Journal of Scientific Computing

, Volume 59, Issue 2, pp 386–411 | Cite as

Variational Dynamics of Free Triple Junctions

  • Hayden Schaeffer
  • Luminita Vese


We propose a level set framework for representing triple junctions either with or without free endpoints. For triple junctions without free endpoints, our method uses two level set functions to represent the three segments that constitute the structure. For free triple junctions, we extend our method using the free curve work of Schaeffer and Vese (J Math Imaging Vis, 1–17, 2013), Smereka (Phys D Nonlinear Phenom 138(3–4):282–301, 2000). For curves moving under length minimizing flows, it is well known that the endpoints either intersect perpendicularly to the boundary, do not intersect the boundary of the domain or the curve itself (free endpoints), or meet at triple junctions. Although many of these cases can be formulated within the level set framework, the case of free triple junctions does not appear in the literature. Therefore, the proposed free triple junction formulation completes the important curve structure representations within the level set framework. We derive an evolution equation for the dynamics of the triple junction under length and area minimizing flow. The resulting system of partial differential equations are both coupled and highly non-linear, so the system is solved numerically using the Sobolev preconditioned descent. Qualitative numerical experiments are presented on various triple junction and free triple junction configurations, as well as an example with a quadruple junction instability. Quantitative results show convergence of the preconditioned algorithm to the correct solutions.


Triple junction Free endpoints Energy minimization Level set method Essentially non-oscillatory  Curve evolution 



This research was made possible by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) and by NSF DMS 1217239. The authors would like to thank Selim Esedoglu for his helpful discussion and the reviewers and editor for their useful comments. The authors would also like to thank Stanley Osher for his helpful comments.


  1. 1.
    Almgren, F., Taylor, J., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31(2), 387–438 (1993)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bar, L., Sapiro, G.: Generalized Newton-type methods for energy formulations in image processing. SIAM J. Imaging Sci. 2(2), 508 (2009)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bucur, D., Buttazzo, G., Varchon, N.: On the problem of optimal cutting. SIAM J. Optim. 13(1), 157–167 (2002)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Caselles, R., Kimmel, V., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)MATHCrossRefGoogle Scholar
  6. 6.
    Chan, T.F., Sandberg, B.Y., Vese, L.A.: Active contours without edges for vector-valued images. J. Vis. Commun. Image Represent. 11(2), 130–141 (2000)CrossRefGoogle Scholar
  7. 7.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)MATHCrossRefGoogle Scholar
  8. 8.
    Chung, G., Vese, L.A.: Energy minimization based segmentation and denoising using a multilayer level set approach. In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 439–455. Springer, Berlin (2005)Google Scholar
  9. 9.
    Cohen, L.D., Kimmel, R.: Global minimum for active contour models: a minimal path approach. Int. J. Comput. Vis. 24(1), 57–78 (1997)CrossRefGoogle Scholar
  10. 10.
    Concus, P., Golub, G.H.: Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. SIAM J. Numer. Anal. 10(6), 1103–1120 (1973)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Crandall, M.G., Evans, L.C., Gariepy, R.F.: Optimal lipschitz extensions and the infinity laplacian. Calc. Var. Partial Differ. Equ. 13(2), 123–139 (2001)MATHMathSciNetGoogle Scholar
  12. 12.
    Dal Maso, G., Toader, R.: A model for the quasi-static growth of brittle fractures based on local minimization. ArXiv Mathematics e-prints, June (2002)Google Scholar
  13. 13.
    Esedoglu, S., Ruuth, S., Tsai, R.: Diffusion generated motion using signed distance functions. J. Comput. Phys. 229(4), 1017–1042 (2010)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Esedoglu, S., Smereka, P.: A variational formulation for a level set representation of multiphase flow and area preserving curvature flow. Commun. Math. Sci. 6(1), 125–148 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL (1992)MATHGoogle Scholar
  16. 16.
    Evans, L.C., Yu, Y.: Various properties of solutions of the Infinity-Laplacian equation. Commun. Partial Differ. Equ. 30(9), 1401–1428 (2005)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Jung, M., Chung, G., Sundaramoorthi, G., Vese, L.A., Yuille, A.L.: Sobolev gradients and joint variational image segmentation, denoising, and deblurring. Proc. SPIE, 7246(1), 72460I–72460I-13 (2009)Google Scholar
  18. 18.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)CrossRefGoogle Scholar
  19. 19.
    Kimmel, R., Bruckstein, A.M.: Regularized laplacian zero crossings as optimal edge integrators. Int. J. Comput. Vis. 53, 225–243 (2001)CrossRefGoogle Scholar
  20. 20.
    Lacoste, C., Descombes, X., Zerubia, J.: Unsupervised line network extraction in remote sensing using a polyline process. Pattern Recogn. 43(4), 1631–1641 (2010)MATHCrossRefGoogle Scholar
  21. 21.
    Larsen, C.J., Richardson, C.L., Sarkis, M.: A level set method for the mumford -Shah functional and fracture. Preprint serie A, Instituto Nacional de Matemática Pura e Aplicada, Brazilian Ministry for Science and Technology (2008)Google Scholar
  22. 22.
    Li, H., X-C T.: Piecewise constant level set methods for multiphase motion. Int. J. Numer. Anal. Mod. 4(2), 291–305 (2007)Google Scholar
  23. 23.
    Lu, G., Wang, P.: Inhomogeneous infinity laplace equation. Adv. Math. 217(4), 1838–1868 (2008)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Melonakos, J., Pichon, E., Angenent, S., Tannenbaum, A.: Finsler active contours. Pattern Anal. Mach. Intell. IEEE Trans. 30(3), 412–423 (2008)Google Scholar
  25. 25.
    Merriman, B., Bence, J.K., Osher, S.J.: Motion of multiple junctions: a level set approach. J. Comput. Phys. 112(2), 334–363 (1994)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Neuberger, J.W.: Sobolev Gradients and Differential Equations. Springer, Berlin (2009)Google Scholar
  28. 28.
    Oberman, A.M.: A convergent difference scheme for the infinity laplacian: construction of absolutely minimizing lipschitz extensions. Math. Comput. 74(251), 1217–1230 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)Google Scholar
  30. 30.
    Reitich, F., Soner, H.M.: Three-phase boundary motion under constant velocities i: the vanishing surface tension limit. Proc. R. Soc. Edinb. 126A, 837–865 (1997)MathSciNetGoogle Scholar
  31. 31.
    Renka, R.J.: A simple explanation of the sobolev gradient method (2006).
  32. 32.
    Richardson, W.B.: Sobolev gradient preconditioning for image-processing PDEs. Commun. Numer. Methods Eng. 24(6), 493–504 (2006)CrossRefGoogle Scholar
  33. 33.
    Ruuth, S.J.: A diffusion-generated approach to multiphase motion. J. Comput. Phys. 145(1), 166–192 (1998)MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Ruuth, S.J.: Efficient algorithms for diffusion-generated motion by mean curvature. J. Comput. Phys. 144(2), 603–625 (1998)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Schaeffer, H.: Active arcs and contours. UCLA CAM Report, pp. 12–54 (2012)Google Scholar
  36. 36.
    Schaeffer, H., Vese, L.: Active contours with free endpoints. J. Math. Imaging Vis. (2013). doi: 10.1007/s10851-013-0437-4
  37. 37.
    Smereka, P.: Spiral crystal growth. Phys. D Nonlinear Phenom. 138(3–4), 282–301 (2000)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Smereka, P.: Semi-implicit level set methods for curvature and surface diffusion motion (english). J. Sci. Comput. 19(1–3), 439–456 (2003)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Smith, K.A., Solis, F.J., Chopp, D.: A projection method for motion of triple junctions by level sets. Interfaces Free Boundaries 4(3), 263–276 (2002)MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Sundaramoorthi, G., Yezzi, A., Mennucci, A.C.: Sobolev active contours. Int. J. Comput. Vis. 73(3), 345–366 (2007)CrossRefGoogle Scholar
  41. 41.
    Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114(1), 146–159 (1994)MATHCrossRefGoogle Scholar
  42. 42.
    Taylor, J.E.: A variational approach to crystalline triple-junction motion. J. Stat. Phys. 95, 1221–1244 (1999). doi: 10.1023/A:1004523005442 Google Scholar
  43. 43.
    Taylor, J.E.: The motion of multiple-phase junctions under prescribed phase-boundary velocities. J. Differ. Equ. 119(1), 109–136 (1995)MATHCrossRefGoogle Scholar
  44. 44.
    Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the mumford and shah model. Int. J. Comput. Vis. 50(3), 271–293 (2002)MATHCrossRefGoogle Scholar
  45. 45.
    Zhao, H.-K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127(1), 179–195 (1996)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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