Abstract
In this work we discuss variants of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead use piecewise constant level set functions, and let interfaces be represented by discontinuities. Some of the properties of the standard level set function are preserved in the proposed method. Using the methods for interface problems, we minimize a smooth locally convex functional under a constraint. We show numerical results using the methods for image segmentation.
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References
Bertsekas, D.P.: Constrained optimization and Lagrange multiplier methods. Academic Press Inc., London (1982)
Chambolle, A.: Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55, 827–863 (1995)
Chan, T.F., Tai, X.-C.: Identification of discontinuous coefficients in elliptic problems using total variation regularization. SIAM J. Sci. Comput. 25, 881–904 (2003)
Chan, T.F., Tai, X.-C.: Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J. Comput. Phys. 193, 40–66 (2003)
Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Im. Proc. 10, 266–277 (2001)
Cremers, D., Kohlberger, T., Schnörr, C.: Shape statistics in kernel space for variational image segmentation. Patt. Recogn. 36, 1929–1943 (2003)
Cremers, D., Sochen, N., Schnörr, C.: Multiphase dynamic labeling for variational recognition-driven image segmentation. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 74–86. Springer, Heidelberg (2004)
Dias Velasco, F.R.: Thresholding using the ISODATA clustering algorithm. IEEE Trans. Systems Man Cybernet. 10, 771–774 (1980)
Esedoglu, S., Tsai, Y.-H.R.: Threshold dynamics for the piecewise constant mumford-shah functional, UCLA-CAM Report 04-63 (2004)
Fedkiw, R.P., Sapiro, G., Shu, C.-W.: Shock capturing, level sets, and PDE based methods in computer vision and image processing: a review of Osher’s contributions. J. Comput. Phys. 185, 309–341 (2003)
Gibou, F., Fedkiw, R.: A fast hybrid k-means level set algorithm for segmentation, Stanford Tech. Rep., 2002 (2002) (in review)
Heath, M.T.: Scientific computing: an introductory survey (2001)
Kunisch, K., Tai, X.-C.: Sequential and parallel splitting methods for bilinear control problems in Hilbert spaces. SIAM J. Numer. Anal. 34, 91–118 (1997)
Lie, J., Lysaker, M., Tai, X.-C.: A variant of the level set method and applications to image segmentation, UCLA, CAM-report, 03-50 (2003)
Lie, J., Lysaker, M., Tai, X.-C.: A binary level set model and some applications for mumford-shah image segmentation, UCLA, CAM-report, 04-31 (2004)
Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)
Nocedal, J., Wright, S.J.: Numerical optimization. Series in Operations Research. Springer, Heidelberg (1999)
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. In: Appl. Math. Sci., vol. 153. Springer, Heidelberg (2003)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithm. Physica D. 60, 259–268 (1992)
Samson, C., Blanc-Feraud, L., Aubert, G., Zerubia, J.: A level set model for image classification. IJCV 40, 187–198 (2000)
Sethian, J.A.: Level set methods and fast marching methods, 2nd edn. Cambridge University Press, Cambridge (1999)
Song, B., Chan, T.F.: A fast algorithm for level set based optimization, Tech. Rep. CAM 02-68, UCLA (2002)
Tai, X.-C., Chan, T.F.: A survey on multiple set methods with applications for identifying piecewise constant functions. Int. J. Num. Anal. and Mod. 1, 25–48 (2004)
Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the mumford and shah model. Int. J. of Comp. Vis. 50, 271–293 (2002)
Weickert, J., Kühne, G.: Fast methods for implicit active contour models. In: Geometric level set methods in imaging, vision, and graphics, pp. 43–57. Springer, Heidelberg (2003)
Zhao, H.-K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127, 179–195 (1996)
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Lie, J., Lysaker, M., Tai, XC. (2005). Piecewise Constant Level Set Methods and Image Segmentation. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds) Scale Space and PDE Methods in Computer Vision. Scale-Space 2005. Lecture Notes in Computer Science, vol 3459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11408031_49
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DOI: https://doi.org/10.1007/11408031_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25547-5
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