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Computational Statistics

, Volume 34, Issue 3, pp 1315–1335 | Cite as

Robust estimation for spatial autoregressive processes based on bounded innovation propagation representations

  • Grisel Maribel Britos
  • Silvia María OjedaEmail author
Original Paper
  • 167 Downloads

Abstract

Robust methods have been a successful approach for dealing with contamination and noise in the context of spatial statistics and, in particular, in image processing. In this paper, we introduce a new robust method for spatial autoregressive models. Our method, called BMM-2D, relies on representing a two-dimensional autoregressive process with an auxiliary model to attenuate the effect of contamination (outliers). We compare the performance of our method with existing robust estimators and the least squares estimator via a comprehensive Monte Carlo simulation study, which considers different levels of replacement contamination and window sizes. The results show that the new estimator is superior to the other estimators, both in accuracy and precision. An application to image filtering highlights the findings and illustrates how the estimator works in practical applications.

Keywords

AR-2D models Robust estimators Image processing Spatial models 

Notes

Acknowledgements

We thank Ph.D. Oscar Bustos and Ph.D. Ronny Vallejos for helpful comments and suggestions. The authors were supported by Secyt-UNC grant (Res. Secyt 313/2016.), Argentina. The first author was partially supported by CIEM-CONICET, Argentina.

Supplementary material

180_2018_845_MOESM1_ESM.pdf (5 mb)
Supplementary material 1 (pdf 5120 KB)
180_2018_845_MOESM2_ESM.r (5 kb)
Supplementary material 2 (R 4 KB)
180_2018_845_MOESM3_ESM.r (12 kb)
Supplementary material 3 (R 11 KB)
180_2018_845_MOESM4_ESM.r (8 kb)
Supplementary material 4 (R 8 KB)

References

  1. Achim A, Kuruoglu E, Zerubia J (2006) SAR image filtering based on the heavy-tailed Rayleigh model. IEEE Trans Image Process 15(9):2686–2693CrossRefGoogle Scholar
  2. Allende H, Galbiati J (2004) A non-parametric filter for digital image restoration, using cluster analysis. Pattern Recognit Lett 25(8):841–847CrossRefGoogle Scholar
  3. Allende H, Galbiati J, Vallejos R (1998) Digital image restoration using autoregressive time series type models. Bull Eur Spat Agency 434:53–59Google Scholar
  4. Allende H, Galbiati J, Vallejos R (2001) Robust image modeling on image processing. Pattern Recognit Lett 22(11):1219–1231CrossRefGoogle Scholar
  5. Aysal T, Barner K (2006) Quadratic weighted median filters for edge enhancement of noisy images. IEEE Trans Image Process 15(11):3294–3310CrossRefGoogle Scholar
  6. Baran S, Pap G, Zuijlen MV (2004) Asymptotic inference for a nearly unstable sequence of stationary spatial AR models. Stat Probab Lett 69(1):53–61MathSciNetzbMATHCrossRefGoogle Scholar
  7. Basu S, Reinsel G (1993) Properties of the spatial unilateral first-order ARMA model. Adv Appl Probab 25(3):631–648MathSciNetzbMATHCrossRefGoogle Scholar
  8. Bhandari A, Kumar A, Singh G (2015) Improved knee transfer function and gamma correction based method for contrast and brightness enhancement of satellite image. Int J Electron Commun 69(2):579–589CrossRefGoogle Scholar
  9. Bustos O (1997) Robust statistics in SAR image processing. In: Processing of the first Latino-American seminar on radar remote sensing, pp 81–89Google Scholar
  10. Bustos O, Yohai V (1986) Robust estimates for ARMA models. J Am Stat Assoc 81(393):155–168MathSciNetCrossRefGoogle Scholar
  11. Bustos O, Ojeda S, Vallejos R (2009a) Spatial ARMA models and its applications to image filtering. Braz J Probab Stat 23(2):141–165MathSciNetzbMATHCrossRefGoogle Scholar
  12. Bustos O, Ruiz M, Ojeda S, Vallejos R, Frery A (2009b) Asymptotic behavior of RA-estimates in autoregressive 2D processes. J Stat Plan Inference 139(10):3649–3664MathSciNetzbMATHCrossRefGoogle Scholar
  13. Comport A, Marchand E, Chaumette F (2006) Statistically robust 2D visual servoing. IEEE Trans Robot 22(2):415–420CrossRefGoogle Scholar
  14. Gottardo R, Raftery A, Yeung KY, Bumgarner R (2006) Bayesian robust inference for differential gene expression in microarrays with multiple samples. Biometrics 62(1):10–18MathSciNetzbMATHCrossRefGoogle Scholar
  15. Hamza A, Krim H (2001) Image denoising: a nonlinear robust statistical approach. IEEE Trans Signal Process 49(12):3045–3054CrossRefGoogle Scholar
  16. Huang H, Lee T (2006) Data adaptive median filters for signal and image denoising using a generalized sure criterion. IEEE Signal Process Lett 13(9):561–564CrossRefGoogle Scholar
  17. Huber P (1964) Robust estimation of a location parameter. Ann Math Stat 35(1):73–101MathSciNetzbMATHCrossRefGoogle Scholar
  18. Ji Z, Huang Y, Xia Y, Zheng Y (2017) A robust modified gaussian mixture model with rough set for image segmentation. Neurocomputing 266:550–565CrossRefGoogle Scholar
  19. Kashyap R, Eom K (1988) Robust image modeling techniques with an image restoration application. IEEE Trans Acoust Speech Signal Process 36(8):1313–1325zbMATHCrossRefGoogle Scholar
  20. Kim J, Han J (2006) Outlier correction from uncalibrated image sequence using the triangulation method. Pattern Recognit 39(3):394–404MathSciNetzbMATHCrossRefGoogle Scholar
  21. Lin T (2007) A new adaptive center weighted median filter for suppressing impulsive noise in images. Inf Sci 177(4):1073–1087CrossRefGoogle Scholar
  22. Maronna R, Martin R, Yohai V (2006) Robust statistics. Wiley, ChichesterzbMATHCrossRefGoogle Scholar
  23. Martin R (1980) Robust estimation of autoregressive models. In: Brillinger DR, Tiao GC (eds) Directions in time series. Institute of Mathematical Statistics, Haywood, pp 228–262Google Scholar
  24. Muler N, Peña D, Yohai V (2009) Robust estimation for ARMA models. Ann Stat 37(2):816–840MathSciNetzbMATHCrossRefGoogle Scholar
  25. Ojeda S (1999) Robust RA estimators for bidimensional autoregressive models. PhD thesis, Universidad Nacional de CórdobaGoogle Scholar
  26. Ojeda S, Vallejos R, Lucini M (2002) Performance of robust RA estimator for bidimensional autoregressive models. J Stat Comput Simul 72(1):47–62MathSciNetzbMATHCrossRefGoogle Scholar
  27. Ojeda S, Vallejos R, Bustos O (2010) A new image segmentation algorithm with applications to image inpainting. Comput Stat Data Anal 54(9):2082–2093MathSciNetzbMATHCrossRefGoogle Scholar
  28. Ojeda S, Vallejos R, Lamberti P (2012) Measure of similarity between images based on the codispersion coefficient. J Electron Imaging 21(2):023019CrossRefGoogle Scholar
  29. Ojeda S, Britos G, Vallejos R (2018) An image quality index based on coefficients of spatial association with an application to image fusion. Spat Stat 23:1–16MathSciNetCrossRefGoogle Scholar
  30. Pistonesi S, Martinez J, Ojeda S, Vallejos R (2015) A novel quality image fusion assessment based on maximum codispersion. In: Pardo A, Kittler J (eds) Proceedings of the Iberoamerican congress on pattern recognition, pp 383–390Google Scholar
  31. Ponomarenko N, Lukin V, Zelensky A, Egiazarian K, Carli M, Battisti F (2009) TID2008—a database for evaluation of full-reference visual quality assessment metrics. Adv Mod Radioelectron 10(4):30–45Google Scholar
  32. Ponomarenko N, Jin L, Ieremeiev O, Lukin V, Egiazarian K, Astola J, Vozel B, Chehdi K, Carli M, Battisti F, Kuo CJ (2015) Image database TID2013: peculiarities, results and perspectives. Signal Process Image Commun 30:57–77CrossRefGoogle Scholar
  33. Prastawa M, Bullitt E, Ho S, Gerig G (2004) A brain tumor segmentation framework based on outlier detection. Med Image Anal 8(3):275–283CrossRefGoogle Scholar
  34. Quintana C, Ojeda S, Tirao G, Valente M (2011) Mammography image detection processing for automatic micro-calcification recognition. Chil J Stat 2(2):69–79MathSciNetzbMATHGoogle Scholar
  35. R Core Team (2017) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. https://www.R-project.org/
  36. Sadabadi M, Shafiee M, Karrari M (2009) Two-dimensional ARMA model order determination. ISA Trans 48(3):247–253zbMATHCrossRefGoogle Scholar
  37. Schabenberger O, Gotway C (2005) Statistical methods for spatial data analysis. Chapman and Hall/CRC, Boca RatonzbMATHGoogle Scholar
  38. Singh M, Arora H, Ahuja N (2004) A robust probabilistic estimation framework for parametric image models. Comput Vis ECCV 2004:508–522zbMATHGoogle Scholar
  39. Tarel J, Ieng S, Charbonnier P (2002) Using robust estimation algorithms for tracking explicit curves. Comput Vis ECCV 2002:492–507zbMATHGoogle Scholar
  40. Theodoridis S, Koutroumbas K (2003) Pattern recognition. Elsevier, BurlingtonzbMATHGoogle Scholar
  41. Tjøstheim D (1978) Statistical spatial series modelling. Adv Appl Probab 10(1):130–154MathSciNetzbMATHCrossRefGoogle Scholar
  42. Vallejos R, García-Donato G (2006) Bayesian analysis of contaminated quarter plane moving average models. J Stat Comput Simul 76(2):131–147MathSciNetzbMATHCrossRefGoogle Scholar
  43. Vallejos R, Mardesic T (2004) A recursive algorithm to restore images based on robust estimation of NSHP autoregressive models. J Comput Graph Stat 13(3):674–682MathSciNetCrossRefGoogle Scholar
  44. Wang Z, Bovik A (2002) A universal image quality index. IEEE Signal Process Lett 9(3):81–84CrossRefGoogle Scholar
  45. Whittle P (1954) On stationary processes in the plane. Biometrika 41:434–449MathSciNetzbMATHCrossRefGoogle Scholar
  46. Yao Q, Brockwell P (2006) Gaussian maximum likelihood estimation for ARMA models II: spatial processes. Bernoulli 12(3):403–429zbMATHCrossRefGoogle Scholar
  47. Zielinski J, Bouaynaya N, Schonfeld D (2010) Two-dimensional ARMA modeling for breast cancer detection and classification. In: Proceedings of the international conference on signal processing and communications. IEEE, pp 1–4Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Facultad de Matemática, Astronomía, Física y ComputaciónUniversidad Nacional de CórdobaCórdobaArgentina

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