Abstract
A typical task of image segmentation is to partition a given image into regions of homogeneous property. In this paper we focus on the problem of further detecting scales of discontinuities of the image. The approach uses a recently developed iterative numerical algorithm for minimizing the Mumford-Shah functional which is based on topological derivatives. For the scale selection we use a squared norm of the gradient at edge points. During the iteration progress, the square norm, as a function varied with iteration numbers, provides information about different scales of the discontinuity sets. For realistic image data, the graph of the norm function is regularized by using total variation minimization to provide stable separation. We present the details of the algorithm and document various numerical experiments.
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References
Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43(8), 999–1036 (1990)
Auroux, D., Belaid, L.J., Masmoudi, M.: A topological asymptotic analysis for the regularized grey-level image classification problem. Math. Model. Numer. Anal. 41(3) (2007)
Auroux, D., Masmoudi, M.: Image processing by topological asymptotic expansion. J. Math. Imaging Vision 33(2) (2009)
Chambolle, A.: Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55(3), 827–863 (1995)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20(1-2), 89–97 (2004)
Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)
Chan, T., Vese, L.: Active Contours without Edges. IEEE Trans. Image Processing 10(2), 266–277 (2001)
Davies, P.L., Kovac, A.: Local extremes, runs, strings and multiresolution. Ann. Statist. 29(1), 1–65 (2001)
Feijóo, R.A., Novotny, A., Padra, C., Taroco, E.: The topological derivative for the Poisson problem. Math. Mod. Meth. Appl. Sci. 13(12), 1825–1844 (2003)
Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for PDE systems: The elasticity case. SIAM J. Control Optimiz. 39(6), 1756–1778 (2000)
Grasmair, M.: The equivalence of the taut string algorithm and BV-regularization. J. Math. Imaging Vision 27(1), 59–66 (2007)
Grasmair, M., Muszkieta, M., Scherzer, O.: An approach to the minimization of the Mumford–Shah functional using Γ-convergence and topological asymptotic expansion. Preprint on ArXiv, arXiv:1103.4722v1, University of Vienna, Austria (2011)
Jung, Y.M., Kang, S.H., Shen, J.: Multiphase Image Segmentation via Modica-Mortola Phase transitio. SIAM Applied Mathematics 67(5), 1213–1232 (2007)
Kimmel, R., Sochen, N.A., Weickert, J. (eds.): Scale Space and PDE Methods in Computer Vision. LNCS, vol. 3459. Springer, Heidelberg (2005)
Mammen, E., van de Geer, S.: Locally adaptive regression splines. Ann. Statist. 25(1), 387–413 (1997)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42(5), 577–685 (1989)
Muszkieta, M.: Optimal edge detection by topological asymptotic analysis. Math. Methods Appl. Sci. 19(11), 2127–2143 (2009)
Pöschl, C., Scherzer, O.: Characterization of minimizers of convex regularization functionals. In: Frames and Operator Theory in Analysis and Signal Processing. Contemp. Math., vol. 451, pp. 219–248. Amer. Math. Soc., Providence (2008)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1-4), 259–268 (1992)
Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational methods in imaging. In: Applied Mathematical Sciences, vol. 167. Springer, New York (2009)
Shen, J.: A stochastic-variational model for Soft Mumford-Shah segmentation. In: International Journal of Biomedical Imaging, ID92329 (2006)
Sokołowski, J., Żochowski, A.: On topological derivative in shape optimization. SIAM J. Control Optimiz 37(4), 1251–1272 (1999)
Steidl, G., Didas, S., Neumann, J.: Relations between higher order TV regularization and support vector regression. In: [14], pp. 515–527 (2005)
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Dong, G., Grasmair, M., Kang, S.H., Scherzer, O. (2013). Scale and Edge Detection with Topological Derivatives. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2013. Lecture Notes in Computer Science, vol 7893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38267-3_34
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DOI: https://doi.org/10.1007/978-3-642-38267-3_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38266-6
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