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W-operator window design by minimization of mean conditional entropy

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Abstract

This paper presents a technique that gives a minimal window W for the estimation of a W-operator from training data. The idea is to choose a subset of variables W that maximizes the information observed in a training set. The task is formalized as a combinatorial optimization problem, where the search space is the powerset of the candidate variables and the measure to be minimized is the mean entropy of the estimated conditional probabilities. As a full exploration of the search space requires prohibitive computational effort, some heuristics of the feature selection literature are applied. The proposed technique is mathematically sound and experimental results including binary image filtering and gray-scale texture recognition show its successful performance in practice.

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References

  1. Barrera J, Terada R, Hirata R Jr, Hirata NST (2000) Automatic programming of morphological machines by pac learning. Fundamenta Informaticae, pp 229–258

  2. Di Ruberto C, Dempster A, Khan S, Jarra B (2002) Analysis of infected blood cell images using morphological operators. Image Vis Comput 20(2):133–146

    Article  Google Scholar 

  3. Dougherty ER (2004) Automatic design of morphological operators. J Electron Imaging 13(3):486–491

    Article  Google Scholar 

  4. Green AC, Marshall S, Greenhalgh D, Dougherty ER (2003) Design of multi-mask aperture filters. Signal Processing 83(9):1961–1971

    Article  Google Scholar 

  5. Pitas I, Venetsanopoulos AN (1990) Nonlinear digital filters. Kluwer, Dordrecht

    MATH  Google Scholar 

  6. Dougherty ER, Barrera J (2002) Pattern recognition theory in nonlinear signal processing. J Math Imaging Vis 16(3):181–197

    Article  MATH  MathSciNet  Google Scholar 

  7. Hirata NST, Dougherty ER, Barrera J (2000) A switching algorithm for design of optimal increasing binary filters over large windows. Pattern Recognit 33:1059–1081

    Article  Google Scholar 

  8. Soille P (2002) On morphological operators based on rank filters. Pattern Recognit 35(2):527–535

    Article  MATH  Google Scholar 

  9. Jain A, Zongker D (1997) Feature selection—evaluation, application, and small sample performance. IEEE Trans Pattern Anal Mach Intell 19(2):153–158

    Article  Google Scholar 

  10. Theodoridis S, Koutroumbas K (1999) Pattern recognition, 1st edn. Academic, New York

    MATH  Google Scholar 

  11. Pudil P, Novovicová J, Kittler J (1994) Floating search methods in feature selection. Pattern Recognit Lett 15:1119–1125

    Article  Google Scholar 

  12. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623–656

    Google Scholar 

  13. Cover TM, Thomas JA (1991) Elements of information theory. In: Wiley series in telecommunications. Wiley, New York

    Google Scholar 

  14. Kullback S (1968) Information theory and statistics. Dover, New York

    MATH  Google Scholar 

  15. Soofi ES (2000) Principal information theoretic approaches. J Am Stat Assoc 95:1349–1353

    Article  MATH  MathSciNet  Google Scholar 

  16. Duda RO, Hart PE, Stork D (2000) Pattern classification. Wiley, New York

    MATH  Google Scholar 

  17. Hall MA, Smith LA (1999) Feature selection for machine learning: comparing a correlation-based filter approach to the wrapper. In: Proceedings of the FLAIRS conference, AAAI Press, pp 235–239

  18. Lewis DD (1992) Feature selection and feature extraction for text categorization. In: Proceedings of speech and natural language workshop, Morgan Kaufmann, San Mateo, pp 212–217

  19. Bonnlander BV, Weigend AS (1994) Selecting input variables using mutual information and nonparametric density estimation. In: Proceedings of the 1994 international symposium on artificial neural networks, Tainan, pp 42–50

  20. Viola P, Wells WM III (1997) Alignment by maximization of mutual information. Int J Comput Vision 24(2):137–154

    Article  Google Scholar 

  21. Zaffalon M, Hutter M (2002) Robust feature selection by mutual information distributions. In: 18th international conference on uncertainty in artificial intelligence (UAI), pp 577–584

  22. Costa LF, Cesar RM Jr (2001) Shape analysis and classification: theory and practice. CRC Press, Boca Raton

    Google Scholar 

  23. Dougherty ER, Kim S, Chen Y (2000) Coefficient of determination in nonlinear signal processing. Signal Processing 80:2219–2235

    Article  MATH  Google Scholar 

  24. Vaquero DA, Barrera J, Hirata R Jr (2005) A maximum-likelihood approach for multiresolution W-operator design. In: Proceedings of XVIII Brazilian symposium on computer graphics and image processing (SIBGRAPI). IEEE Computer Society Press, pp 71–78

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Acknowledgments

The authors are grateful to FAPESP (99/12765-2, 01/094 01-0, 04/03967-0 and 05/00587-5), CNPq (300722/98-2, 468 413/00-6, 521097/01-0 474596/04-4 and 491323/05-0) and CAPES for financial support. This work was partially supported by grant 1 D43 TW07015-01 from the National Institutes of Health, USA. We also thank Daniel O. Dantas by his complementing post-processing idea for texture recognition (mode filter applied more than once).

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  1. Departamento de Ciência da Computação, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo, SP, Brasil

    David C. Martins Jr, Roberto M. Cesar Jr & Junior Barrera

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  1. David C. Martins Jr
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  2. Roberto M. Cesar Jr
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  3. Junior Barrera
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Correspondence to David C. Martins Jr.

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Martins, D.C., Cesar, R.M. & Barrera, J. W-operator window design by minimization of mean conditional entropy. Pattern Anal Applic 9, 139–153 (2006). https://doi.org/10.1007/s10044-006-0031-0

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