Abstract
A novel numerical method for multilabel segmentation of vector-valued images is presented. The algorithm seeks minimizers for a generalization of the piecewise-constant Mumford–Shah energy and is particularly appropriate for energies with a fitting (or fidelity) term that is computationally expensive to evaluate. The framework for the algorithm is the standard alternating-minimization scheme in which the update of the partition is alternated with the update of the vector-valued constants associated with each part of the segmentation. The update of the partition is based on the distance function-based diffusion-generated motion algorithms for mean curvature flow. The update of the vector-valued constants is based on an Augmented Lagrangian method. The scheme automatically chooses the appropriate number of segments in the partition. It is initialized with a partition of many more segments than are expected to be necessary. Adjacent segmentations of the partition are merged when energetically advantageous. The utility of the algorithm is demonstrated in the context of atomic-resolution polycrystalline image segmentation.
Similar content being viewed by others
References
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000)
Bae, E., Yuan, J., Tai, X.C.: Global minimization for continuous multiphase partitioning problems using a dual approach. Int. J. Comput. Vis. 92(1), 112–129 (2010). doi:10.1007/s11263-010-0406-y
Berkels, B., Rätz, A., Rumpf, M., Voigt, A.: Extracting grain boundaries and macroscopic deformations from images on atomic scale. J. Sci. Comput. 35(1), 1–23 (2008). doi:10.1007/s10915-007-9157-5
Boerdgen, M., Berkels, B., Rumpf, M., Cremers, D.: Convex relaxation for grain segmentation at atomic scale. In: Fellner, D. (ed.) VMV 2010—Vision, Modeling and Visualization, pp. 179–186. Eurographics Association (2010)
Bresson, X., Esedoḡlu, S., Vandergheynst, P., Thirau, J.-P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28, 151–167 (2007). doi:10.1007/s10851-007-0002-0
Chan, T.F., Esedoglu, S., Nikolova, M.: Finding the global minimum for binary image restoration. In: Proceedings of the International Conference on Image Processing, vol. 1, pp. 121–124 (2005). doi:10.1109/ICIP.2005.1529702
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). doi:10.1006/jvci.1999.0442
Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001). doi:10.1109/83.902291
Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. SIAM, Philadelphia (2000)
Delong, A., Boykov, Y.: Globally optimal segmentation of multi-region objects. In: Proceedings of the International Conference on Computer Vision, pp. 285–292 (2009)
El-Zehiry, N., Sahoo, P., Xu, S., Elmaghraby, A.: Graph cut optimization for the Mumford-Shah model. In: Proceedings of the International Conference on Visualization, Imaging and Image Processing (IASTED), pp. 182–187 (2007)
El-Zehiry, N.Y., Elmaghraby, A.: A graph cut based active contour for multiphase image segmentation. In: Proceedings of the International Conference on Image Processing (ICIP), pp. 3188–3191 (2008). doi:10.1109/ICIP.2008.4712473
El-Zehiry, N.Y., Grady, L.: Combinatorial optimization of the discretized multiphase Mumford–Shah functional. Int. J. Comput. Vis. 104, 270–285 (2013)
Elder, K.R., Grant, M.: Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 70, 051,605 (2004)
Elsey, M., Esedoḡlu, S.: Fast and accurate redistancing by directional optimization. SIAM J. Sci. Comput. 36(1), A219–A231 (2014). doi:10.1137/120889447
Elsey, M., Esedoḡlu, S., Smereka, P.: Diffusion generated motion for grain growth in two and three dimensions. J. Comput. Phys. 228(21), 8015–8033 (2009). doi:10.1016/j.jcp.2009.07.020
Elsey, M., Esedoḡlu, S., Smereka, P.: Simulations of anisotropic grain growth: efficient algorithms and misorientation distributions. Acta Mater. 61, 2033–2043 (2013)
Elsey, M., Wirth, B.: Fast automated detection of crystal distortion and crystal defects in polycrystal images. SIAM Multiscale Model. Simul. 12(1), 1–24 (2014)
Esedoḡlu, S., Otto, F.: Threshold dynamics for networks with arbitrary surface tensions. Commun. Pure Appl. Math. (2014). doi:10.1002/cpa.21527
Esedoḡlu, S., Ruuth, S., Tsai, R.: Diffusion generated motion using signed distance functions. J. Comput. Phys. 229(4), 1017–1042 (2010)
Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009). doi:10.1137/080725891
Jeon, M., Alexander, M., Pedrycz, W., Pizzi, N.: Unsupervised hierarchical image segmentation with level set and additive operator splitting. Pattern Recognit. Lett. 26, 1461–1469 (2005). doi:10.1016/j.patrec.2004.11.023
Merriman, B., Bence, J., Osher, S.: Diffusion generated motion by mean curvature. In: Taylor, J.E. (ed.) Computational Crystal Growers Workshop, pp. 73–83. American Mathematical Society, Providence (1992)
Merriman, B., Bence, J.K., Osher, S.: Motion of multiple junctions: a level set approach. J. Comput. Phys. 112(2), 334–363 (1994)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989). doi:10.1002/cpa.3160420503
Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)
Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)
Patala, S., Mason, J.K., Schuh, C.A.: Improved representations of misorientation information for grain boundary science and engineering. Prog. Mater. Sci. 57, 1383–1425 (2012)
Potts, R.B.: Some generalized order-disorder transformations. Proc. Camb. Philos. Soc. 48, 106–109 (1952)
Read, W.T., Shockley, W.: Dislocation models of crystal grain boundaries. Phys. Rev. 78(3), 275–289 (1950)
Ring, W., Wirth, B.: Optimization methods on Riemannian manifolds and their application to shape space. SIAM J. Optim. 22(2), 596–627 (2012)
Ruuth, S.J.: A diffusion-generated approach to multiphase motion. J. Comput. Phys. 145, 166–192 (1998)
Ruuth, S.J.: Efficient algorithms for diffusion-generated motion by mean curvature. J. Comput. Phys. 144, 603–625 (1998)
Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93, 1591–1595 (1996)
Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Autom. Control 40(9), 1528–1538 (1995)
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). doi:10.1023/A:1020874308076
Acknowledgments
The authors thank Robert V. Kohn for many valuable discussions on the subject. M.E. gratefully acknowledges the support of National Science Foundation Grant OISE-0967140.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Elsey, M., Wirth, B. Redistancing Dynamics for Vector-Valued Multilabel Segmentation with Costly Fidelity: Grain Identification in Polycrystal Images. J Sci Comput 63, 279–306 (2015). https://doi.org/10.1007/s10915-014-9892-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-014-9892-3
Keywords
- Image segmentation
- Mumford–Shah
- Costly fidelity
- Diffusion-generated motion
- Polycrystal
- Multilabel segmentation