Abstract
To improve the quality of image restoration methods directional information has recently been involved in the restoration process. In this paper, we propose a two step procedure for denoising images that is particularly suited to recover sharp vertices and X junctions in the presence of heavy noise. In the first step, we estimate the (smoothed) orientations of the image structures, where we find the double orientations at vertices and X junctions using a model of Aach et al. Based on shape preservation considerations this directional information is then applied to establish an energy functional which is minimized in the second step. We discuss the behavior of our new method in comparison with single direction approaches appearing, e.g., when using the classical structure tensor of Förstner and Gülch and demonstrate the very good performance of our method by numerical examples.
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References
Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 136–152. Springer, Heidelberg (2005)
Hintermüller, M., Kunisch, K.: Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM J. Appl. Math. 4(64), 1311–1333 (2004)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Esedoglu, S., Osher, S.: Decomposition of images by the anisotropic Rudin-Osher-Fatemi model. Comm. Pure and Applied Mathematics 57(12), 1609–1626 (2004)
Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numerische Mathematik 76, 167–188 (1997)
Aach, T., Mota, C., Stuke, I., Mühlich, M., Barth, E.: Analysis of superimposed oriented patterns. IEEE Trans. on Image Processing 15(12), 3690–3700 (2006)
Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: Proc. ISPRS Intercommission Conf. on Fast Processing of Photogrammetric Data, pp. 281–305 (1987)
Teuber, T.: Anisotropic smoothing using double orientations. Preprint University of Mannheim (2009)
Tschumperlé, D.: Fast anisotropic smoothing of multivalued images using curvature preserving PDEs. International Journal of Computer Vision 68(1), 65–82 (2006)
Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)
Tschumperlé, D., Deriche, R.: Vector-valued image regularization with PSDs: A common framework for different applications. IEEE Trans. on Pattern Analysis and Machine Intelligence 27(4) (2005)
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147. Springer, New York (2002)
Steidl, G., Teuber, T.: Diffusion tensors for denoising sheared and rotated rectangles (submitted) (2008)
Tschumperlé, D.: The CImg library. C++ Template Image Processing Library, http://cimg.sourceforge.net
Cabral, B., Leedom, L.C.: Imaging vector fields using line integral convolution. In: SIGGRAPH 1993, Computer Graphics, vol. 27, pp. 263–272 (1993)
Weickert, J.: Anisotropic diffusion filters for image processing based quality control. In: Fasano, A., Primicerio, M. (eds.) Proc. Seventh European Conference on Mathematics in Industry, pp. 355–362. Teubner, Stuttgart (1994)
Goldfarb, D., Wen, Z., Yin, W.: A curvilinear search method for p-harmonic flows on spheres. SIAM Journal on Imaging Sciences 2(1), 84–109 (2009)
Kimmel, R., Sochen, N.: Orientation diffusion or how to comb a porcupine? Journal of Visual Communication and Image Representation 13(1-2), 238–248 (2002)
Lysaker, O., Osher, S., Tai, X.C.: Noise removal using smoothed normals and surface fitting. IEEE Trans. on Image Processing 13(10), 1345–1357 (2004)
Vese, L., Osher, S.: Numerical methods for p-harmonic flows and applications to image processing. SIAM Journal on Numerical Analysis 40(6), 2085–2104 (2002)
Yuan, J., Schnörr, C., Steidl, G.: Convex Hodge decomposition and regularization of image flows. Journal of Mathematical Imaging and Vision 33(2), 169–177 (2009)
Rahman, T., Tai, X.C., Osher, S.: A TV-Stokes denoising algorithm. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 473–483. Springer, Heidelberg (2007)
Spira, A., Kimmel, R., Sochen, N.: A short-time Beltrami kernel for smoothing images and manifolds. IEEE Trans. on Image Processing 16(6), 1628–1636 (2007)
Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proc. Sixth Intern. Conf. on Computer Vision, pp. 839–846. Narosa Publishing House (1998)
Berkels, B., Burger, M., Droske, M., Nemitz, O., Rumpf, M.: Cartoon extraction based on anisotropic image classification. In: Vision, Modeling, and Visualization Proceedings, pp. 293–300 (2006)
Setzer, S., Steidl, G., Teuber, T.: Restoration of images with rotated shapes. Numerical Algorithms 48(1-3), 49–66 (2008)
Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: IEEE Int. Conf. on Comp. Vision and Pattern Recognition., vol. 2, pp. 60–65 (2005)
Manjón, J.V., Buades, A.: NL means. MATLAB Software, http://dmi.uib.es/~abuades/software.html
The MOSEK Optimization Toolbox, http://www.mosek.com
Scharr, H.: Diffusion-like reconstruction schemes from linear data models. In: Franke, K., Müller, K.-R., Nickolay, B., Schäfer, R. (eds.) DAGM 2006. LNCS, vol. 4174, pp. 51–60. Springer, Heidelberg (2006)
Mühlich, M., Aach, T.: A theory for multiple orientation estimation. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 69–82. Springer, Heidelberg (2006)
Setzer, S., Steidl, G.: Variational methods with higher-order derivatives in image processing. In: Neamtu, M., Schumaker, L.L. (eds.) Approximation Theory XII: San Antonio 2007, pp. 360–385. Nashboro Press (2008)
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Steidl, G., Teuber, T. (2009). Anisotropic Smoothing Using Double Orientations. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_40
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DOI: https://doi.org/10.1007/978-3-642-02256-2_40
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