On an effective multigrid solver for solving a class of variational problems with application to image segmentation

References

• L. Adams and Z. Li, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput. 24(2) (2002), pp. 463–479.
   Web of Science ®Google Scholar
• U.M. Ascher, E. Haber, and H. Huang, On effective methods for implicit piecewise smooth surface recovery, SIAM J. Sci. Comput. 28(1) (2006), pp. 339–358.
   Web of Science ®Google Scholar
• N. Badshah and K Chen, Multigrid method for the Chan-Vese model in variational segmentation, Commun. Comput. Phys. 4(2) (2008), pp. 294–316.
   Web of Science ®Google Scholar
• N. Badshah and K. Chen, On two multigrid algorithms for modeling variational multiphase image segmentation, IEEE Trans. Image Process. 18(5) (2009), pp. 1097–1106.
   PubMed Web of Science ®Google Scholar
• N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach, Commun. Comput. Phys. 7(4) (2010), pp. 759–778.
   Web of Science ®Google Scholar
• N. Badshah, K. Chen, H. Ali, and G. Murtaza, A coefficient of variation based image selective segmentation model using active contours, East Asian J. Appl. Math. 2 (2012), pp. 150–169.
   Web of Science ®Google Scholar
• R.E. Bank and D.J. Rose, Analysis of a multilevel iterative method for nonlinear finite element equations, Math. Comput. 39(160) (1982), pp. 453–465.
   Web of Science ®Google Scholar
• K.J. Brabazon, M.E. Hubbard, and P.K. Jimack, Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient, Comput. Math. Appl. 68(12, Part A) (2014), pp. 1619–1634.
   Web of Science ®Google Scholar
• D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the v-cycle, SIAM J. Numer. Anal. 20(5) (1983), pp. 967–975.
   Web of Science ®Google Scholar
• J.H. Bramble, J.E. Pasciak, J.P. Wang, and J. Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comput. 57(195) (1991), pp. 23–45.
   Web of Science ®Google Scholar
• J.H. Bramble, J.E. Pasciak, J.P. Wang, and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comput.57(195) (1991), pp. 1–21.
   Web of Science ®Google Scholar
• A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comput. 31(138) (1977), pp. 333–390.
   Web of Science ®Google Scholar
• A. Brandt, Multi-level adaptive techniques for singular-perturbation problems. NASA-TM-80966, ICASE report 78-18, 1978.
   Google Scholar
• A. Brandt, Rigorous quantitative analysis of multigrid, I. Constant coefficients two-level cycle with ℓ2-norm, SIAM J. Numer. Anal. 31(6) (1994), pp. 1695–1730.
   Web of Science ®Google Scholar
• X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran, and S. Osher, Fast global minimization of the active contour/snake model, J. Math. Imaging Vis.28(2) (2007), pp. 151–167.
   Web of Science ®Google Scholar
• W.L. Briggs, V. E. Henson, S.F. McCormick, A Multigrid Tutorial Vol. 72, SIAM, Philadelphia, USA, 2000.
   Google Scholar
• C. Brito-Loeza and K. Chen, Multigrid algorithm for high order denoising, SIAM J. Imaging Sci. 3(3) (2010), pp. 363–389.
   Web of Science ®Google Scholar
• E.S. Brown, T.F. Chan, X. Bresson, A convex relaxation method for a class of vector-valued minimization problems with applications to mumford-shah segmentation. UcCLA Cam Rep., USA 10(43) (2010), pp. 1–34. ftp://ftp.math.ucla.edu/pub/camreport/cam10-43.pdf.
   Google Scholar
• V. Caselles, R. Kimmel, and G. Sapiro, Geodesic active contours, Int. J. Comput. Vis. 22(1) (1997), pp. 61–79.
   Web of Science ®Google Scholar
• F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 29(1) (1992), pp. 182–193.
   Web of Science ®Google Scholar
• T.F. Chan and K. Chen, On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation, Numer. Algorithms 41(4) (2006), pp. 387–411.
   Web of Science ®Google Scholar
• T.F. Chan and L.A. Vese, Active contours without edges, IEEE Trans. Image Process. 10(2) (2001), pp. 266–277.
   PubMed Web of Science ®Google Scholar
• T.F. Chan, S. Esedoglu, and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math. 66(5) (2006), pp. 1632–1648.
   Web of Science ®Google Scholar
• K. Chen, Matrix Preconditioning Techniques and Applications, Cambridge University Press, Cambridge, UK, 2005.
   Google Scholar
• D. Cremers, F.R. Schmidt, F. Barthel, Shape priors in variational image segmentation: Convexity, Lipschitz continuity and globally optimal solutions. IEEE conference on Computer Vision and Pattern Recognition. CVPR 2008, IEEE, Anchorage, Alaska, USA, 2008, pp. 1–6.
   Google Scholar
• P. D'Ambra and G. Tartaglione, Solution of ambrosio-tortorelli model for image segmentation by generalized relaxation method, Commun. Nonlinear Sci. Numer. Simul. 20 (2014), pp. 819–831.
   Web of Science ®Google Scholar
• C.K. Filelis-Papadopoulos and G.A. Gravvanis, On the multigrid method based on finite difference approximate inverses, Comput. Model Eng. Sci. 90 (2013), pp. 233–253.
   Web of Science ®Google Scholar
• C. Frohn-Schauf, S. Henn, and K. Witsch, Nonlinear multigrid methods for total variation image denoising, Comput. Vis. Sci. 7(3–4) (2004), pp. 199–206.
   Google Scholar
• B. Goldluecke, D. Cremers, Convex relaxation for multilabel problems with product label spaces. European Conference on Computer Vision, Springer, Crete, Greece, 2010, pp. 225–238.
   Google Scholar
• T. Goldstein and S. Osher, The split bregman method for l1-regularized problems, SIAM J. Imaging Sci. 2(2) (2009), pp. 323–343.
   Web of Science ®Google Scholar
• D.G. Gordeziani and G.V. Meladze, Simulation of the third boundary value problem for multidimensional parabolic equations in an arbitrary domain by one-dimensional equations, USSR Comput. Math. Math. Phys. 14 (1974), pp. 249–253.
   Google Scholar
• C. Gout, C. Le Guyader, and L. Vese, Image segmentation under interpolation conditions. Preprint CAM-IPAM, 2003, p. 44.
   Google Scholar
• G.A. Gravvanis and C.K. Filelis-Papadopoulos, On the multigrid cycle strategy with approximate inverse smoothing, Eng. Comput. 31 (2014), pp. 110–122.
   Web of Science ®Google Scholar
• W. Hackbusch, Multi-grid Methods and Applications, Springer Science & Business Media, Berlin Heidelberg, Germany, 2013.
   Google Scholar
• W. Hackbusch and A. Reusken, Analysis of a damped nonlinear multilevel method, Numer. Math.55(2) (1989), pp. 225–246.
   Web of Science ®Google Scholar
• W. Hackbusch, A. Reusken, On global multigrid convergence for nonlinear problems. in Robust Multi-grid Methods, Proceedings of 4th GAMM-Seminar, Kiel 1988, W. Hackbusch, ed., Springer Vieweg Teubner Verlag, 1989, pp. 105–113.
   Google Scholar
• V.E. Henson, Multigrid methods nonlinear problems: An overview, Comput. Imaging 5016 (2003), pp. 36–48.
   Google Scholar
• P. Jaccard, The distribution of the flora in the alpine zone. 1, New Phytologist 11(2) (1912), pp. 37–50.
   Google Scholar
• A.K. Jumaat and K. Chen, An optimization-based multilevel algorithm for variational image segmentation models, Electron. Trans. Numer. Anal. 46 (2017), pp. 474–504.
   Web of Science ®Google Scholar
• T. Kim, J.E. Pasciak, and P.S. Vassilevski, Mesh-independent convergence of the modified inexact newton method for a second order non-linear problem, Numer. Linear Algebra Appl. 13(1) (2006), pp. 23–47.
   Web of Science ®Google Scholar
• M. Klodt, F. Steinbrücker, D. Cremers, Moment constraints in convex optimization for segmentation and tracking. Adv. Topics Comput. Vis., Advances in Computer Vision and Pattern Recognition (2013), pp. 1–29.
   Google Scholar
• C. Le Guyader and C. Gout, Geodesic active contour under geometrical conditions: Theory and 3D applications, Numer. Algorithms 48(1–3) (2008), pp. 105–133.
   Web of Science ®Google Scholar
• R.J. Leveque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31(4) (1994), pp. 1019–1044.
   Web of Science ®Google Scholar
• C. Liu, M.K.-P. Ng, T. Zeng, Weighted variational model for selective image segmentation with application to medical images. Pattern Recognit. 76 (2018), pp. 367–379.
   Web of Science ®Google Scholar
• T. Lu, P. Neittaanmäki, and X-C. Tai, A parallel splitting up method and its application to navier-stokes equations, Appl. Math. Lett. 4(2) (1991), pp. 25–29.
   Web of Science ®Google Scholar
• S.P. MacLachlan and C.W. Oosterlee, Local fourier analysis for multigrid with overlapping smoothers applied to systems of pdes, Numer. Linear Algebra Appl. 18(4) (2011), pp. 751–774.
   Web of Science ®Google Scholar
• A. Napov and Y. Notay, Smoothing factor, order of prolongation and actual multigrid convergence, Numer. Math. 118(3) (2011), pp. 457–483.
   Web of Science ®Google Scholar
• T.N.A. Nguyen, J. Cai, J. Zhang, and J. Zheng, Robust interactive image segmentation using convex active contours, IEEE Trans. Image Process 21(8) (2012), pp. 3734–43.
   PubMed Web of Science ®Google Scholar
• P. Ochs, T. Brox, Object segmentation in video: A hierarchical variational approach for turning point trajectories into dense regions. International Conference on Computer Vision, Barcelona, 2011, pp. 1583–1590. doi:10.1109/ICCV.2011.6126418.
   Google Scholar
• S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79(1) (1988), pp. 12–49.
   Web of Science ®Google Scholar
• P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12(7) (1990), pp. 629–639.
   Web of Science ®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. Rada and K. Chen, Improved selective segmentation model using one level-set, J. Algorithms Comput. Technol. 7(4) (2013), pp. 509–540.
   Google Scholar
• A. Reusken, Convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems, Numer. Math. 52(3) (1987), pp. 251–277.
   Web of Science ®Google Scholar
• A. Reusken, Convergence of the multilevel full approximation scheme including thev-cycle, Numer. Math. 53(6) (1988), pp. 663–686.
   Web of Science ®Google Scholar
• M. Roberts, Selective image segmentation models and fast multigrid methods. PhD thesis, University of Liverpool, 2019.
   Google Scholar
• M. Roberts and J. Spencer, Chan-Vese reformulation for selective image segmentation, J. Math. Imaging Vis. (2019), To appear.
   PubMed Web of Science ®Google Scholar
• M. Roberts, K. Chen, K.L. Irion, Multigrid algorithm based on hybrid smoothers for variational and selective segmentation models. Int. J. Comput. Math. 96(8) (2019), pp. 1623–1647. doi:10.1080/00207160.2018.1494827.
   Web of Science ®Google Scholar
• M. Roberts, K. Chen, and K.L. Irion, A convex geodesic selective model for image segmentation, J. Math. Imaging Vis. 61(4) (2019), pp. 482–503.
   Web of Science ®Google Scholar
• V.V. Shaidurov, Multigrid Methods for Finite Elements, Springer Science & Business Media, Dordrecht, Netherlands, 2013.
   Google Scholar
• J. Spencer and K. Chen, A convex and selective variational model for image segmentation, Commun. Math. Sci. 13(6) (2015), pp. 1453–1472.
   Web of Science ®Google Scholar
• C. Tai, X. Zhang, and Z. Shen, Wavelet frame based multiphase image segmentation, SIAM. J. Imaging Sci. 6(4) (2013), pp. 2521–2546.
   Web of Science ®Google Scholar
• U. Trottenberg, C.W. Oosterlee, A. Schüller, Multigrid, Academic Press, London, UK, 2001.
   Google Scholar
• L.A. Vese and T.F. Chan, A multiphase level set framework for image segmentation using the MS model, Int. J. Comput. Vis. 50(3) (2002), pp. 271–293.
   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.
   PubMed Web of Science ®Google Scholar
• R. Wienands, W. Joppich, Practical Fourier Analysis for Multigrid Methods, CRC press, Boca Raton, Florida, USA, 2004.
   Google Scholar
• D. Xie, New nonlinear multigrid analysis. NASA Conference Publication 3339, Proceedings of Copper Mountain Conference on Multigrid Methods, N. D. Melson, T. A. Manteuffel, S. F. McCormick and C. C. Douglas, eds., Copper Mountain, Colorado, USA, 1996, pp. 793–808. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.38.4442&rep=rep1&type=pdf.
   Google Scholar
• J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34(4) (1992), pp. 581–613.
   Web of Science ®Google Scholar
• J. Zhang, K. Chen, and D.A. Gould, A fast algorithm for automatic segmentation and extraction of a single object by active surfaces, Int. J. Comput. Math.92(6) (2015), pp. 1251–1274.
   Web of Science ®Google Scholar
• W. Zhu, X.-C. Tai, and T.F. Chan, Image segmentation using euler's elastica as the regularization, J. Sci. Comput. 57 (2013), pp. 414–438.
   Web of Science ®Google Scholar