s00707-006-0430-8
   Web of Science ®Google Scholar
• Jo. Savage and K. Chen, An improved and accelerated non-linear multigrid method for total-variation denoising, Int. J. Comput. Math. 82(8) (2005), pp. 1001–1015. doi: 10.1080/00207160500069904
   Web of Science ®Google Scholar
• J. Savage and 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. Chan, and S. Osher, eds., Springer, Berlin, Heidelberg, 2007, pp. 69–94.
   Google Scholar
• J.A. Sethian. Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science, Cambridge University Press, Cambridge, 1996.
   Google Scholar
• A. Tsai, A.J. YezziJr., and A.S. Willsky, Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, IEEE Trans. Image Process. 10(8) (2001), pp. 1169–1186. doi: 10.1109/83.935033
   PubMed Web of Science ®Google Scholar
• Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process. 9(10) (2000), pp. 1723–1730. doi: 10.1109/83.869184
   PubMed Web of Science ®Google Scholar
• L.A. Vese, Variational Methods in Image Processing, Chapman & Hall, CRC Press, Boca Raton, FL, 2012.
   Google Scholar
• L.A. Vese and T.F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Comput. Vis. 50(3) (2002), pp. 271–293. doi: 10.1023/A:1020874308076
   Web of Science ®Google Scholar
• C.R. Vogel, A multigrid method for total variation-based image denoising, in Computation and Control IV, Birkhauser, Boston, MA, 1995, pp. 323–331.
   Google Scholar
• C.R. Vogel and M.E. Oman, Iterative methods for total variation denoising, SIAM J. Sci. Comput. 17 (1996), pp. 227–238. doi: 10.1137/0917016
   Web of Science ®Google Scholar
• J. Weickert, B.M. Ter Haar Romeny, and M.A. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Process. 7(3) (1998), pp. 398–410. doi: 10.1109/83.661190
   PubMed Web of Science ®Google Scholar
• J. Weickert, B.M. Romeny, and M.A. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Process. 7(3) (1998), pp. 398–410. doi: 10.1109/83.661190
   PubMed Web of Science ®Google Scholar
• F. Yang, K. Chen, and B. Yu, Homotopy method for a mean curvature-based denoising model, Appl. Numer. Math. 62(3) (2012), pp. 185–200. doi: 10.1016/j.apnum.2011.12.001
   Web of Science ®Google Scholar
• S.-P. Zhu, An exact and explicit solution for the valuation of American put options, Quant. Finance 6(3) (2006), pp. 229–242. doi: 10.1080/14697680600699811
   Web of Science ®Google Scholar
• W. Zhu and T.F. Chan, Image denoising using mean curvature, Unpublished article, 2008. Available at http://www.math.nyu.edu/~wzhu/.
   Google Scholar
• W. Zhu and T.F. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci. 5 (2012), pp. 1–32. doi: 10.1137/110822268
   Web of Science ®Google Scholar
• H. Zhu, H. Shu, and M. Ding, Numerical solutions of partial differential equations by discrete homotopy analysis method, Appl. Math. Comput. 216(12) (2010), pp. 3592–3605. doi: 10.1016/j.amc.2010.05.005
   Web of Science ®Google Scholar