Skip to main content

Compressive Sensing Inspired Multivariate Median


A new form of the multivariate median is introduced. It is defined as a point in the multidimensional space whose sum of distances from a set of multidimensional hyperplanes is minimal. This median can be used to formulate and solve the problem of sparse signal reconstruction. Application of the proposed multivariate median is illustrated on examples.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. G.R. Arce, J.L. Parades, Recursive weighted median filters admitting negative weights and their optimization. IEEE Trans. Signal Process. 48(3), 768–779 (2000)

    Article  Google Scholar 

  2. J. Astola, P. Haavisto, Y. Neuvo, Vector median filters. Proc. IEEE 78(4), 678–689 (1990)

    Article  Google Scholar 

  3. R. Chelouaha, P. Siarry, Genetic and Nelder–Mead algorithms hybridized for a more accurate global optimization of continuous multiminima functions. Eur. J. Oper. Res. 148(2), 335–348 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  4. I. Djurović, L. Stanković, J.F. Böhme, Robust L-estimation based forms of signal transforms and time-frequency representations. IEEE Trans. Signal Process. 51(7), 1753–1761 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  5. D.L. Donoho, M. Gasko, Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Stat. 20(4), 1803–1827 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  6. W. Hu, K. Wu, P.P. Shum, N.I. Zheludev, C. Soci, All-optical implementation of the ant colony optimization algorithm. Sci. Reports (2016).

    Google Scholar 

  7. P.J. Huber, E.M. Ronchetti, Robust statistics, 2nd edn. (Wiley, Hoboken, 2009)

    Book  MATH  Google Scholar 

  8. V. Katkovnik, Robust M-periodogram. IEEE Trans. Signal Process. 46, 3104–3109 (1998)

    MathSciNet  Article  Google Scholar 

  9. S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Optimization by simulated annealing. Science 220(4598), 671–680 (1983).

    MathSciNet  Article  MATH  Google Scholar 

  10. J. Larson, S.M. Wild, A batch, derivative-free algorithm for finding multiple local minima. Optim. Eng. 17(1), 205–228 (2016).

    MathSciNet  Article  MATH  Google Scholar 

  11. V.V. Lukin, S.K. Abramov, V.V. Abramova, J.T. Astola, K.O. Egiazarian, DCT-based denoising in multichannel imaging with refrence. Telecommun. Radio Eng. 75(13), 1167–1191 (2016)

    Article  Google Scholar 

  12. A. Neumaier, Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 13(1), 271–369 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  13. D. Pršić, N. Nedić, V. Stojanović, A nature inspired optimal control of pneumatic-driven parallel robot platform. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 231(1), 59–71 (2017)

    Article  Google Scholar 

  14. A. A. Roenko, V. V. Lukin, and I. Djurović, An overview of the adaptive robust DFT, EURASIP J Advances Signal Process., Special issue robust processing of nonstationary signals, 2010, (Article ID 595071), 17 pages, (2010)

  15. J.C. Spall, Introduction to stochastic search and optimization (Wiley, Hoboken, 2005). (ISBN 0-471-33052-3)

    Google Scholar 

  16. L. Stanković, M. Daković, On a gradient-based algorithm for sparse signal reconstruction in the signal/measurements domain. Math. Probl. Eng. 2016, 1–11 (2016). (Article ID 6212674)

    MathSciNet  MATH  Google Scholar 

  17. L. Stanković, M. Daković, S. Vujović, Reconstruction of sparse signals in impulsive disturbance environments. Circuits Syst. Signal Process. 36(2), 767–794 (2017)

    Article  MATH  Google Scholar 

  18. L. Stanković, D. Mandić, M. Daković, M. Brajović, Time-frequency decomposition of multivariate multicomponent signals. Signal Process. 142, 468–479 (2018)

    Article  Google Scholar 

  19. V. Stojanović, V. Filipovic, Adaptive input design for identification of output error model with constrained output. Circuits Syst. Signal Process. 33(1), 97–113 (2014)

    MathSciNet  Article  Google Scholar 

  20. V. Stojanovic, N. Nedic, A nature inspired parameter tuning approach to cascade control for hydraulically driven parallel robot platform. J.Optim. Theory Appl. 168(1), 332–347 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  21. V. Stojanović, N. Nedić, Robust identification of OE model with constrained output using optimal input design. J. Franklin Inst. 353(2), 576–593 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  22. Y. Vardi , C.H. Zhang The multivariate L1-median and associated data depth. in Proceedings of the national academy of sciences of the United States (PNAS)

  23. M. Welk Multivariate median filters and partial differential equations (March 2016). arXiv:1509.08082v2

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ljubiša Stanković.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stanković, L., Daković, M. Compressive Sensing Inspired Multivariate Median. Circuits Syst Signal Process 38, 2369–2379 (2019).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Compressive sensing
  • Sparse signals
  • Median