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Comparison and Evaluation of First Derivatives Estimation

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Computer Vision and Graphics (ICCVG 2016)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9972))

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Abstract

Computing derivatives from observed integral data is known as an ill-posed inverse problem. The ill-posed qualifier refers to the noise amplification that can occur in the numerical solution if appropriate measures are not taken (small errors for measurement values on specified points may induce large errors in the derivatives). For example, the accurate computation of the derivatives is often hampered in medical images by the presence of noise and a limited resolution, affecting the accuracy of segmentation methods. In our case, we want to obtain an upper airways segmentation, so it is necessary to compute the first derivatives as accurately as possible, in order to use gradient-based segmentation techniques. For this reason, the aim of this paper is to present a comparative analysis of several methods (finite differences, interpolation, operators and regularization), that have been developed for numerical differentiation. Numerical results are presented for artificial and real data sets.

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Notes

  1. 1.

    Insight Segmentation and Registration Toolkit, http://www.itk.org/.

References

  1. Anderssen, R.S., Hegland, M.: For numerical differentiation, dimensionality can be a blessing!. Math. Comput. 68, 1121–1141 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bouma, H., Vilanova, A., Bescós, J.O., Haar Romeny, B.M., Gerritsen, F.A.: Fast and accurate gaussian derivatives based on B-splines. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 406–417. Springer, Heidelberg (2007). doi:10.1007/978-3-540-72823-8_35

    Chapter  Google Scholar 

  3. Burden, R.L., Faires, J.D.: Numerical Analysis. Brooks Colem, USA (2011)

    MATH  Google Scholar 

  4. Cullum, J.: Numerical differentiation and regularization. SIAM J. Numer. Anal. 8, 254–265 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dahlquist, G., Björck, Å.: Numerical Methodsin Scientific Computing, vol. I. Prentice-Hall, USA (2007)

    Google Scholar 

  6. Deriche, R.: Recursively implementating the gaussian and its derivatives. Research report 1893, INRIA, France (1993)

    Google Scholar 

  7. Farid, H., Simoncelli, E.P.: Differentiation of discrete multidimensional signals. IEEE Trans. Image Process. 13, 496–508 (2004)

    Article  MathSciNet  Google Scholar 

  8. Gambaruto, A.M.: Processing the image gradient field using a topographic primal sketch approach. Int. J. Numer. Methods Biomed. Eng. 28, 72–86 (2015)

    MathSciNet  Google Scholar 

  9. Gerald, C.F., Wheatley, P.O.: Applied Numerical Analysis. Pearson, USA (2004)

    MATH  Google Scholar 

  10. Getreuer, P.: A survey of gaussian convolution algorithms. Image Process. Line 3, 276–300 (2013)

    Google Scholar 

  11. Griewank, A., Walther, A.: Evaluating derivatives: principles and techniques of algorithmic differentiation. Society for Industrial and Applied Mathematics, Philadelphia (2008)

    Book  MATH  Google Scholar 

  12. Hamming, R.W.: Numerical Methods for Scientists and Engineers. Dover Publications Inc., New York (1986)

    MATH  Google Scholar 

  13. Jauberteau, F., Jauberteau, J.L.: Numerical differentiation with noisy signal. Appl. Math. Comput. 215, 2283–2297 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Jin, J.S., Gao, Y.: Recursive implementation of LoG filtering. Real-Time Imaging 3, 59–65 (1997)

    Article  Google Scholar 

  15. Khan, I.R., Ohba, R., Hozumi, N.: Mathematical proof of closed form expressions for finite difference approximations based on Taylor series. J. Comput. Appl. Math. 150, 303–309 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Khan, I.R., Ohba, R.: Closed-form expressions for the finite difference approximations of first and higher derivatives based on Taylor series. J. Comput. Appl. Math. 107, 179–193 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Khan, I.R., Ohba, R.: Digital differentiators based on Taylor series. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E82–A, 2822–2824 (1999)

    Google Scholar 

  18. Khan, I.R., Ohba, R.: Taylor series based finite difference approximations of higher-degree derivatives. J. Comput. Appl. Math. 154, 115–124 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Knowles, I., Wallace, R.: A variational method for numerical differentiation. Numer. Math. 70, 91–111 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Krueger, W.M., Phillips, K.: The geometry of differential operators with application to image processing. IEEE Trans. Pattern Anal. Mach. Intell. 11, 1252–1264 (1989)

    Article  Google Scholar 

  21. Li, J.: General explicit difference formulas for numerical differentiation. J. Comput. Appl. Math. 183, 29–52 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lindeberg, T.: Scale-space: a framework for handling image structures at multiple scales. Cern Eur. Organ. Nucl. Res. 96, 1–12 (1996)

    Google Scholar 

  23. Lu, S., Pereverzev, S.V.: Numerical differentiation from a viewpoint of regularization theory. Math. Comput. 75, 1853–1870 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Luo, J., Ying, K., He, P., Bai, J.: Properties of Savitzky-Golay digital differentiators. Digit. Signal Process. A Rev. J. 15, 122–136 (2005)

    Article  Google Scholar 

  25. Luo, J., Ying, K., Bai, J.: Savitzky-Golay smoothing and differentiation filter for even number data. Signal Process. 85, 1429–1434 (2005)

    Article  MATH  Google Scholar 

  26. Macia, I.: Generalized computation of gaussian derivatives using ITK. Insight J. 1–14 (2007)

    Google Scholar 

  27. Poggio, T., Koch, C.: Ill-Posed problems in early vision: from computational theory to analogue networks. In: Proceedings of the Royal Society of London. Series B, Biological Sciences, pp. 303–323 (1985)

    Google Scholar 

  28. Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipesin C: The Art of Scientific Computing. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  29. Ramm, A.G., Smirnova, A.B.: On stable numerical differentiation. Math. Comput. 70, 1131–1153 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Silvester, P.: Numerical formation of finite-difference operators. IEEE Trans. Microwave Theory Tech. 18, 740–743 (1970)

    Article  Google Scholar 

  31. Spontón, H., Cardelino, J.: A review of classic edge detectors. Image Process. Line. 5, 90–123 (2015)

    Article  MathSciNet  Google Scholar 

  32. ter Haar Romeny, B.M., Florack, L.M.J.: Front end vision: a multiscale geometry engine. In: Lee, S.-W., Bülthoff, H.H., Poggio, T. (eds.) BMCV 2000. LNCS, vol. 1811, pp. 297–307. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  33. Unser, M.: Splines: a perfect fit for medical imaging. Med. Imaging Process. Proc. Spie. 4684, 225–236 (2002)

    Google Scholar 

  34. Unser, M.: Splines: a perfect fit for signal and image processing. IEEE Signal Process. Mag. 16, 22–38 (1999)

    Article  Google Scholar 

  35. van Vliet, L.J., Young, I.T., Verbeek, P.W.: Recursive gaussian derivative filters. In: Proceedings of Fourteenth International Conference Pattern Recognition, pp. 509–514 (1998)

    Google Scholar 

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Authors and Affiliations

  1. Pontificia Universidad Javeriana, Bogotá, DC, Colombia

    César Bustacara-Medina & Leonardo Flórez-Valencia

Authors
  1. César Bustacara-Medina
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  2. Leonardo Flórez-Valencia
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Correspondence to César Bustacara-Medina .

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Editors and Affiliations

  1. Faculty of Applied Informatics and Math, Warsaw University of Life Sciences , Warsaw, Poland

    Leszek J. Chmielewski

  2. Computer Sci and Software Engineering, University of Western Australia , Perth, Australia

    Amitava Datta

  3. Faculty of Applied Informatics and Math, Warsaw University of Life Sciences SGGW , Warsaw, Poland

    Ryszard Kozera

  4. Institute of Computer Science, Silesian University of Technology , Gliwice, Poland

    Konrad Wojciechowski

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Bustacara-Medina, C., Flórez-Valencia, L. (2016). Comparison and Evaluation of First Derivatives Estimation. In: Chmielewski, L., Datta, A., Kozera, R., Wojciechowski, K. (eds) Computer Vision and Graphics. ICCVG 2016. Lecture Notes in Computer Science(), vol 9972. Springer, Cham. https://doi.org/10.1007/978-3-319-46418-3_11

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  • DOI: https://doi.org/10.1007/978-3-319-46418-3_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-46417-6

  • Online ISBN: 978-3-319-46418-3

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