Regularization with Sparse Vector Fields: From Image Compression to TV-type Reconstruction

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)

Abstract

This paper introduces a novel variational approach for image compression motivated by recent PDE-based approaches combining edge detection and Laplacian inpainting. The essential feature is to encode the image via a sparse vector field, ideally concentrating on a set of measure zero. An equivalent reformulation of the compression approach leads to a variational model resembling the ROF-model for image denoising, hence we further study the properties of the effective regularization functional introduced by the novel approach and discuss similarities to TV and TGV functionals. Moreover we computationally investigate the behaviour of the model with sparse vector fields for compression in particular for high resolution images and give an outlook towards denoising.

Keywords

Image compression Denoising Reconstruction Diffusion inpainting Sparsity Total variation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andreu, F., Ballester, C., Caselles, V., Mazon, J.M.: Minimizing total variation flow. Differential and integral equations 14, 321–360 (2001)MATHMathSciNetGoogle Scholar
  2. 2.
    Benning, M., Brune, C., Burger, M., Müller, J.: Higher-order TV methods: Enhancement via Bregman iteration. J Sci Comput 54, 269–310 (2013)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Benning, M., Burger, M.: Ground states and singular vectors of convex variational regularization methods. Methods and Applications of Analysis 20, 295–334 (2013)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3, 492–526 (2010)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bredies, K., Pikkarainen, H.K.: Inverse problems in spaces of measures. ESAIM: Control, Optimisation and Calculus of Variations 19, 190–218 (2013)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Burger, M., Gilboa, G., Osher, S., Xu, J.: Nonlinear inverse scale space methods. Communications in Mathematical Sciences 4, 179–212 (2006)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. In: Fornasier, M. (ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery, pp. 263–340. DeGruyter, Berlin (2010)Google Scholar
  8. 8.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM Journal on Scientific Computing 22, 503–516 (2000)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Delfour, M.C., Zolesio, J.-P.: Shapes and geometries: metrics, analysis, differential calculus, and optimization. SIAM, Philadelphia (2011)CrossRefGoogle Scholar
  11. 11.
    Esser, E., Zhang, X., Chan, T.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015–1046 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Goldstein, T., Esser, E., Baraniuk, R.: Adaptive Primal-Dual Hybrid GradientMethods for Saddle-Point Problems. arXiv Preprint arxiv:1305.0546v1 (2013)Google Scholar
  13. 13.
    Mainberger, M., Bruhn, A., Weickert, J., Forchhammer, S.: Edge-based compression of cartoon-like images with homogeneous diffusion. Pattern Recognition 44(9), 1859–1873 (2011)CrossRefGoogle Scholar
  14. 14.
    Mainberger, M., Weickert, J.: Edge-based image compression with homogeneous diffusion. In: Jiang, X., Petkov, N. (eds.) CAIP 2009. LNCS, vol. 5702, pp. 476–483. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  15. 15.
    Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. AMS, Providence (2001)CrossRefMATHGoogle Scholar
  16. 16.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An Iterative Regularization Method for Total Variation-Based Image Restoration. Multiscale Model. Simul. 4, 460–489 (2005)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Ring, W.: Structural properties of solutions to total variation regularization problems. ESAIM: Math. Modelling Numer. Analysis 34, 799–810 (2000)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)CrossRefMATHGoogle Scholar
  19. 19.
    Valkonen, T.: The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising. arXiv preprint arXiv:1407.1531 (2014)
  20. 20.
    Weickert, J., Hagenburg, K., Breuß, M., Vogel, O.: Linear osmosis models for visual computing. In: Heyden, A., Kahl, F., Olsson, C., Oskarsson, M., Tai, X.-C. (eds.) EMMCVPR 2013. LNCS, vol. 8081, pp. 26–39. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Eva-Maria Brinkmann
    • 1
  • Martin Burger
    • 1
  • Joana Grah
    • 1
    • 2
  1. 1.Institute for Computational and Applied MathematicsWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Department of Applied Mathematics and Theoretical PhysicsUniversity of Cambridge, Centre for Mathematical SciencesCambridgeUK

Personalised recommendations