Abstract
The Mumford-Shah functional minimization, and related algorithms for image segmentation, involve a tradeoff between a two-dimensional image structure and one-dimensional parametric curves (contours) that surround objects or distinct regions in the image.
We propose an alternative functional that is independent of parameterization; it is a geometric functional which is given in terms of the geometry of surfaces representing the data and image in a feature space. The Γ-convergence technique is combined with the minimal surface theory in order to yield a global generalization of the Mumford-Shah segmentation functional.
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References
Bertalmio, M., et al.: Image inpainting. In: Computer Graphics (SIGGRAPH 2000), July 2000, pp. 417–424 (2000)
Canny, J.: A Computational Approach to Edge Detection. IEEE Transactions on PAMI 8(6), 679–698 (1986)
Dierkes, U., et al.: Minimal Surfaces I. Grundlehren der mathematischen Wissenschaften, vol. 295. Springer, Heidelberg (1991)
Kimmel, R., Malladi, R., Sochen, N.: On the geometry of texture. Report LBNL 39640, LBNL, UC-405 UC Berkeley (November 1996)
Kluzner, V., Wolansky, G., Zeevi, Y.Y.: Minimal surfaces, measure-based metric and image segmentation. CCIT Report No. 605, EE Pub. No. 1562 (November 2006)
Morel, J.M., Solimini, S.: Variational Methods in Image Segmentation. Birkhauser, Boston (1995)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)
Otsu, N.: A Threshold Selection Method from Gray-Level Histograms. IEEE Transactions on Systems, Man, and Cybernetics 9(1), 62–66 (1979)
Richardson, T., Mitter, S.: Approximation, computation and distortion in the variational formulation. In: ter Haar Romeny, M. (ed.) Comm. Geometric-Driven Diffusion in Computer Vision, Kluwer Academic Publishers, Dordrecht (1994)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Sochen, N.A., Kimmel, R., Malladi, R.: A general framework for low-level vision. IEEE Trans. Image Processing 7, 310–318 (1998)
Vogel, C., Oman, M.: Iterative methods for total variation denoising. SIAM J. Sci. Statist. Comput. 17(1), 227–238 (1996)
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Kluzner, V., Wolansky, G., Zeevi, Y.Y. (2007). A Geometric-Functional-Based Image Segmentation and Inpainting. In: Sgallari, F., Murli, A., Paragios, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer Science, vol 4485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72823-8_15
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DOI: https://doi.org/10.1007/978-3-540-72823-8_15
Publisher Name: Springer, Berlin, Heidelberg
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