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Convex Lifting-Type Methods for Curvature Regularization

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Abstract

Human visual perception is able to complete contours of objects even if they are disrupted or occluded in images. A possible mathematical imitation of this property is to represent object contours in the higher-dimensional space of positions and directions, the so-called roto-translation space, and to use this representation to promote contours with small curvature and separate overlapping objects. Interpreting image level lines as contours then leads to curvature-penalizing regularization functionals for image processing, which become convex through the additional dimension. We present the basic concept, some of its properties, as well as numerical discretization approaches for those functionals.

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References

  1. Allard, W.K.: On the first variation of a varifold. Ann. Math. (2) 95, 417–491 (1972)

    Google Scholar 

  2. Almgren Jr., F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. (2) 87, 321–391 (1968)

    Google Scholar 

  3. Ambrosio, L., Masnou, S.: A direct variational approach to a problem arising in image reconstruction. Interfaces Free Bound. 5, 63–81 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anzellotti, G., Serapioni, R., Tamanini, I.: Curvatures, functionals, currents. Indiana Univ. Math. J. 39(3), 617–669 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bae, E., Tai, X.C., Zhu, W.: Augmented Lagrangian method for an Euler’s elastica based segmentation model that promotes convex contours. Inverse Prob. Imaging 11(1), 1–23 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ballester, C., Bertalmio, M., Caselles, V., Sapiro, G., Verdera, J.: Filling-in by joint interpolation of vector fields and gray levels. IEEE Trans. Image Process. 10(8), 1200–1211 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ballester, C., Caselles, V., Verdera, J.: Disocclusion by joint interpolation of vector fields and gray levels. Multiscale Model. Simul. 2(1), 80–123 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Balzani, N., Rumpf, M.: A nested variational time discretization for parametric Willmore flow. Interfaces Free Bound. 14(4), 431–454 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bellettini, G., Mugnai, L.: Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. Henri Poincare (C) Non Linear Anal. 21(6), 839–880 (2004)

    Google Scholar 

  10. Bellettini, G., Dal Maso, G., Paolini, M.: Semicontinuity and relaxation properties of a curvature depending functional in 2d. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 20(2), 247–297 (1993)

    Google Scholar 

  11. Berkels, B., Effland, A., Rumpf, M.: A posteriori error control for the binary Mumford-Shah model. Math. Comput. 86(306), 1769–1791 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Boscain, U., Chertovskih, R.A., Gauthier, J.P., Remizov, A.O.: Hypoelliptic diffusion and human vision: a semidiscrete new twist. SIAM J. Imaging Sci. 7(2), 669–695 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bredies, K., Pock, T., Wirth, B.: Convex relaxation of a class of vertex penalizing functionals. J. Math. Imaging Vis. 47(3), 278–302 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bredies, K., Pock, T., Wirth, B.: A convex, lower semicontinuous approximation of Euler’s elastica energy. SIAM J. Math. Anal. 47(1), 566–613 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bretin, E., Masnou, S., Oudet, É.: Phase-field approximations of the Willmore functional and flow. Numer. Math. 131(1), 115–171 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chambolle, A.: Convex representation for lower semicontinuous envelopes of functionals in L1. J. Convex Anal. 8(1), 149–170 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chambolle, A., Pock, T.: Total roto-translational variation. Numer. Math. 142, 611–666 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Citti, G., Sarti, A.: A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24(3), 307–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Citti, G., Franceschiello, B., Sanguinetti, G., Sarti, A.: Sub-Riemannian mean curvature flow for image processing. SIAM J. Imaging Sci. 9(1), 212–237 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Delladio, S.: Minimizing functionals depending on surfaces and their curvatures: a class of variational problems in the setting of generalized Gauss graphs. Pac. J. Math. 179(2), 301–323 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Delladio, S.: Special generalized Gauss graphs and their application to minimization of functional involving curvatures. J. Reine Angew. Math. 1997(486), 17–44 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Droske, M., Rumpf, M.: A level set formulation for Willmore flow. Interfaces Free Bound. 6, 361–378 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. El-Zehiry, N.Y., Grady, L.: Contrast driven elastica for image segmentation. IEEE Trans. Image Process. 25(6), 2508–2518 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Lausanne (1744)

    Google Scholar 

  26. Hubel, D.H., Wiesel, T.N.: Receptive fields of single neurones in the cat’s striate cortex. J. Physiol. 148(3), 574–591 (1959)

    Article  Google Scholar 

  27. Hutchinson, J.: Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J. 35(1), 45 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kanizsa, G.: Organization in Vision: Essays on Gestalt Perception. Praeger, New York (1979)

    Google Scholar 

  29. Mantegazza, C.: Curvature varifolds with boundary. J. Differ. Geom. 43(4), 807–843 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Masnou, S.: Disocclusion: a variational approach using level lines. IEEE Trans. Image Process. 11(2), 68–76 (2002)

    Article  MathSciNet  Google Scholar 

  31. Mekes, B.: Konvexe Krümmungsregularisierung für Linienmaße in drei Dimensionen. Master’s thesis, University of Münster, 681–682 (2016). https://www.uni-muenster.de/AMM/wirth/theses/

  32. MOSEK ApS: The MOSEK optimization toolbox for MATLAB manual. Version 9.0. (2019). http://docs.mosek.com/9.0/toolbox/index.html

  33. Natterer, F.: The Mathematics of Computerized Tomography. B. G. Teubner/Wiley, Stuttgart/Chichester (1986)

    Google Scholar 

  34. Nitzberg, M., Mumford, D., Shiota, T.: Filtering, Segmentation and Depth. Springer, Berlin (1993)

    Google Scholar 

  35. Prandi, D., Boscain, U., Gauthier, J.P.: Image processing in the semidiscrete group of rototranslations. In: Lecture Notes in Computer Science, pp. 627–634. Springer International Publishing, Cham (2015)

    Google Scholar 

  36. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Lecture Notes in Mathematics, pp. 292–315. Springer, Berlin (1977)

    Google Scholar 

  37. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D Nonlinear Phenom. 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sarti, A., Citti, G., Petitot, J.: The symplectic structure of the primary visual cortex. Biol. Cybern. 98(1), 33–48 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Schoenemann, T., Kahl, F., Masnou, S., Cremers, D.: A linear framework for region-based image segmentation and inpainting involving curvature penalization. Int. J. Comput. Vis. 99(1), 53–68 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schoenemann, T., Masnou, S., Cremers, D.: On a linear programming approach to the discrete Willmore boundary value problem and generalizations. In: Curves and Surfaces, pp. 629–646. Springer, Berlin (2012)

    Google Scholar 

  41. Sharma, U., Duits, R.: Left-invariant evolutions of wavelet transforms on the similitude group. Appl. Comput. Harmon. Anal. 39(1), 110–137 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Shen, J., Kang, S.H., Chan, T.F.: Euler’s elastica and curvature-based inpainting. SIAM J. Appl. Math. 63(2), 564–592 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tai, X.C., Hahn, J., Chung, G.J.: A fast algorithm for Euler’s elastica model using augmented Lagrangian method. SIAM J. Imaging Sci. 4(1), 313–344 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tinius, D.: Discretization via line measures for a curvature regularization framework. Master’s thesis, University of Münster, 711–712 (2015). https://www.uni-muenster.de/AMM/wirth/theses/

  45. Wertheimer, M.: Untersuchungen zur Lehre von der Gestalt. Psychol. Forsch. 4, 301–350 (1923)

    Article  Google Scholar 

  46. Willmore, T.J.: Riemannian Geometry. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  47. Yashtini, M., Kang, S.H.: Alternating direction method of multiplier for Euler’s elastica-based denoising. In: Lecture Notes in Computer Science, pp. 690–701. Springer International Publishing, Cham (2015)

    Google Scholar 

  48. Yashtini, M., Kang, S.H.: A fast relaxed normal two split method and an effective weighted TV approach for Euler’s elastica image inpainting. SIAM J. Imaging Sci. 9(4), 1552–1581 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  49. Young, L.C.: Generalized surfaces in the calculus of variations. Ann. Math. (2) 43, 84–103 (1942)

    Google Scholar 

  50. Zhu, W., Tai, X.C., Chan, T.: Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Prob. Imaging 7(4), 1409–1432 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Cells-in-Motion Cluster of Excellence (EXC 1003—CiM), University of Münster, Germany, by Germany’s Excellence Strategy—EXC 2044—, Mathematics Münster: Dynamics—Geometry—Structure, and by the Alfried Krupp Prize for Young University Teachers awarded by the Alfried Krupp von Bohlen und Halbach-Stiftung.

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Authors and Affiliations

  1. Westfälische Wilhelms-Universität Münster, Applied Mathematics Münster, Münster, Germany

    Ulrich Böttcher & Benedikt Wirth

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  1. Ulrich Böttcher
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  2. Benedikt Wirth
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Correspondence to Ulrich Böttcher .

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Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Böttcher, U., Wirth, B. (2020). Convex Lifting-Type Methods for Curvature Regularization. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_7

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