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Section B

A 3D multi-grid algorithm for the Chan–Vese model of variational image segmentation

, &
Pages 160-189 | Received 18 Oct 2010, Accepted 11 Oct 2011, Published online: 06 Dec 2011

References

  • Ambrosio , L. and Tortorelli , V. M. 1990 . Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence . Comm. Pure Appl. Math , 43 : 999 – 1036 .
  • Aubert , G. and Kornprobst , P. 2002 . Mathematical Problems in Image Processing , New York : Springer .
  • Badshah , N. and Chen , K. 2008 . Multigrid method for the Chan–Vese model in variational segmentation . Commun. Comput. Phys , 4 ( 2 ) : 294 – 316 .
  • Badshah , N. and Chen , K. 2009 . On two multigrid algorithms for modeling variational multiphase image segmentation . IEEE Trans. Image Process , 18 ( 5 ) : 1097 – 1106 .
  • Badshah , N. and Chen , K. 2010 . Image selective segmentation under geometrical constraints using an active contour approach . Commun. Comput. Phys , 7 ( 4 ) : 759 – 778 .
  • Brandt , A. 1977 . Multi-level adaptive solution to boundary-value problems . Math. Comp , 31 : 333 – 390 .
  • Brandt , A. 1981 . Multigrid Solvers for Non-elliptic and Singular-perturbation Steady-state Problems , Rehovot, , Israel : Weizmann Institute of Science .
  • Brandt , A. 1994 . Rigorous local quantitative analysis of multigrid: I. Constant coefficients two level cycle with L2 norm . SIAM J. Numer. Anal , 31 : 1695 – 1730 .
  • Bresson , X. , Esedoglu , S. , Vandergheynst , P. , Thiran , J. P. and Osher , S. 2007 . Global minimizers of the active contour/snake model . J. Math. Imaging Vision , 28 ( 2 ) : 151 – 167 .
  • Brown , E. S. , Chan , T. F. and Bresson , X. “ Completely convex formulation of the Chan-Vese image segmentation model ” . Tech. Rep. CAM 10-44, Department of Mathematics, University of California, Los Angeles, USA, 2010
  • Caselles , V. , Kimmel , R. , Sapiro , G. and Sbert , C. 1997 . Minimal surfaces based object segmentation . IEEE Trans. Pattern Anal. Mach. Intell , 19 : 394 – 398 .
  • Chan , T. F. and Shen , J. H. 2005 . Image Processing and Analysis – Variational PDE, Wavelet, and Stochastic Methods , Philadelphia : SIAM Publications .
  • Chan , T. F. and Vese , L. A. 2001 . Active contours without edges . IEEE Trans. Image Process , 10 ( 2 ) : 266 – 277 .
  • Chan , T. F. and Vese , L. A. 2002 . “ Active contour and segmentation models using geometric PDE's for medical imaging ” . In Geometric Methods in Bio-medical Image Processing , Edited by: Malladi , R. 63 – 75 . New York : Springer . Mathematics and Visualization
  • Chang , Q. S. , Tai , X.-C. and Xing , L. 2009 . A compound algorithm of denoising using second-order and fourth-order partial differential equations . Numer. Math. Theor. Method Appl , 2 ( 4 ) : 353 – 376 .
  • Chen , K. 2005 . Matrix Preconditioning Techniques and Applications , Cambridge : Cambridge University Press .
  • Chen , K. and Tai , X. C. 2007 . A nonlinear multigrid method for total variation minimization from image restoration . J. Sci. Comput , 33 ( 2 ) : 115 – 138 .
  • Cobzas , D. , Birkbeck , N. , Schmidt , M. , Jagersand , M. and Murtha , A. 3D variational brain tumor segmentation using a high dimensional feature set . IEEE 11th International Conference on Computer Vision, ICCV 2007, Rio de Janeiro, Brazil . pp. 1 – 8 . New York : IEEE .
  • Goldenberg , R. , Kimmel , R. , Rivlin , E. and Rudzsky , M. 2001 . Fast geodesic active contoers . IEEE Trans. Image Process , 10 ( 10 ) : 1467 – 1475 .
  • Goldstein , R. and Osher , S. 2009 . The split Bregman method for L1-regularized problems . SIAM J. Imaging Sci , 2 ( 2 ) : 323 – 343 .
  • Goldstein , R. , Bresson , X. and Osher , S. 2010 . Geometric applications of the split Bregman method: Segmentaion and surface reconstruction . J. Sci. Comput , 45 ( 1–3 ) : 272 – 293 .
  • Haralick , R. 1984 . Digital step edges from zero crossing of second directional derivatives . IEEE Trans. Pattern Anal. Mach. Intell , 6 ( 1 ) : 58 – 68 .
  • Kass , M. , Witkin , A. and Terzopoulos , D. 1987 . Snake: Active contour models . Int. J. Comput. Vis , 1 ( 4 ) : 321 – 331 .
  • Kim , J. , Fisher , J. , Yezzi , A. , Cetin , M. and Willsky , A. 2005 . A nonparametric statistical method for image segmentation using information theory and curve evolution . IEEE Trans. Image Process , 14 : 1486 – 1502 .
  • Kimmel , R. 2003 . “ Fast edge integration ” . In Geometric Level Set Methods in Imaging, Vision and Graphics , Edited by: Osher , S. and Paragios , N. 59 – 77 . Berlin : Springer Verlag .
  • Kimmel , R. 2003 . Geometric Segmentation of 3D Structures . Proceedings of the 2003 International Conference on Image Processing (ICIP 2003) . September 14–17 2003 , Barcelona, Catalonia, Spain. Vol. 2 , pp. 639 – 642 .
  • Lu , T. , Neittaanmäki , P. and Tai , X.-C. 1991 . A parallel splitting up method and its application to Navier–Stokes equations . Appl. Math. Lett , 4 ( 2 ) : 25 – 29 .
  • Marr , D. and Hildreth , E. 1988 . Theory of edge detection . Proc. R. Soc. Lond. Ser. B , 207 : 187 – 217 .
  • Mumford , D. and Shah , J. 1989 . Optimal approximation by piecewise smooth functions and associated variational problem . Comm. Pure Appl. Math , 42 : 577 – 685 .
  • Osher , S. and Fedkiw , R. 2003 . Level Set Methods and Dynamic Implicit Surfaces , New York : Springer-Verlag .
  • Osher , S. and Sethian , J. A. 1989 . Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations . J. Comput. Phys , 79 : 12 – 49 .
  • Rudin , L. , Osher , S. and Fatemi , E. 1992 . Nonlinear total variation based noise removal algorithms . Phys. D , 60 : 259 – 268 .
  • Scherzer , O. , Grasmair , M. , Grossauer , H. , Haltmeier , M. and Lenzen , F. 2009 . Variational Methods in Imaging , New York : Springer-Verlag .
  • Shen , J. H. 2005 . Γ-Convergence approximation to piecewise constant Mumford-Shah segmentation . Adv. Concepts Intell. Vis. Syst , 3708 : 499 – 506 .
  • Strang , G. 1963 . Accurate partial differrence methods I: Linear Cauchy problems . Arch. Ration. Mech. Anal , 12 ( 1 ) : 392 – 402 .
  • Tao , W. B. and Tai , X.-C. “ Multiple piecewise constant active contours for image segmentation using graph cuts optimization ” . Tech. Rep. CAM 09-13, Department of Mathematics, University of California, Los Angeles, USA, 2009
  • Trottenberg , U. , Oostelee , C. W. and Schuller , A. 2001 . Multigrid , San Diego, CA : Academic Press .
  • Vese , L. A. and Chan , T. F. 2002 . A multiphase level set framework for image segmentation using the Mumford and Shah model . Int. J. Comput. Vis , 50 ( 3 ) : 271 – 293 .
  • Weickert , J. , Romeny , B. M. and Viergever , M. A. 1998 . Efficient and reliable scheme for nonlinear diffusion filtering . IEEE Trans. Image Process , 7 : 398 – 410 .
  • Wienands , R. and Joppich , W. 2005 . Practical Fourier Analysis for Multigrid Method , Boca Raton, FL : Chapman & Hall/CRC .
  • Xu , C. Y. and Prince , J. L. 1998 . Snakes, shapes, and gradient vector flow . IEEE Trans. Image Process , 7 ( 3 ) : 359 – 369 .
  • Yuan , J. , Bae , E. , Tai , X.-C. and Boykov , Y. 2010 . “ A study on continuous max-flow and min-cut approaches ” . Tech. Rep. CAM 10-61, Department of Mathematics, University of California, Los Angeles, USA
  • Zhu , S. C. and Yuille , A. 1996 . Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation . IEEE Trans. PAMI , 18 : 884 – 900 .

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