Skip to main content
SpringerLink
Log in

Extracting Grain Boundaries and Macroscopic Deformations from Images on Atomic Scale

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Nowadays image acquisition in materials science allows the resolution of grains at atomic scale. Grains are material regions with different lattice orientation which are frequently in addition elastically stressed. At the same time, new microscopic simulation tools allow to study the dynamics of such grain structures. Single atoms are resolved experimentally as well as in simulation results on the data microscale, whereas lattice orientation and elastic deformation describe corresponding physical structures mesoscopically. A qualitative study of experimental images and simulation results and the comparison of simulation and experiment requires the robust and reliable extraction of mesoscopic properties from the microscopic image data. Based on a Mumford–Shah type functional, grain boundaries are described as free discontinuity sets at which the orientation parameter for the lattice jumps. The lattice structure itself is encoded in a suitable integrand depending on a local lattice orientation and one global elastic displacement. For each grain a lattice orientation and an elastic displacement function are considered as unknowns implicitly described by the image microstructure. In addition the approach incorporates solid–liquid interfaces. The resulting Mumford–Shah functional is approximated with a level set active contour model following the approach by Chan and Vese. The implementation is based on a finite element discretization in space and a step size controlled, regularized gradient descent algorithm.

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. Aujol, J.-F., Aubert, G., Blanc-Feáaud, L.: Wavelet-based level set evolution for classification of textured images. IEEE Trans. Image Process. 12(12), 1634–1641 (2003)

    Article  MathSciNet  Google Scholar 

  2. Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)

    Article  MathSciNet  Google Scholar 

  3. Backofen, R., Rätz, A., Voigt, A.: Nucleation and growth by a phase field crystal (PFC) model. Phil. Mag. Lett. (2007, accepted)

  4. Aujol, J.-F., Chan, T.F.: Combining geometrical and textured information to perform image classification. J. Vis. Commun. Image Represent. 17(5), 1004–1023 (2006)

    Article  Google Scholar 

  5. Berkels, B., Rätz, A., Rumpf, R., Voigt, A.: Identification of grain boundary contours at atomic scale. In: Proceedings of the First International Conference on Scale Space Methods and Variational Methods in Computer Vision, pp. 765–776. Springer, Berlin (2007)

  6. Berthod, M., Kato, Z., Yu, S., Zerubia, J.B.: Bayesian image classification using Markov random fields. Image Vis. Comput. 14(4), 285–295 (1996)

    Article  Google Scholar 

  7. Bouman, C., Shapiro, M.: Multiscale random field model for bayesian image segmentation. IEEE Trans. Image Process. 3(2), 162–177 (1994)

    Article  Google Scholar 

  8. Caselles, V., Catté, F., Coll, T., Dibos, F.: A geometric model for active contours in image processing. Numer. Math. 66, 1–31 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  10. Cremers, D., Schnörr, C.: Statistical shape knowledge in variational motion segmentation. Image Vis. Comput. 21(1), 77–86 (2003)

    Article  Google Scholar 

  11. Doretto, G., Cremers, D., Favaro, P., Soatto, S.: Dynamic texture segmentation. In: Triggs, B., Zisserman, A. (eds.) IEEE International Conference on Computer Vision (ICCV), vol. 2, pp. 1236–1242. Nice, October 2003

  12. Droske, M., Rumpf, M.: A variational approach to non-rigid morphological registration. SIAM Appl. Math. 64(2), 668–687 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Elder, K.R., Grant, M.: Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 70(5), 051605 (2004)

    Google Scholar 

  14. Heiler, M., Schnörr, C.: Natural image statistics for natural image segmentation. Int. J. Comput. Vis. 63(1), 5–19 (2005)

    Article  Google Scholar 

  15. King, W.E., Campbell, G.H., Foiles, S.M., Cohen, D., Hanson, K.M.: Quantitative HREM observation of the \({\Sigma}11(113)/[\bar{1}00]\) grain-boundary structure in aluminium and comparison with atomistic simulation. J. Microsc. 190(1-2), 131–143 (1998)

    Article  Google Scholar 

  16. Kosmol, P.: Optimierung und Approximation. de Gruyter, Berlin (1991)

    MATH  Google Scholar 

  17. Lakkis, O., Nochetto, R.H.: A posteriori error analysis for the mean curvature flow of graphs. SIAM J. Numer. Anal. 42(5), 1875–1898 (2004)

    Article  MathSciNet  Google Scholar 

  18. Lakshmanan, S., Derin, H.: Simultaneous parameter estimation and segmentation of Gibbs random fields using simulated annealing. IEEE Trans. Pattern Anal. Mach. Intell. 11(8), 799–813 (1989)

    Article  Google Scholar 

  19. Manjunath, B.S., Chellappa, R.: Unsupervised texture segmentation using Markov random field models. IEEE Trans. Pattern Anal. Mach. Intell. 13(5), 478–482 (1991)

    Article  Google Scholar 

  20. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures. American Mathematical Society, Boston (2001)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2002)

    Google Scholar 

  23. Paragios, N., Deriche, R.: Geodesic active regions and level set methods for motion estimation and tracking. Comput. Vis. Image Underst. 97(3), 259–282 (2005)

    Article  Google Scholar 

  24. Sandberg, B., Chan, T., Vese, L.: A level-set and Gabor-based active contour algorithm for segmenting textured images. Technical Report 02-39, UCLA CAM Reports, 2002

  25. Schryvers, D., et al.: Measuring strain fields and concentration gradients around Ni4Ti3 precipitates. Mater. Sci. Eng. A: Struct. Mater. Prop. Microstruct. Process. 438, 485–488 (2006) (Special Issue)

    Google Scholar 

  26. Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  27. Sikolowski, J., Zolésio, J.-P.: Introduction to shape optimization. In: Shape Sensitivity Analysis. Springer, Berlin (1992)

    Google Scholar 

  28. Singh, Y.: Density-functional theory of freezing and properties of the ordered phase. Phys. Rep. 207(6), 351–444 (1991)

    Article  Google Scholar 

  29. Unser, M.: Texture classification and segmentation using wavelet frames. IEEE Trans. Image Process. 4(11), 1549–1560 (1995)

    Article  MathSciNet  Google Scholar 

  30. Vese, L., 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 

  31. Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Benjamin Berkels & Martin Rumpf

  2. Crystal Growth Group, Research Center Caesar, Bonn, Germany

    Andreas Rätz

  3. Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, Dresden, Germany

    Axel Voigt

Authors
  1. Benjamin Berkels
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Rätz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin Rumpf
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Axel Voigt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benjamin Berkels.

Additional information

This work was supported by the DFG priority program 1114.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berkels, B., Rätz, A., Rumpf, M. et al. Extracting Grain Boundaries and Macroscopic Deformations from Images on Atomic Scale. J Sci Comput 35, 1–23 (2008). https://doi.org/10.1007/s10915-007-9157-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-007-9157-5

Keywords

Search

Navigation