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Performance Bounds for Cosparse Multichannel Signal Recovery via Collaborative-TV

  • Lukas Kiefer
  • Stefania PetraEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)

Abstract

We consider a new class of regularizers called collaborative total variation (CTV) to cope with the ill-posed nature of multichannel image reconstruction. We recast our reconstruction problem in the analysis framework from compressed sensing. This allows us to derive theoretical measurement bounds that guarantee successful recovery of multichannel signals via CTV regularization. We derive new measurement bounds for two types of CTV from Gaussian measurements. These bounds are proved for multichannel signals of one and two dimensions. We compare them to empirical phase transitions of one-dimensional signals and obtain a good agreement especially when the sparsity of the analysis representation is not very small.

Keywords

Gaussian Measurement Measurement Bound Gaussian Width Gaussian Random Matrix Finite Difference Operator 
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.

References

  1. 1.
    Duran, J., Möller, M., Sbert, C., Cremers, D.: Collaborative total variation: a general framework for vectorial TV models. SIAM J. Imaging Sci. 9(1), 116–151 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Shikhaliev, P., Fritz, S.: Photon counting spectral CT versus conventional CT: comparative evaluation for breast imaging application. Phys. Med. Biol. 56(7), 1905–1930 (2011)CrossRefGoogle Scholar
  3. 3.
    Semerci, O., Hao, N., Kilmer, M., Miller, E.: Tensor-based formulation and nuclear norm regularization for multienergy computed tomography. IEEE Trans. Image Process. 23(4), 1678–1693 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Birkhäuser, Basel (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chandrasekaran, V., Recht, B., Parrilo, P., Willsky, A.: The convex geometry of linear inverse problems. Found. Comput. Math. 12(6), 805–849 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kabanava, M., Rauhut, H., Zhang, H.: Robust analysis \(\ell _1\)-recovery from Gaussian measurements and total variation minimization. Eur. J. Appl. Math. 26(06), 917–929 (2015)CrossRefGoogle Scholar
  7. 7.
    Amelunxen, D., Lotz, M., McCoy, M., Tropp, J.: Living on the edge: phase transitions in convex programs with random data. Inf. Inference 3(3), 224–294 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ahsanullah, M., Kibria, B., Shakil, M.: Normal and Student’s \(t\) Distributions and Their Applications. Atlantis Studies in Probability and Statistics. Atlantis Press, Paris (2014)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cody, W.J.: Rational Chebyshev approximations for the error function. Math. Comput. 23(107), 631–637 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nam, S., Davies, M.E., Elad, M., Gribonval, R.: The cosparse analysis model and algorithms. Appl. Comput. Harmon. 34(1), 30–56 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deniţiu, A., Petra, S., Schnörr, C., Schnörr, C.: Phase transitions and cosparse tomographic recovery of compound solid bodies from few projections. Fundam. Inform. 135(1–2), 73–102 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.MIG, Institute of Applied MathematicsHeidelberg UniversityHeidelbergGermany

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