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Projections onto convex sets parameter estimation through harmony search and its application for image restoration

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Abstract

Image restoration is a research field that attempts to recover a blurred and noisy image. Although we have one-step algorithms that are often fast for image restoration, iterative formulations allow a better control of the trade-off between the enhancement of high frequencies (image details) and noise amplification. Projections onto convex sets (POCS) is an iterative—and parametric-based approach that employs a priori knowledge about the blurred image to guide the restoration process, with promising results in different application domains. However, a proper choice of its parameters is a high computational burden task, since they are continuous-valued and there are an infinity of possible values to be checked. In this paper, we propose to optimize POCS parameters by means of harmony search-based techniques, since they provide elegant and simple formulations for optimization problems. The proposed approach has been validated in synthetic and real images, being able to select suitable parameters in a reasonable amount of time.

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Notes

  1. Image restoration and reconstruction attempt at the same purpose, they are just used under different names depending on the field of knowledge.

  2. Under-projections refer to RAP constrained to \(\lambda < 1\), and over-projections refer to RAP constrained to \(\lambda > 1\).

  3. The better is the quality of the restored image \({\hat{f}}\), we have that \(({\hat{f}}-f)^2\rightarrow 0\). Thus, \(\frac{(g-f)^2}{({\hat{f}}-f)^2}\rightarrow \infty \), which means that high ISNR values stand for better restored images.

  4. The original (uncorrupted) image that has been degraded is the so-called “phantom image”.

  5. We have used 60 agents and 30 iterations for both BHA and HS techniques.

  6. The search range for \(\lambda \) was executed within [0, 2] with steps of 0.05, and the search range of K was executed within the range [1, 20] with steps of 1. This means we have 800 evaluations of the fitness function considering the near-exhaustive search. Although one could meta-optimize the step of grid-search, we do not recommend that, since it might be too costly.

References

  • Cauchy AL (1847) Méthode générale pour la résolution des systèmes d’équations simultanées. Compte Rendu des S’eances de L’Acad’emie des Sciences XXV S’erie A(25):536–538

    Google Scholar 

  • Chen Y, Pock T, Ranftl R, Bischof H (2013) Revisiting loss-specific training of filter-based mrfs for image restoration. In: Weickert J, Hein M, Schiele B (eds) Pattern recognition, lecture notes in computer science, vol. 8142, pp. 271–281. Springer, Berlin. doi:10.1007/978-3-642-40602-7_30

  • Dong W, Zhang L, Shi G, Li X (2013) Nonlocally centralized sparse representation for image restoration. IEEE Trans Image Process 22(4):1620–1630

    Article  MathSciNet  Google Scholar 

  • Fletcher R (1970) A new approach to variable metric algorithms. Comput J 13(3):317–322

    Article  MathSciNet  MATH  Google Scholar 

  • Fletcher R, Powell MJD (1963) A rapidly convergent descent method for minimization. Comput J 6(2):163–168

    Article  MathSciNet  MATH  Google Scholar 

  • Geem ZW (2009) Music-inspired harmony search algorithm: theory and applications, 1st edn. Springer Publishing Company, Incorporated

  • Gonzalez R, Woods R (2006) Digital image processing, 3rd edn. Prentice-Hall Inc, Upper Saddle River

    Google Scholar 

  • Hatamlou A (2013) Black hole: a new heuristic optimization approach for data clustering. Inf Sci 222:175–184

  • Kaczmarz S (1937) Angenäherte auflösung von systemen linearer gleichungen. Bulletin International de l’Académie Polonaise des Sciences et des Lettres. Classe des Sciences Mathmatiques et Naturelles. Série A, Sciences Mathématiques 35:355–357

  • Katsaggelos A (1991) Digital image restoration. Springer, New York

    Book  Google Scholar 

  • Kennedy J, Eberhart R (2001) Swarm intelligence. M. Kaufman, Los Altos

    Google Scholar 

  • Liu H, Xiong R, Ma S, Fan X, Gao W (2014) Non-local extension of total variation regularization for image restoration. In: IEEE international symposium on circuits and systems, pp 1102–1105

  • Lucy LB (1974) An iterative technique for the rectification of observed distributions. Astron J 79(6):745–754

    Article  Google Scholar 

  • Lv XG, Song YZ, Wang SX, Le J (2013) Image restoration with a high-order total variation minimization method. Appl Math Model 37(16–17):8210–8224

    Article  MathSciNet  Google Scholar 

  • Mahdavi M, Fesanghary M, Damangir E (2007) An improved harmony search algorithm for solving optimization problems. Appl Math Comput 188(2):1567–1579

    MathSciNet  MATH  Google Scholar 

  • Mammone R (1992) Computation methods of signal recovery and recognition. Wiley, London

    Google Scholar 

  • Omran MG, Mahdavi M (2008) Global-best harmony search. Appl Math Comput 198(2):643–656

    MathSciNet  MATH  Google Scholar 

  • Pan QK, Suganthan P, Tasgetiren MF, Liang J (2010) A self-adaptive global best harmony search algorithm for continuous optimization problems. Appl Math Comput 216(3):830–848

    MathSciNet  MATH  Google Scholar 

  • Papa JP, Fonseca LMG, de Carvalho LAS (2010) Projections onto convex sets through particle swarm optimization and its application for remote sensing image restoration. Pattern Recognit Lett 31(13):1876–1886

    Article  Google Scholar 

  • Pires RG, Pereira LAM, Mansano AF, Papa JP (2013) A hybrid image restoration algorithm based on projections onto convex sets and harmony search. In: IEEE international symposium on circuits and systems, pp 2824–2827

  • Polak E, Ribiere G (1969) Note sur la convergence de methode de directions conjuguees. ESAIM Math Model Numer Anal 3(R1):35–43

    MathSciNet  MATH  Google Scholar 

  • Richardson WH (1972) Bayesian-based iterative method of image restoration. J Opt Soc Am 62(1):55–59

    Article  Google Scholar 

  • Samaddar S, Mammone R (1995) Image restoration using a row action projection method with adaptative smoothing. Opt Eng 34(4):1147

    Google Scholar 

  • Stark H, Yang Y, Yang Y (1998) Vector space projections: a numerical approach to signal and image processing, neural nets, and optics. Wiley, London

    MATH  Google Scholar 

  • Wang Z, Bovik A (2002) A universal image quality index. IEEE Signal Process Lett 9(3):81–84

    Article  Google Scholar 

  • Wiener N (1949) Extrapolation, interpolation, and smoothing of stationary time series. Wiley, London

    MATH  Google Scholar 

  • Wilcoxon F (1945) Individual comparisons by ranking methods. Biom Bull 1(6):80–83

    Article  Google Scholar 

  • Zou D, Gao L, Wu J, Li S (2010) Novel global harmony search algorithm for unconstrained problems. Neurocomputing 73(1618):3308–3318

    Article  Google Scholar 

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Acknowledgments

The authors would like to thank Capes and FAPESP Grants #2009/16206-1, #2011/14094-1, #2013/20387-7 and #2014/16250-9, as well as CNPq Grants #303182/2011-3, #470571/2013-6 and #306166/2014-3.

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Authors and Affiliations

  1. Department of Computing, UNESP - Univ Estadual Paulista, Bauru, Brazil

    Rafael G. Pires, Danillo R. Pereira, Luís A. M. Pereira, Alex F. Mansano & João P. Papa

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  1. Rafael G. Pires
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  2. Danillo R. Pereira
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  3. Luís A. M. Pereira
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  4. Alex F. Mansano
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  5. João P. Papa
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Correspondence to João P. Papa.

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Pires, R.G., Pereira, D.R., Pereira, L.A.M. et al. Projections onto convex sets parameter estimation through harmony search and its application for image restoration. Nat Comput 15, 493–502 (2016). https://doi.org/10.1007/s11047-015-9507-4

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