- Volume 18 Issue 1
DOI QR Code
A NUMERICAL METHOD FOR THE MODIFIED VECTOR-VALUED ALLEN-CAHN PHASE-FIELD MODEL AND ITS APPLICATION TO MULTIPHASE IMAGE SEGMENTATION
- Lee, Hyun Geun (INSTITUTE OF MATHEMATICAL SCIENCES, EWHA WOMANS UNIVERSITY) ;
- Lee, June-Yub (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY)
- Received : 2013.10.31
- Accepted : 2013.11.15
- Published : 2014.03.25
In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen-Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for the vector-valued Allen-Cahn phase-field equation. We split the modified vector-valued Allen-Cahn equation into a nonlinear equation and a linear diffusion equation with a source term. The linear diffusion equation is discretized using an implicit scheme and the resulting implicit discrete system of equations is solved by a multigrid method. The nonlinear equation is solved semi-analytically using a closed-form solution. And by treating the source term of the linear diffusion equation explicitly, we solve the modified vector-valued Allen-Cahn equation in a decoupled way. By decoupling the governing equation, we can speed up the segmentation process with multiple phases. We perform some characteristic numerical experiments for multiphase image segmentation.
Supported by : National Research Foundation of Korea(NRF)
- S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095. https://doi.org/10.1016/0001-6160(79)90196-2
- A. A. Wheeler, W. J. Boettinger and G. B. McFadden, Phase-field model for isothermal phase transitions in binary alloys, Physical Review A, 45 (1992), 7424-7439. https://doi.org/10.1103/PhysRevA.45.7424
- M. Cheng and J. A. Warren, An efficient algorithm for solving the phase field crystal model, Journal of Computational Physics, 227 (2008), 6241-6248. https://doi.org/10.1016/j.jcp.2008.03.012
- Y. Li, H. G. Lee and J. Kim, A fast, robust, and accurate operator splitting method for phase-field simulations of crystal growth, Journal of Crystal Growth, 321 (2011), 176-182. https://doi.org/10.1016/j.jcrysgro.2011.02.042
- Y. Li, D. Lee, H. G. Lee, D. Jeong, C. Lee, D. Yang and J. Kim, A robust and accurate phase-field simulation of snow crystal growth, Journal of the Korean Society for Industrial and Applied Mathematics, 16 (2012), 15-29. https://doi.org/10.12941/jksiam.2012.16.1.015
- L.-Q. Chen and W. Yang, Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: The grain-growth kinetics, Physical Review B, 50 (1994), 15752-15756. https://doi.org/10.1103/PhysRevB.50.15752
- I. Steinbach, F. Pezzolla, B. Nestler, M. SeeSSelberg, R. Prieler, G. J. Schmitz and J. L. L. Rezende, A phase field concept for multiphase systems, Physica D, 94 (1996), 135-147. https://doi.org/10.1016/0167-2789(95)00298-7
- D. Fan, C. Geng and L.-Q. Chen, Computer simulation of topological evolution in 2-D grain growth using a continuum diffuse-interface field model, Acta Materialia, 45 (1997), 1115-1126. https://doi.org/10.1016/S1359-6454(96)00221-2
- M. T. Lusk, A phase-field paradigm for grain growth and recrystallization, Proceedings of the Royal Society of London A, 455 (1999), 677-700. https://doi.org/10.1098/rspa.1999.0329
- R. Kobayashi, J. A.Warren andW. C. Carter, A continuum model of grain boundaries, Physica D, 140 (2000), 141-150. https://doi.org/10.1016/S0167-2789(00)00023-3
- M. Benes, V. Chalupecky and K. Mikula, Geometrical image segmentation by the Allen-Cahn equation, Applied Numerical Mathematics, 51 (2004), 187-205. https://doi.org/10.1016/j.apnum.2004.05.001
- J. A. Dobrosotskaya and A. L. Bertozzi, A wavelet-Laplace variational technique for image deconvolution and inpainting, IEEE Transactions on Image Processing, 17 (2008), 657-663. https://doi.org/10.1109/TIP.2008.919367
- L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Communications on Pure and Applied Mathematics, 45 (1992), 1097-1123. https://doi.org/10.1002/cpa.3160450903
- T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, Journal of Differential Geometry, 38 (1993), 417-461. https://doi.org/10.4310/jdg/1214454300
- M. Katsoulakis, G. T. Kossioris and F. Reitich, Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions, The Journal of Geometric Analysis, 5 (1995), 255-279. https://doi.org/10.1007/BF02921677
- L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Computer Physics Communications, 108 (1998), 147-158. https://doi.org/10.1016/S0010-4655(97)00115-X
- M. Benes and K. Mikula, Simulation of anisotropic motion by mean curvature-comparison of phase-field and sharp-interface approaches, Acta Mathematica Universitatis Comenianae, 67 (1998), 17-42.
- X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numerische Mathematik, 94 (2003), 33-65. https://doi.org/10.1007/s00211-002-0413-1
- T. Ohtsuka, Motion of interfaces by an Allen-Cahn type equation with multiple-well potentials, Asymptotic Analysis, 56 (2008), 87-123.
- X. Yang, J. J. Feng, C. Liu and J. Shen, Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, Journal of Computational Physics, 218 (2006), 417-428. https://doi.org/10.1016/j.jcp.2006.02.021
- Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, Journal of Computational Physics, 198 (2004), 450-468. https://doi.org/10.1016/j.jcp.2004.01.029
- J. A. Sethian and J. Strain, Crystal growth and dendritic solidification, Journal of Computational Physics, 98 (1992), 231-253. https://doi.org/10.1016/0021-9991(92)90140-T
- S. Li, J. S. Lowengrub and P. H. Leo, Nonlinear morphological control of growing crystals, Physica D, 208 (2005), 209-219. https://doi.org/10.1016/j.physd.2005.06.021
- D. Li, R. Li and P. Zhang, A cellular automaton technique for modelling of a binary dendritic growth with convection, Applied Mathematical Modelling, 31 (2007), 971-982. https://doi.org/10.1016/j.apm.2006.04.004
- H. Yin and S. D. Felicelli, A cellular automaton model for dendrite growth in magnesium alloy AZ91, Modelling and Simulation in Materials Science and Engineering, 17 (2009), 075011. https://doi.org/10.1088/0965-0393/17/7/075011
- D. Juric and G. Tryggvason, A front-tracking method for dendritic solidification, Journal of Computational Physics, 123 (1996), 127-148. https://doi.org/10.1006/jcph.1996.0011
- D. Stafford, M. J. Ward and B. Wetton, The dynamics of drops and attached interfaces for the constrained Allen-Cahn equation, European Journal of Applied Mathematics, 12 (2001), 1-24.
- S. Chen, B. Merriman, S. Osher and P. Smereka, A simple level set method for solving Stefan problems, Journal of Computational Physics, 135 (1997), 8-29. https://doi.org/10.1006/jcph.1997.5721
- L.-L. Wang and Y. Gu, Efficient dual algorithms for image segmentation using TV-Allen-Cahn type models, Communications in Computational Physics, 9 (2011), 859-877. https://doi.org/10.4208/cicp.221109.290710a
- Z. Xu, H. Huang, X. Li and P. Meakin, Phase field and level set methods for modeling solute precipitation and/or dissolution, Computer Physics Communications, 183 (2012), 15-19. https://doi.org/10.1016/j.cpc.2011.08.005
- A. E. Lobkovsky and J. A.Warren, Phase-field model of crystal grains, Journal of Crystal Growth, 225 (2001), 282-288. https://doi.org/10.1016/S0022-0248(01)00867-3
- B. Nestler and A. A. Wheeler, Phase-field modeling of multi-phase solidification, Computer Physics Communications, 147 (2002), 230-233. https://doi.org/10.1016/S0010-4655(02)00252-7
- J.-W. Choi, H. G. Lee, D. Jeong and J. Kim, An unconditionally gradient stable numerical method for solving the Allen-Cahn equation, Physica A, 388 (2009), 1791-1803. https://doi.org/10.1016/j.physa.2009.01.026
- Y. Li, H. G. Lee, D. Jeong and J. Kim, An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation, Computers and Mathematics with Applications, 60 (2010), 1591-1606. https://doi.org/10.1016/j.camwa.2010.06.041
- Y. Li and J. Kim, Multiphase image segmentation using a phase-field model, Computers & Mathematics with Applications, 62 (2011), 737-745. https://doi.org/10.1016/j.camwa.2011.05.054
- H. Garcke, B. Nestler and B. Stoth, On anisotropic order parameter models for multi-phase systems and their sharp interface limits, Physica D, 115 (1998), 87-108. https://doi.org/10.1016/S0167-2789(97)00227-3
- H. Garcke, B. Nestler and B. Stinner, A diffuse interface model for alloys with multiple components and phases, SIAM Journal on Applied Mathematics, 64 (2004), 775-799. https://doi.org/10.1137/S0036139902413143
- H. Garcke and V. Styles, Bi-directional diffusion induced grain boundary motion with triple junctions, Interfaces and Free Boundaries, 6 (2004), 271-294.
- B. Nestler, H. Garcke and B. Stinner, Multicomponent alloy solidification: Phase-field modeling and simulations, Physical Review E, 71 (2005), 041609. https://doi.org/10.1103/PhysRevE.71.041609
- N. Moelans, B. Blanpain and P. Wollants, A phase field model for the simulation of grain growth in materials containing finely dispersed incoherent second-phase particles, Acta Materialia, 53 (2005), 1771-1781. https://doi.org/10.1016/j.actamat.2004.12.026
- R. Kornhuber and R. Krause, Robust multigrid methods for vector-valued Allen-Cahn equations with logarithmic free energy, Computing and Visualization in Science, 9 (2006), 103-116. https://doi.org/10.1007/s00791-006-0020-2
- D. A. Kay and A. Tomasi, Color image segmentation by the vector-valued Allen-Cahn phase-field model: a multigrid solution, IEEE Transactions on Image Processing, 18 (2009), 2330-2339. https://doi.org/10.1109/TIP.2009.2026678
- L. Vanherpe, F. Wendler, B. Nestler and S. Vandewalle, A multigrid solver for phase field simulation of microstructure evolution, Mathematics and Computers in Simulation, 80 (2010), 1438-1448. https://doi.org/10.1016/j.matcom.2009.10.007
- H. G. Lee and J. Kim, An efficient and accurate numerical algorithm for the vector-valued Allen-Cahn equations, Computer Physics Communications, 183 (2012), 2107-2115. https://doi.org/10.1016/j.cpc.2012.05.013
- L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, International Journal of Computer Vision, 50 (2002), 271-293. https://doi.org/10.1023/A:1020874308076
- C. Samson, L. Blanc-Feraud, G. Aubert and J. Zerubia, A level set model for image classification, International Journal of Computer Vision, 40 (2000), 187-197. https://doi.org/10.1023/A:1008183109594
- J. Lie, M. Lysaker and X.-C. Tai, A variant of the level set method and applications to image segmentation, Mathematics of computation, 75 (2006), 1155-1174. https://doi.org/10.1090/S0025-5718-06-01835-7
- Y. M. Jung, S. H. Kang and J. Shen, Multiphase image segmentation via Modica-Mortola phase transition, SIAM Journal on Applied Mathematics, 67 (2007), 1213-1232. https://doi.org/10.1137/060662708
- W. L. Briggs, A Multigrid Tutorial, SIAM, Philadelphia, PA, 1987.
- U. Trottenberg, C. Oosterlee and A. Schuller, Multigrid, Academic Press, London, 2001.
- D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685. https://doi.org/10.1002/cpa.3160420503
- S. C. Zhu and A. Yuille, Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 18 (1996), 884-900. https://doi.org/10.1109/34.537343
- M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331. https://doi.org/10.1007/BF00133570
- N. Paragios and R. Deriche, Geodesic active regions for supervised texture segmentation, The Proceedings of the Seventh IEEE International Conference on Computer Vision, Corfu, Greece 1999.
- T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277. https://doi.org/10.1109/83.902291