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Image Segmentation with Shape Priors: Explicit Versus Implicit Representations

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Handbook of Mathematical Methods in Imaging

Abstract

Image segmentation is among the most studied problems in image understanding and computer vision. The goal of image segmentation is to partition the image plane into a set of meaningful regions. Here meaningful typically refers to a semantic partitioning where the computed regions correspond to individual objects in the observed scene. Unfortunately, generic purely low-level segmentation algorithms often do not provide the desired segmentation results, because the traditional low-level assumptions like intensity or texture homogeneity and strong edge contrast are not sufficient to separate objects in a scene.

To overcome these limitations, researchers have proposed to impose prior knowledge into low-level segmentation methods. In the following, we will review methods which allow to impose knowledge about the shape of objects of interest into segmentation processes.

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References

  1. Amini, A.A., Weymouth, T.E., Jain, R.C.: Using dynamic programming for solving variational problems in vision. IEEE Trans. Pattern Anal. Mach. Intell. 12(9), 855–867 (1990)

    Article  Google Scholar 

  2. Awate, S.P., Tasdizen, T., Whitaker, R.T.: Unsupervised texture segmentation with nonparametric neighborhood statistics. In: European Conference on Computer Vision (ECCV), pp. 494–507, Graz. Springer (2006)

    Google Scholar 

  3. Blake, A., Isard, M.: Active Contours. Springer, London (1998)

    Book  Google Scholar 

  4. Blake, A., Zisserman, A.: Visual Reconstruction. MIT, Cambridge (1987)

    Google Scholar 

  5. Bookstein, F.L.: The Measurement of Biological Shape and Shape Change. Lecture Notes in Biomath, vol, 24. Springer, New York (1978)

    Google Scholar 

  6. Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: IEEE International Conference on Computer Vision, Nice, pp. 26–33 (2003)

    Google Scholar 

  7. Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-ow algorithms for energy minimization in vision. IEEE Trans. Pattern Anal. Mach. Intell. 26(9), 1124–1137 (2004)

    Article  Google Scholar 

  8. Brox, T., Rousson, M., Deriche, R., Weickert, J.: Unsupervised segmentation incorporating colour, texture, and motion. In: Petkov, N., Westenberg, M.A. (eds.) Computer Analysis of Images and Patterns. LNCS, vol. 2756, pp. 353–360. Springer, Groningen (2003)

    Chapter  Google Scholar 

  9. Brox, T., Weickert, J.: A TV flow based local scale measure for texture discrimination. In: Pajdla, T., Hlavac, V. (eds.) European Conference on Computer Vision. LNCS, vol. 3022, pp. 578–590. Springer, Prague (2004)

    Google Scholar 

  10. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: Proceedings of the IEEE International Conference on Computer Vision, Boston, pp. 694–699 (1995)

    Google Scholar 

  11. Chan, T., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Chan, T.F., Vese, L.A.: A level set algorithm for minimizing the Mumford–Shah functional in image processing. In: IEEE Workshop on Variational and Level Set Methods, Vancouver, pp. 161–168 (2001)

    Google Scholar 

  14. Chan, T., Zhu, W.: Level set based shape prior segmentation. Technical report 03-66, Computational Applied Mathematics, UCLA, Los Angeles (2003)

    Google Scholar 

  15. Charpiat, G., Faugeras, O., Keriven, R.: Approximations of shape metrics and application to shape warping and empirical shape statistics. J. Found. Comput. Math. 5(1), 1–58 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Charpiat, G., Faugeras, O., Pons, J.-P., Keriven, R.: Generalized gradients: priors on minimization flows. Int. J. Comput. Vis. 73(3), 325–344 (2007)

    Article  Google Scholar 

  17. Chen, Y., Tagare, H., Thiruvenkadam, S., Huang, F., Wilson, D., Gopinath, K.S., Briggs, R.W., Geiser, E.: Using shape priors in geometric active contours in a variational framework. Int. J. Comput. Vis. 50(3), 315–328 (2002)

    Article  MATH  Google Scholar 

  18. Chow, Y.S., Geman, S., Wu, L.D.: Consistent cross-validated density estimation. Ann. Stat. 11, 25–38 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cipolla, R., Blake, A.: The dynamic analysis of apparent contours. In: IEEE International Conference on Computer Vision, Osaka, pp. 616–625. Springer (1990)

    Google Scholar 

  20. Cohen, L., Kimmel, R.: Global minimum for active contour models: a minimal path approach. Int. J. Comput. Vis. 24(1), 57–78 (1997)

    Article  Google Scholar 

  21. Cootes, T.F., Taylor, C.J.: A mixture model for representing shape variation. Image Vis. Comput. 17(8), 567–574 (1999)

    Article  Google Scholar 

  22. Cootes, T.F., Taylor, C.J., Cooper, D.M., Graham, J.: Active shape models – their training and application. Comput. Vis. Image Underst. 61(1), 38–59 (1995)

    Article  Google Scholar 

  23. Coughlan, J., Yuille, A., English, C., Snow, D.: Efficient deformable template detection and localization without user initialization. Comput. Vis. Image Underst. 78(3), 303–319 (2000)

    Article  Google Scholar 

  24. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Interscience, New York (1953)

    Google Scholar 

  25. Cremers, D.: Statistical shape knowledge in variational image segmentation. PhD thesis, Department of Mathematics and Computer Science, University of Mannheim, Germany (2002)

    Google Scholar 

  26. Cremers, D.: Dynamical statistical shape priors for level set based tracking. IEEE Trans. Pattern Anal. Mach. Intell. 28(8), 1262–1273 (2006)

    Article  Google Scholar 

  27. Cremers, D., Kohlberger, T., Schnörr, C.: Nonlinear shape statistics in Mumford–Shah based segmentation. In: Heyden, A., et al. (eds.) European Conference on Computer Vision, Copenhagen, pp 93–108. LNCS, vol. 2351. Springer (2002)

    Google Scholar 

  28. Cremers, D., Kohlberger, T., Schnörr, C.: Shape statistics in kernel space for variational image segmentation. Pattern Recognit. 36(9), 1929–1943 (2003)

    Article  MATH  Google Scholar 

  29. Cremers, D., Osher, S.J., Soatto, S.: Kernel density estimation and intrinsic alignment for shape priors in level set segmentation. Int. J. Comput. Vis. 69(3), 335–351 (2006)

    Article  Google Scholar 

  30. Cremers, D., Rousson, M., Deriche, R.: A review of statistical approaches to level set segmentation: integrating color, texture, motion and shape. Int. J. Comput. Vis. 72(2), 195–215 (2007)

    Article  Google Scholar 

  31. Cremers, D., Schmidt, F.R., Barthel, F.: Shape priors in variational image segmentation: convexity, lipschitz continuity and globally optimal solutions. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Anchorage (2008)

    Google Scholar 

  32. Cremers, D., Soatto, S.: A pseudo-distance for shape priors in level set segmentation. In: Paragios, N. (ed.) IEEE 2nd International Workshop on Variational, Geometric and Level Set Methods, Nice, pp. 169–176 (2003)

    Google Scholar 

  33. Cremers, D., Soatto, S.: Motion competition: a variational framework for piecewise parametric motion segmentation. Int. J. Comput. Vis. 62(3), 249–265 (2005)

    Article  Google Scholar 

  34. Cremers, D., Sochen, N., Schnörr, C.: A multiphase dynamic labeling model for variational recognition-driven image segmentation. In: Pajdla, T., Hlavac, V. (eds.) European Conference on Computer Vision, Graz. LNCS, vol. 3024, pp 74–86. Springer (2006)

    Google Scholar 

  35. Cremers, D., Sochen, N., Schnörr, C.: A multiphase dynamic labeling model for variational recognition-driven image segmentation. Int. J. Comput. Vis. 66(1), 67–81 (2006)

    Article  MATH  Google Scholar 

  36. Cremers, D., Tischhäuser, F., Weickert, J., Schnörr, C.: Diffusion snakes: introducing statistical shape knowledge into the Mumford–Shah functional. Int. J. Comput. Vis. 50(3), 295–313 (2002)

    Article  MATH  Google Scholar 

  37. Deheuvels, P.: Estimation non paramétrique de la densité par histogrammes généralisés. Revue de Statistique Appliquée 25, 5–42 (1977)

    MathSciNet  Google Scholar 

  38. Delingette, H., Montagnat, J.: New algorithms for controlling active contours shape and topology. In: Vernon, D. (ed.) Proceedings of the European Conference on Computer Vision, Dublin. LNCS, vol. 1843, pp. 381–395. Springer (2000)

    Google Scholar 

  39. Dervieux, A., Thomasset, F.: A finite element method for the simulation of Raleigh–Taylor instability. Springer Lect. Notes Math. 771, 145–158 (1979)

    MathSciNet  Google Scholar 

  40. Devroye, L., Györfi, L.: Nonparametric Density Estimation: The L1 View. Wiley, New York (1985)

    Google Scholar 

  41. Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)

    MATH  Google Scholar 

  42. Duin, R.P.W.: On the choice of smoothing parameters for Parzen estimators of probability density functions. IEEE Trans. Comput. 25, 1175–1179 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  43. Farin, G.: Curves and Surfaces for Computer–Aided Geometric Design. Academic, San Diego (1997)

    MATH  Google Scholar 

  44. Franchini, E., Morigi, S., Sgallari, F.: Segmentation of 3D tubular structures by a PDE-based anisotropic diffusion model. In: International Conference on Scale Space and Variational Methods, Voss. LNCS, vol. 5567, pp. 75–86. Springer (2009)

    Google Scholar 

  45. Fréchet, M.: Les courbes aléatoires. Bull. Int. Stat. Inst. 38, 499–504 (1961)

    MATH  Google Scholar 

  46. Fundana, K., Overgaard, N.C., Heyden, A.: Variational segmentation of image sequences using region-based active contours and deformable shape priors. Int. J. Comput. Vis. 80(3), 289–299 (2008)

    Article  Google Scholar 

  47. Gdalyahu, Y., Weinshall, D.: Flexible syntactic matching of curves and its application to automatic hierarchical classication of silhouettes. IEEE Trans. Pattern Anal. Mach. Intell. 21(12), 1312–1328 (1999)

    Article  Google Scholar 

  48. Geiger, D., Gupta, A., Costa, L.A., Vlontzos, J.: Dynamic programming for detecting, tracking and matching deformable contours. IEEE Trans. Pattern Anal. Mach. Intell. 17(3), 294–302 (1995)

    Article  Google Scholar 

  49. Greig, D.M., Porteous, B.T., Seheult, A.H.: Exact maximum a posteriori estimation for binary images. J. R. Stat. Soc. B 51(2), 271–279 (1989)

    Google Scholar 

  50. Grenander, U., Chow, Y., Keenan, D.M.: Hands: A Pattern Theoretic Study of Biological Shapes. Springer, New York (1991)

    Book  Google Scholar 

  51. Heiler, M., Schnörr, C.: Natural image statistics for natural image segmentation. In: IEEE International Conference on Computer Vision, Nice, pp. 1259–1266 (2003)

    Google Scholar 

  52. Ising, E.: Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik 23, 253–258 (1925)

    Article  Google Scholar 

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

    Article  Google Scholar 

  54. Kendall, D.G.: The diffusion of shape. Adv. Appl. Probab. 9, 428–430 (1977)

    Article  Google Scholar 

  55. Kervrann, C., Heitz, F.: Statistical deformable model-based segmentation of image motion. IEEE Trans. Image Process. 8, 583–588 (1999)

    Article  Google Scholar 

  56. Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., Yezzi, A.J.: Gradient ows and geometric active contour models. In: IEEE International Conference on Computer Vision, Cambridge, pp. 810–815 (1995)

    Google Scholar 

  57. Kim, J., Fisher, J.W., Yezzi, A., Cetin, M., Willsky, A.: Nonparametric methods for image segmentation using information theory and curve evolution. In: International Conference on Image Processing, Rochester, vol. 3, pp. 797–800 (2002)

    Google Scholar 

  58. Kohlberger, T., Cremers, D., Rousson, M., Ramaraj, R.: 4D shape priors for level set segmentation of the left myocardium in SPECT sequences. In: Medical Image Computing and Computer Assisted Intervention. LNCS, vol. 4190, pp. 92–100. Springer, Heidelberg (2006)

    Google Scholar 

  59. Kolev, K., Klodt, M., Brox, T., Cremers, D.: Continuous global optimization in multview 3D reconstruction. Int. J. Comput. Vis. 84, 80–96 (2009)

    Article  Google Scholar 

  60. Lachaud, J.-O., Montanvert, A.: Deformable meshes with automated topology changes for coarse-to-fine three-dimensional surface extraction. Med. Image Anal. 3(2), 187–207 (1999)

    Article  Google Scholar 

  61. Leitner, F., Cinquin, P.: Complex topology 3D objects segmentation. In: SPIE Conference on Advances in Intelligent Robotics Systems, Boston, vol. 1609 (1991)

    Google Scholar 

  62. Leventon, M., Grimson, W., Faugeras, O.: Statistical shape inuence in geodesic active contours. In: International Conference on Computer Vision and Pattern Recognition, Hilton Head Island, vol. 1, pp. 316–323 (2000)

    Google Scholar 

  63. Malladi, R., Sethian, J.A., Vemuri, B.C.: Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Anal. Mach. Intell. 17(2), 158–175 (1995)

    Article  Google Scholar 

  64. Matheron, G.: Random sets and integral geometry. Wiley, New York (1975)

    MATH  Google Scholar 

  65. McInerney, T., Terzopoulos, D.: Topologically adaptable snakes. In: Proceedings of the 5th International Conference on Computer Vision, Cambridge, 20–23 June 1995, pp. 840–845. IEEE Computer Society, Los Alamitos (1995)

    Google Scholar 

  66. Menet, S., Saint-Marc, P., Medioni, G.: B–snakes: implementation and application to stereo. In: Proceedings of the DARPA Image Understanding Workshop, Pittsburgh, 6–8 Apr 1990, pp. 720–726 (1990)

    Google Scholar 

  67. Mercer, J.: Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. R. Soc. Lond. A 209, 415–446 (1909)

    Article  MATH  Google Scholar 

  68. Moghaddam, B., Pentland, A.: Probabilistic visual learning for object representation. IEEE Trans. Pattern Anal. Mach. Intell. 19(7), 696–710 (1997)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  70. Nain, D., Yezzi, A., Turk, G.: Vessel segmentation using a shape driven flow. In: MICCAI, Montréal, pp. 51–59 (2003)

    Google Scholar 

  71. Neumaier, A., Schneider, T.: Estimation of parameters and eigenmodes of multivariate autoregressive models. ACM Trans. Math. Softw. 27(1), 27–57 (2001)

    Article  MATH  Google Scholar 

  72. Osher, S.J., Sethian, J.A.: Fronts propagation with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  73. Paragios, N., Deriche, R.: Geodesic active regions and level set methods for supervised texture segmentation. Int. J. Comput. Vis. 46(3), 223–247 (2002)

    Article  MATH  Google Scholar 

  74. Parent, P., Zucker, S.W.: Trace inference, curvature consistency, and curve detection. IEEE Trans. Pattern Anal. Mach. Intell. 11(8), 823–839 (1989)

    Article  Google Scholar 

  75. Parzen, E.: On the estimation of a probability density function and the mode. Ann. Math. Stat. 33, 1065–1076 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  76. Rasmussen, C.-E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT, Cambridge (2006)

    MATH  Google Scholar 

  77. Riklin-Raviv, T., Kiryati, N., Sochen, N.: Unlevel sets: geometry and prior-based segmentation. In: Pajdla, T., Hlavac, V. (eds.) European Conference on Computer Vision. LNCS, vol. 3024, pp. 50–61. Springer, Prague (2004)

    Google Scholar 

  78. Rochery, M., Jermyn, I., Zerubia, J.: Higher order active contours. Int. J. Comput. Vis. 69, 27–42 (2006)

    Article  Google Scholar 

  79. Rosenblatt, F.: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27, 832–837 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  80. Rosenfeld, A., Zucker, S.W., Hummel, R.A.: An application of relaxation labeling to line and curve enhancement. IEEE Trans. Comput. 26(4), 394–403 (1977)

    Google Scholar 

  81. Rousson, M., Brox, T., Deriche, R.: Active unsupervised texture segmentation on a diffusion based feature space. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Madison, pp. 699–704 (2003)

    Google Scholar 

  82. Rousson, M., Cremers, D.: Efficient kernel density estimation of shape and intensity priors for level set segmentation. In: MICCAI, Palm Springs, vol. 1, pp. 757–764 (2005)

    Google Scholar 

  83. Rousson, M., Paragios, N.: Shape priors for level set representations. In: Heyden, A., et al. (eds.) European Conference on Computer Vision, Copenhagen. LNCS, vol. 2351, pp. 78–92. Springer (2002)

    Google Scholar 

  84. Rousson, M., Paragios, N., Deriche, R.: Implicit active shape models for 3D segmentation in MRI imaging. In: MICCAI, Saint-Malo. LNCS, vol. 2217, pp. 209–216. Springer (2004)

    Google Scholar 

  85. Schmidt, F.R., Farin, D., Cremers, D.: Fast matching of planar shapes in sub-cubic runtime. In: IEEE International Conference on Computer Vision, Rio de Janeiro (2007)

    Book  Google Scholar 

  86. Schoenemann, T., Cremers, D.: Globally optimal image segmentation with an elastic shape prior. In: IEEE International Conference on Computer Vision, Rio de Janeiro (2007)

    Google Scholar 

  87. Schoenemann, T., Cremers, D.: Introducing curvature into globally optimal image segmentation: minimum ratio cycles on product graphs. In: IEEE International Conference on Computer Vision, Rio de Janeiro (2007)

    Google Scholar 

  88. Schoenemann, T., Cremers, D.: Matching non-rigidly deformable shapes across images: a globally optimal solution. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Anchorage (2008)

    Google Scholar 

  89. Schoenemann, T., Cremers, D.: A combinatorial solution for model-based image segmentation and real-time tracking. IEEE Trans. Pattern Anal. Mach. Intell. 32(7), 1153–1164 (2010)

    Article  Google Scholar 

  90. Schoenemann, T., Kahl, F., Cremers, D.: Curvature regularity for region-based image segmentation and inpainting: a linear programming relaxation. In: IEEE International Conference on Computer Vision, Kyoto (2009)

    Google Scholar 

  91. Schoenemann, T., Schmidt, F.R., Cremers, D.: Image segmentation with elastic shape priors via global geodesics in product spaces. In: British Machine Vision Conference, Leeds (2008)

    Book  Google Scholar 

  92. Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)

    Article  MATH  Google Scholar 

  93. Sebastian, T., Klein, P., Kimia, B.: On aligning curves. IEEE Trans. Pattern Anal. Mach. Intell. 25(1), 116–125 (2003)

    Article  Google Scholar 

  94. Serra, J.: Image Analysis and Mathematical Morophology. Academic, London (1982)

    Google Scholar 

  95. Silverman, B.W.: Choosing the window width when estimating a density. Biometrika 65, 1–11 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  96. Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London (1992)

    Google Scholar 

  97. Sundaramoorthi, G., Yezzi, A., Mennucci, A., Sapiro, G.: New possibilities with sobolev active contours. Int. J. Comput. Vis. 84(2), 113–129 (2009)

    Article  Google Scholar 

  98. Sussman, M., Smereka, P., Osher, S.J.: A level set approach for computing solutions to incompressible twophase flow. J. Comput. Phys. 94, 146–159 (1994)

    Article  Google Scholar 

  99. Tsai, A., Wells, W., Warfield, S.K., Willsky, A.: Level set methods in an EM framework for shape classification and estimation. In: MICCAI, Saint-Malo (2004)

    Book  Google Scholar 

  100. Tsai, A., Yezzi, A., Wells, W., Tempany, C., Tucker, D., Fan, A., Grimson, E., Willsky, A.: Model–based curve evolution technique for image segmentation. In: IEEE Conference on Computer Vision Pattern Recognition, Kauai, pp. 463–468 (2001)

    Google Scholar 

  101. Tsai, A., Yezzi, A.J., Willsky, A.S.: Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. Image Process. 10(8), 1169–1186 (2001)

    Article  MATH  Google Scholar 

  102. Unal, G., Krim, H., Yezzi, A.Y.: Information-theoretic active polygons for unsupervised texture segmentation. Int. J. Comput. Vis. 62(3), 199–220 (2005)

    Article  Google Scholar 

  103. Unger, M., Pock, T., Cremers, D., Bischof, H.: TVSeg – interactive total variation based image segmentation. In: British Machine Vision Conference (BMVC), Leeds (2008)

    Google Scholar 

  104. Wagner, T.J.: Nonparametric estimates of probability densities. IEEE Trans. Inf. Theory 21, 438–440 (1975)

    Article  MATH  Google Scholar 

  105. Zhu, S.C., Yuille, A.: Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 18(9), 884–900 (1996)

    Article  Google Scholar 

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  1. Department of Computer Science, Technische Universität München, 110 8th Street, Garching, Germany

    Daniel Cremers

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  1. Computational Science Center, University of Vienna, Vienna, Austria

    Otmar Scherzer

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Cremers, D. (2015). Image Segmentation with Shape Priors: Explicit Versus Implicit Representations. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0790-8_40

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