A Discretized Newton Flow for Time-Varying Linear Inverse Problems

  • Martin KleinsteuberEmail author
  • Simon Hawe
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 30)


The reconstruction of a signal from only a few measurements, deconvolving, or denoising are only a few interesting signal processing applications that can be formulated as linear inverse problems. Commonly, one overcomes the ill-posedness of such problems by finding solutions that match some prior assumptions on the signal best. These are often sparsity assumptions as in the theory of Compressive Sensing. In this paper, we propose a method to track the solutions of linear inverse problems, and consider the two conceptually different approaches based on the synthesis and the analysis signal model. We assume that the corresponding solutions vary smoothly over time. A discretized Newton flow allows to incorporate the time varying information for tracking and predicting the subsequent solution. This prediction requires to solve a linear system of equations, which is in general computationally cheaper than solving a new inverse problem. It may also serve as an additional prior that takes the smooth variation of the solutions into account, or as an initial guess for the preceding reconstruction. We exemplify our approach with the reconstruction of a compressively sampled synthetic video sequence.


Sparse Representation Compressive Sensing Measurement Matrix Blind Signal Synthesis Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work has partially been supported by the Cluster of Excellence CoTeSys – Cognition for Technical Systems, funded by the German Research Foundation (DFG).


  1. 1.
    Baumann, M., Helmke, U., Manton, J.: Reliable tracking algorithms for principal and minor eigenvector computations. In: 44th IEEE Conference on Decision and Control and European Control Conference, Institute of Electrical and Electronics Engineers pp. 7258–7263 (2005)Google Scholar
  2. 2.
    Benke, G.: Generalized rudin-shapiro systems. J. Fourier Anal. Appl. 1(1), 87–101 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bertalmìo, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: ACM SIGGRAPH, Association for Computing Machinery. pp. 417–424 (2000)Google Scholar
  4. 4.
    Bronstein, M., Bronstein, A., Zibulevsky, M., Zeevi, Y.: Blind deconvolution of images using optimal sparse representations. IEEE Trans. Image Process. 14(6), 726–736 (2005)CrossRefGoogle Scholar
  5. 5.
    Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theor. 52(2), 489–509 (2006)zbMATHCrossRefGoogle Scholar
  6. 6.
    Chartrand, R., Staneva, V.: Restricted isometry properties and nonconvex compressive sensing. Inverse Probl. 24(3), 1–14 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inform. Theor. 52(4), 1289–1306 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization. Proc. Nat. Acad. Sci. USA 100(5), 2197–2202 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Elad, M., Figueiredo, M., Ma, Y.: On the role of sparse and redundant representations in image processing. Proc. IEEE 98(6), 972–982 (2010)CrossRefGoogle Scholar
  11. 11.
    Elad, M., Milanfar, P., Rubinstein, R.: Analysis versus synthesis in signal priors. Inverse Probl. 3(3), 947–968 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hawe, S., Kleinsteuber, M., Diepold, K.: Cartoon-like image reconstruction via constrained p-minimization. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, Institute of Electrical and Electronics Engineers pp. 717–720 (2012)Google Scholar
  13. 13.
    Lustig, M., Donoho, D., Pauly, J.M.: Sparse MRI: The application of compressed sensing for rapid MR imaging. Mag. Reson. Med. 58(6), 1182–1195 (2007)CrossRefGoogle Scholar
  14. 14.
    Nam, S., Davies, M., Elad, M., Gribonval, R.: Cosparse analysis modeling – uniqueness and algorithms. In: IEEE International Conference on Acoustics, Speech and Signal Processing, Institute of Electrical and Electronics Engineers pp. 5804–5807 (2011)Google Scholar
  15. 15.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  16. 16.
    Romberg, J.: Imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 14–20 (2008)CrossRefGoogle Scholar
  17. 17.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)zbMATHCrossRefGoogle Scholar
  18. 18.
    Selesnick, I.W., Figueiredo, M.A.T.: Signal restoration with overcomplete wavelet transforms: Comparison of analysis and synthesis priors. In: Proceedings of SPIE Wavelets XIII, The International Society for Optical Engineering. (2009)Google Scholar
  19. 19.
    Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., Knight, K.: Sparsity and smoothness via the fused lasso. J. Roy. Statist. Soc. Ser. B 67(1), 91–108 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Tropp, J.A., Wright, S.J.: Computational methods for sparse solution of linear inverse problems. Proc. IEEE 98(6), 948–958 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Information TechnologyTechnische Universität MünchenMünchenGermany

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