Skip to main content
SpringerLink
Log in

Adaptive wavelet collocation methods for image segmentation using TV–Allen–Cahn type models

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

An adaptive wavelet-based method is proposed for solving TV(total variation)–Allen–Cahn type models for multi-phase image segmentation. The adaptive algorithm integrates (i) grid adaptation based on a threshold of the sparse wavelet representation of the locally-structured solution; and (ii) effective finite difference on irregular stencils. The compactly supported interpolating-type wavelets enjoy very fast wavelet transforms, and act as a piecewise constant function filter. These lead to fairly sparse computational grids, and relax the stiffness of the nonlinear PDEs. Equipped with this algorithm, the proposed sharp interface model becomes very effective for multi-phase image segmentation. This method is also applied to image restoration and similar advantages are observed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allen, S.M., Cahn, J.W.: A microscope theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27(6), 1085–1095 (1979)

    Article  Google Scholar 

  2. Alpert, B., Beylkin, G., Gines, D., Vozovoi, L.: Adaptive solution of partial differential equations in multiwavelet bases. J. Comput. Phys. 182(1), 149–190 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bae, E., Tai, X.C.: Graph cuts for the multiphase Mumford–Shah model using piecewise constant level set methods. UCLA CAM Report 08, 36 (2008)

  4. Bae, E., Tai, X.C.: Graph cut optimization for the piecewise constant level set method applied to multiphase image segmentation. In: Scale Space and Variational Methods in Computer Vision, pp. 1–13 (2009)

  5. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Boston (2003)

    MATH  Google Scholar 

  6. Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53(3), 484–512 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beylkin, G., Keiser, J.M.: On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases. J. Comput. Phys. 132(2), 233–259 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cai, W., Wang, J.: Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear PDEs. SIAM J. Numer. Anal. 33(3), 937–970 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cai, W., Zhang, W.: An adaptive spline wavelet ADI (SW-ADI) method for two-dimensional reaction-diffusion equations. J. Comput. Phys. 139(1), 92–126 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ceniceros, H.D., Hou, T.Y.: An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 172(2), 609–639 (2001)

    Article  MATH  Google Scholar 

  11. Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Article  MATH  Google Scholar 

  12. Chan, T.F., Zhou, H.M.: Total variation improved wavelet thresholding in image compression. In: Proc. Seventh International Conference on Image Processing, vol. 2, pp. 391–394. Citeseer (2000)

  13. Chan, T.F., Zhou, H.M.: ENO-wavelet transforms for piecewise smooth functions. SIAM J. Numer. Anal. 40(4), 1369–1404 (2003)

    Article  MathSciNet  Google Scholar 

  14. Chan, T.F., Zhou, H.M.: Total variation wavelet thresholding. J. Sci. Comput. 32(2), 315–341 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chan, T.F., Chen, K., Tai, X.C.: Nonlinear multilevel schemes for solving the total variation image minimization problem. In: Image Processing Based on Partial Differential Equations, Math. Vis., pp. 265–288. Springer, Berlin (2007)

    Chapter  Google Scholar 

  16. Chen, Z.Y., Micchelli, C.A., Xu, Y.S.: A construction of interpolating wavelets on invariant sets. Math. Comput. 68(228), 1569–1587 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Christiansen, O., Tai, X.C.: Fast implementation of piecewise constant level set methods. In: Image Processing Based on Partial Differential Equations, Math. Vis., pp. 289–308. Springer, Berlin (2007)

    Chapter  Google Scholar 

  18. Cohen, A., Daubechies, I.: Non-separable bidimensional wavelet bases. Rev. Mat. Iberoam. 9(1), 51–137 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  20. Demkowicz, L., Oden, J.T., Rachowicz, W., Hardy, O.: Toward a universal hp adaptive finite element strategy. I: Constrained approximation and data structure. Comput. Methods Appl. Mech. Eng. 77(1–2), 79–112 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5(1), 49–68 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Donoho, D.L.: Interpolating wavelet transforms. Preprint, Department of Statistics, Stanford University (1992)

  23. Dubuc, S.: Interpolation through an iterative scheme. J. Math. Anal. Appl. 114(1), 185–204 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Feng, W.M., Yu, P., Hu, S.Y., Liu, Z.K., Du, Q., Chen, L.Q.: Spectral implementation of an adaptive moving mesh method for phase-field equations. J. Comput. Phys. 220(1), 498–510 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Harten, A.: Adaptive multiresolution schemes for shock computations. J. Comput. Phys. 115(2), 319–338 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Holmström, M.: Solving hyperbolic PDEs using interpolating wavelets. SIAM J. Sci. Comput. 21(2), 405–420 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Holmström, M., Waldén, J.: Adaptive wavelet methods for hyperbolic PDEs. J. Sci. Comput. 13(1), 19–49 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Huang, W., Ren, Y., Russell, R.D.: Moving mesh methods based on moving mesh partial differential equations. J. Comput. Phys. 113(2), 279–290 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jain, S., Tsiotras, P., Zhou, H.M.: A hierarchical multiresolution adaptive mesh refinement for the solution of evolution PDEs. SIAM J. Sci. Comput. 31(2), 1221 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jameson, L.: A wavelet-optimized, very high order adaptive grid and order numerical method. SIAM J. Sci. Comput. 19(6), 1980–2013 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)

    Article  Google Scholar 

  32. Kevlahan, N.K.R., Vasilyev, O.V.: An adaptive wavelet collocation method for fluid-structure interaction at high Reynolds numbers. SIAM J. Sci. Comput. 26(6), 1894–1915 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, H., Tai, X.C.: Piecewise constant level set methods for multiphase motion. Int. J. Numer. Anal. Model. 4, 274–293 (2007)

    MathSciNet  Google Scholar 

  34. Li, R., Tang, T., Zhang, P.: Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys. 170(2), 562–588 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liandrat, J., Tchamitchian, P.: Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation algorithm and numerical results. ICASE report 90-83 (1990)

  36. Lie, J., Lysaker, M., Tai, X.C.: A binary level set model and some applications to Mumford-Shah image segmentation. IEEE Trans. Image Process. 15(5), 1171–1181 (2006)

    Article  Google Scholar 

  37. Lie, J., Lysaker, M., Tai, X.C.: A variant of the level set method and applications to image segmentation. Math. Comput. 75(255), 1155–1174 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lu, T., Neittaanmaki, P., Tai, X.C.: A parallel splitting up method and its application to Navier–Stokes equations. Appl. Math. Lett. 4(2), 25–29 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lu, T., Neittaanmaki, P., Tai, X.C.: A parallel splitting up method for partial diferential equations and its application to Navier–Stokes equations. RAIRO Math. Model. Numer. Anal. 26(6), 673–708 (1992)

    MathSciNet  MATH  Google Scholar 

  40. Luo, Z., Tong, L.Y., Luo, J.Z., Wei, P., Wang, M.Y.: Design of piezoelectric actuators using a multiphase level set method of piecewise constants. J. Comput. Phys. 228, 2643–2659 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Marchuk, G.I.: Splitting and alternating direction methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. I (1990)

  42. Micchelli, C.A., Xu, Y.S.: Using the matrix refinement equation for the construction of wavelets on invariant sets. Appl. Comput. Harmon. Anal. 1(4), 391–401 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  44. Oden, J.T., Demkowicz, L., Rachowicz, W., Westermann, T.A.: Toward a universal hp adaptive finite element strategy. II: A posteriori error estimation. Comput. Methods Appl. Mech. Eng. 77(1–2), 113–180 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  45. Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003)

    MATH  Google Scholar 

  46. 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)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MATH  Google Scholar 

  48. Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on the sphere. In: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, pp. 161–172. ACM, New York (1995)

    Google Scholar 

  49. Shen, J., Yang, X.F.: An efficient moving mesh spectral method for the phase-field model of two-phase flows. J. Comput. Phys. 228(8), 2978–2992 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  51. Sweldens, W.: The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3(2), 186–200 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sweldens, W.: The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29(2), 511–546 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sweldens, W., Schroder, P.: Building your own wavelets at home. Lect. Notes Earth Sci. 9, 72–107 (2000)

    Article  Google Scholar 

  54. Tai, X.C., Christiansen, O., Lin, P., Skjælaaen, I.: Image segmentation using some piecewise constant level set methods with MBO type of projection. Int. J. Comput. Vis. 73(1), 61–76 (2007)

    Article  Google Scholar 

  55. Vasilyev, O.V., Bowman, C.: Second-generation wavelet collocation method for the solution of partial differential equations. J. Comput. Phys. 165(2), 660–693 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  56. Vasilyev, O.V., Paolucci, S.: A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain. J. Comput. Phys. 125(2), 498–512 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  57. 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)

    Article  MATH  Google Scholar 

  58. Weickert, J., Romeny, B., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7(3), 398–410 (1998)

    Article  Google Scholar 

  59. Xu, J.C., Shann, W.C.: Galerkin-wavelet methods for two-point boundary value problems. Numer. Math. 63(1), 123–144 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  60. Xu, Y.S., Zou, Q.S.: Adaptive wavelet methods for elliptic operator equations with nonlinear terms. Adv. Comput. Math. 19(1–3), 99–146 (2003). Challenges in Computational Mathematics (Pohang, 2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Zhijian Rong, Li-Lian Wang & Xue-Cheng Tai

  2. School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, China

    Zhijian Rong

  3. Department of Mathematics, University of Bergen, Bergen, Norway

    Xue-Cheng Tai

Authors
  1. Zhijian Rong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li-Lian Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xue-Cheng Tai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li-Lian Wang.

Additional information

Communicated by Lixin Shen.

This research is supported by Singapore MOE Grant T207B2202, Singapore NRF2007IDM-IDM002-010, and the Fundamental Research Funds for the Central Universities of China 2011121041.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rong, Z., Wang, LL. & Tai, XC. Adaptive wavelet collocation methods for image segmentation using TV–Allen–Cahn type models. Adv Comput Math 38, 101–131 (2013). https://doi.org/10.1007/s10444-011-9227-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-011-9227-y

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation