Multi-class Graph Mumford-Shah Model for Plume Detection Using the MBO scheme

  • Huiyi Hu
  • Justin Sunu
  • Andrea L. Bertozzi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8932)


We focus on the multi-class segmentation problem using the piecewise constant Mumford-Shah model in a graph setting. After formulating a graph version of the Mumford-Shah energy, we propose an efficient algorithm called the MBO scheme using threshold dynamics. Theoretical analysis is developed and a Lyapunov functional is proven to decrease as the algorithm proceeds. Furthermore, to reduce the computational cost for large datasets, we incorporate the Nyström extension method which efficiently approximates eigenvectors of the graph Laplacian based on a small portion of the weight matrix. Finally, we implement the proposed method on the problem of chemical plume detection in hyper-spectral video data.


graph segmentation Mumford-Shah total variation MBO Nyström hyper-spectral image 


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  1. 1.
    Barles, G., Georgelin, C.: A Simple Proof of Convergence for an Approximation Scheme for Computing Motions by Mean Curvature. SIAM J. Numer. Anal. 32(2), 484–500 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bertozzi, A.L., Flenner, A.: Diffuse Interface Models on Graphs for Classification of High Dimensional Data. Multiscale Modeling Sim. 10(3), 1090–1118 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Broadwater, J.B., Limsui, D., Carr, A.K.: A Primer for Chemical Plume Detection using LWIR Sensors. Tech. Rep., National Security Technology Department (2011)Google Scholar
  4. 4.
    Cai, X., Chan, R., Zeng, T.: A Two-Stage Image Segmentation Method Using a Convex Variant of the Mumford-Shah Model and Thresholding. SIAM J. Imaging Sci. 6(1), 368–390 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chan, T., Vese, L.A.: Active Contours without Edges. IEEE Trans. Image Process. 10, 266–277 (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chung, F.R.K.: Spectral Graph Theory. CBMS Reg. Conf. Ser. Math., vol. 92. AMS (1997)Google Scholar
  7. 7.
    Esedoglu, S., Otto, F.: Threshold Dynamics for Networks with Arbitrary Surface Tensions (2013) (submitted)Google Scholar
  8. 8.
    Esedoglu, S., Tsai, Y.R.: Threshold Dynamics for the Piecewise Constant Mumford-Shah Functional. J. Comput. Phys. 211, 367–384 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Evans, L.C.: Convergence of an Algorithm for Mean Curvature Motion. Indiana Univ. Math. J. 42, 553–557 (1993)CrossRefGoogle Scholar
  10. 10.
    Fowlkes, C., Belongie, S., Chung, F., Malik, J.: Spectral Grouping using the Nystrom Method. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 214–225 (2004)CrossRefGoogle Scholar
  11. 11.
    Garcia-Cardona, C., Merkurjev, E., Bertozzi, A. L., Flenner, A., Percus, A.: Fast Multiclass Segmentation using Diffuse Interface Methods on Graphs. IEEE Trans. Pattern Anal. Mach. Intell. (2014)Google Scholar
  12. 12.
    Gerhart, T., Sunu, J., Lieu, L., Merkurjev, E., Chang, J.-M., Gilles, J., Bertozzi, A.L.: Detection and Tracking of Gas Plumes in LWIR Hyperspetral Video Sequence Data. In: SPIE Conference on Defense, Security, and Sensing (2013)Google Scholar
  13. 13.
    Hu, H., Laurent, T., Porter, M.A., Bertozzi, A.L.: A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme. SIAM J. Appl. Math. 73(6), 2224–2246 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Merkurjev, E., Kostic, T., Bertozzi, A.L.: An MBO Scheme on Graphs for Segmentation and Image Processing. SIAM J. Imaging Sci. 6, 1903–1930 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Merkurjev, E., Sunu, J., Bertozzi, A.L.: Graph MBO Method for Multiclass Segmentation of Hyper Spectral Stand-off Detection Video. In: Proceedings of the International Conference on Image Processing (accepted, 2014)Google Scholar
  16. 16.
    Merriman, B., Bence, J.K., Osher, S.J.: Diffusion Generated Motion by Mean Curvature. In: Proceedings of the Computational Cristal Growers Workshop, pp. 73–83. AMS, Providence (1992)Google Scholar
  17. 17.
    Merriman, B., Bence, J.K., Osher, S.J.: Motion of Multiple Junctions: A Level Set Approach. J. Comput. Phys. 112, 334–363 (1994)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Mumford, D., Shah, J.: Optimal Approximation by Piecewise Smooth Functions and Associated Variational Problems. Comm. Pure Appl. Math. 42, 577–685 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Sunu, J., Chang, J.-M., Bertozzi, A.L.: Simultaneous Spectral Analysis of Multiple Video Sequence Data for LWIR Gas Plumes. In: SPIE Conference on Defense, Security, and Sensing (2014)Google Scholar
  21. 21.
    Tai, X.C., Christiansen, O., Lin, P., Skjælaaen, I.: Image Segmentation using Some Piecewise Constant Level Set Methods with MBO Type of Projection. International Journal of Computer Vision 73(1), 61–76 (2007)CrossRefGoogle Scholar
  22. 22.
    Tochon, G., Chanussot, J., Gilles, J., Dalla Mura, M., Chang, J.-M., Bertozzi, A.L.: Gas Plume Detection and Tracking in Hyperspectral Video Sequences using Binary Partition Trees (preprint, 2014)Google Scholar
  23. 23.
    van Gennip, Y., Guillen, N., Osting, B., Bertozzi, A.L.: Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs. Milan J. Math. 82(1), 3–65 (2014)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Vese, L.A., Chan, T.F.: A New Multiphase Level Set Framework for Image Segmentation via the Mumford and Shahs model. International Journal of Computer Vision 50, 271–293 (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Huiyi Hu
    • 1
  • Justin Sunu
    • 2
  • Andrea L. Bertozzi
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  2. 2.Institute of Mathematical ScienceClaremont Graduate UniversityUSA

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