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Inpainting of Cyclic Data Using First and Second Order Differences

  • Ronny Bergmann
  • Andreas Weinmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8932)

Abstract

Cyclic data arise in various image and signal processing applications such as interferometric synthetic aperture radar, electroencephalogram data analysis, and color image restoration in HSV or LCh spaces. In this paper we introduce a variational inpainting model for cyclic data which utilizes our definition of absolute cyclic second order differences. Based on analytical expressions for the proximal mappings of these differences we propose a cyclic proximal point algorithm (CPPA) for minimizing the corresponding functional. We choose appropriate cycles to implement this algorithm in an efficient way. We further introduce a simple strategy to initialize the unknown inpainting region. Numerical results both for synthetic and real-world data demonstrate the performance of our algorithm.

Keywords

Inpainting variational models with higher order differences cyclic data phase-valued data cyclic proximal point algorithm 

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References

  1. 1.
    Almeida, M., Figueiredo, M.: Deconvolving images with unknown boundaries using the alternating direction method of multipliers. IEEE Trans. on Image Process. 22(8), 3074–3086 (2013)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bačák, M.: The proximal point algorithm in metric spaces. Isr. J. Math. 194(2), 689–701 (2013)CrossRefMATHGoogle Scholar
  3. 3.
    Bačák, M.: Computing medians and means in Hadamard spaces. SIAM J. Optim. (to appear, 2014)Google Scholar
  4. 4.
    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)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bergmann, R., Laus, F., Steidl, G., Weinmann, A.: Second order differences of cyclic data and applications in variational denoising (Preprint, 2014)Google Scholar
  6. 6.
    Bertalmío, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of SIGGRAPH, New Orleans, USA, pp. 417–424 (2000)Google Scholar
  7. 7.
    Bertsekas, D.P.: Incremental gradient, subgradient, and proximal methods for convex optimization: a survey. Technical Report LIDS-P-2848, Laboratory for Information and Decision Systems, MIT, Cambridge, MA (2010)Google Scholar
  8. 8.
    Bertsekas, D.P.: Incremental proximal methods for large scale convex optimization. Math. Program., Ser. B 129(2), 163–195 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 1–42 (2009)MathSciNetGoogle Scholar
  10. 10.
    Bugeau, A., Bertalmío, M., Caselles, V., Sapiro, G.: A comprehensive framework for image inpainting. IEEE Trans. Signal Process. 19, 2634–2645 (2010)Google Scholar
  11. 11.
    Bürgmann, R., Rosen, P.A., Fielding, E.J.: Synthetic aperture radar interferometry to measure earth’s surface topography and its deformation. Annu. Rev. Earth Planet. Sci. 28(1), 169–209 (2000)CrossRefGoogle Scholar
  12. 12.
    Cai, J.-F., Dong, B., Osher, S., Shen, Z.: Image restoration: Total variation, wavelet frames, and beyond. J. Amer. Math. Soc. 25(4), 1033–1089 (2012)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Caselles, V., Morel, J.-M., Sbert, C.: An axiomatic approach to image interpolation. IEEE Trans. on Image Process. 7(3), 376–386 (1998)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Chan, T., Shen, J.: Local inpainting models and TV inpainting. SIAM J. Appl. Math. 62(3), 1019–1043 (2001)MathSciNetGoogle Scholar
  16. 16.
    Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Chan, T.F., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM (2005)Google Scholar
  18. 18.
    Chefd’Hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Regularizing flows for constrained matrix-valued images. J. Math. Imaging Vis. 20(1-2), 147–162 (2004)CrossRefGoogle Scholar
  19. 19.
    Didas, S., Weickert, J., Burgeth, B.: Properties of higher order nonlinear diffusion filtering. J. Math. Imaging Vis. 35, 208–226 (2009)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51(2), 257–270 (2002)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Fisher, N.I.: Statistical Analysis of Circular Data. Cambridge University Press (1995)Google Scholar
  22. 22.
    Fletcher, P.: Geodesic regression and the theory of least squares on Riemannian manifolds. Int. J. Comput. Vision 105(2), 171–185 (2013)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Fletcher, P., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87(2), 250–262 (2007)CrossRefMATHGoogle Scholar
  24. 24.
    Ghiglia, D.C., Pritt, M.D.: Two-dimensional phase unwrapping: theory, algorithms, and software. Wiley (1998)Google Scholar
  25. 25.
    Giaquinta, M., Modica, G., Souček, J.: Variational problems for maps of bounded variation with values in S 1. Calc. Var. 1(1), 87–121 (1993)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Giaquinta, M., Mucci, D.: The BV-energy of maps into a manifold: relaxation and density results. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(4), 483–548 (2006)MATHMathSciNetGoogle Scholar
  27. 27.
    Giaquinta, M., Mucci, D.: Maps of bounded variation with values into a manifold: total variation and relaxed energy. Pure Appl. Math. Q. 3(2), 513–538 (2007)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Grohs, P., Hardering, H., Sander, O.: Optimal a priori discretization error bounds for geodesic finite elements. Technical Report 2013-16, Seminar for Applied Mathematics, ETH Zürich, Switzerland (2013)Google Scholar
  29. 29.
    Grohs, P., Wallner, J.: Interpolatory wavelets for manifold-valued data. Appl. Comput. Harmon. Anal. 27(3), 325–333 (2009)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Guillemot, C., Le Meur, O.: Image inpainting: Overview and recent advances. IEEE Signal Process. Mag. 31(1), 127–144 (2014)CrossRefGoogle Scholar
  31. 31.
    Hinterberger, W., Scherzer, O.: Variational methods on the space of functions of bounded Hessian for convexification and denoising. Computing 76(1), 109–133 (2006)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Jammalamadaka, S.R., SenGupta, A.: Topics in Circular Statistics. World Scientific Publishing Company (2001)Google Scholar
  33. 33.
    Lefkimmiatis, S., Bourquard, A., Unser, M.: Hessian-based norm regularization for image restoration with biomedical applications. IEEE Trans. Image Process. 21(3), 983–995 (2012)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Lellmann, J., Strekalovskiy, E., Koetter, S., Cremers, D.: Total variation regularization for functions with values in a manifold. In: IEEE ICCV 2013, pp. 2944–2951 (2013)Google Scholar
  35. 35.
    Lysaker, M., Lundervold, A., Tai, X.-C.: Noise removal using fourth-order partial differential equations with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)CrossRefMATHGoogle Scholar
  36. 36.
    März, T.: Image inpainting based on coherence transport with adapted distance functions. SIAM J. Imaging Sci. 4(4), 981–1000 (2011)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    März, T.: A well-posedness framework for inpainting based on coherence transport. Found. Comput. Math. (to appear, 2014)Google Scholar
  38. 38.
    Masnou, S., Morel, J.-M.: Level lines based disocclusion. In: IEEE ICIP 1998, pp. 259–263 (1998)Google Scholar
  39. 39.
    Massonnet, D., Feigl, K.L.: Radar interferometry and its application to changes in the Earth’s surface. Rev. Geophys. 36(4), 441–500 (1998)CrossRefGoogle Scholar
  40. 40.
    Papafitsoros, K., Schönlieb, C.B.: A combined first and second order variational approach for image reconstruction. J. Math. Imaging Vis. 2(48), 308–338 (2014)CrossRefGoogle Scholar
  41. 41.
    Pennec, X.: Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Rahman, I.U., Drori, I., Stodden, V.C., Donoho, D.L.: Multiscale representations for manifold-valued data. Multiscale Model. Simul. 4(4), 1201–1232 (2005)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Rocca, F., Prati, C., Guarnieri, A.M.: Possibilities and limits of SAR interferometry. In: Proc. Int. Conf. Image Process. Techn., pp. 15–26 (1997)Google Scholar
  44. 44.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1), 259–268 (1992)CrossRefMATHGoogle Scholar
  46. 46.
    Scherzer, O.: Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing 60, 1–27 (1998)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Setzer, S., Steidl, G.: Variational methods with higher order derivatives in image processing. In: Approximation Theory XII: San Antonio 2007, pp. 360–385 (2008)Google Scholar
  48. 48.
    Setzer, S., Steidl, G., Teuber, T.: Infimal convolution regularizations with discrete l1-type functionals. Commun. Math. Sci. 9(3), 797–872 (2011)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Strekalovskiy, E., Cremers, D.: Total variation for cyclic structures: Convex relaxation and efficient minimization. In: IEEE CVPR 2011, pp. 1905–1911 (2011)Google Scholar
  50. 50.
    Strekalovskiy, E., Cremers, D.: Total cyclic variation and generalizations. J. Math. Imaging Vis. 47(3), 258–277 (2013)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Valkonen, T., Bredies, K., Knoll, F.: Total generalized variation in diffusion tensor imaging. SIAM J. Imag. Sci. 6(1), 487–525 (2013)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Weinmann, A.: Interpolatory multiscale representation for functions between manifolds. SIAM J. Math. Anal. 44(1), 162–191 (2012)CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data (2013) (preprint)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ronny Bergmann
    • 1
  • Andreas Weinmann
    • 2
  1. 1.Department of MathematicsTechnische Universität KaiserslauternKaiserslauternGermany
  2. 2.Department of MathematicsTechnische Universität München and Fast Algorithms for Biomedical Imaging Group, Helmholtz-Zentrum MünchenMünchenGermany

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