Estimating the size of an object captured with error

  • Safet Hamedović
  • Mirta Benšić
  • Kristian Sabo
  • Petar Taler
Original Paper


In many applications we are faced with the problem of estimating object dimensions from a noisy image. Some devices like a fluorescent microscope, X-ray or ultrasound machines, etc., produce imperfect images. Image noise comes from a variety of sources. It can be produced by the physical processes of imaging, or may be caused by the presence of some unwanted structures (e.g. soft tissue captured in images of bones). In the proposed models we suppose that the data are drawn from uniform distribution on the object of interest, but contaminated by an additive error. Here we use two one-dimensional parametric models to construct confidence intervals and statistical tests pertaining to the object size and suggest the possibility of application in two-dimensional problems. Normal and Laplace distributions are used as error distributions. Finally, we illustrate ability of the R-programs we created for these problems on a real-world example.


Noisy image Additive error Maximum likelihood estimator Uniform distribution Normal distribution Laplace distribution 



We are very grateful to the anonymous referees for the valuable comments which have improved considerably the first version of the manuscript. This work is supported by the Croatian Science Foundation through research Grants IP-2016-06-6545.


  1. Benšić M, Sabo K (2007) Estimating the width of a uniform distribution when data are measured with additive normal errors with known variance. Comput Stat Data Anal 51:4731–4741CrossRefGoogle Scholar
  2. Benšić M, Sabo K (2007) Border estimation of a two-dimensional uniform distribution if data are measured with additive error. Stat J Theor Appl Stat 41:311–319Google Scholar
  3. Benšić M, Sabo K (2010) Estimating a uniform distribution when data are measured with a normal additive error with unknown variance. Stat J Theor Appl Stat 44:235–246Google Scholar
  4. Benšić M, Sabo K (2016) Uniform distribution width estimation from data observed with Laplace additive error. J Korean Stat Soc 45:505–517CrossRefGoogle Scholar
  5. Canny J (1986) A computational approach to edge detection. IEEE Trans Pattern Anal Mach Intell 8:679–698CrossRefGoogle Scholar
  6. Carroll RJ, Hall P (1988) Optimal rates of convergence for deconvoluting a density. J Am Stat Assoc 83:1184–1186CrossRefGoogle Scholar
  7. Delaigle A, Gijbels I (2006) Estimation of boundary and discontinuity points in deconvolution problems. Stat Sin 16:773–788Google Scholar
  8. Fan J (1991) On the optimal rates of convergence for nonparametric deconvolution problems. Ann Stat 19:1257–1272CrossRefGoogle Scholar
  9. Jurun E, Ratković N, Ujević I (2017) A cluster analysis of Croatian counties as the base for an active demographic policy. Croat Oper Res Rev 8:221–236CrossRefGoogle Scholar
  10. Kneip A, Simar L (1996) A general framework for frontier estimation with panel data. J Prod Anal 7:187–212CrossRefGoogle Scholar
  11. Meister A (2006) Estimating the support of multivariate densities under measurement error. J Multivar Anal 97:1702–1717CrossRefGoogle Scholar
  12. Meister A (2009) Deconvolution problems in nonparametric statistics. Springer, BerlinCrossRefGoogle Scholar
  13. Neumann MH (1997) Optimal change-point estimation in inverse problems. Scand J Stat 24:503–521CrossRefGoogle Scholar
  14. Qiu P (2005) Image processing and jump regression analysis. Wiley, HobokenCrossRefGoogle Scholar
  15. R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.
  16. Ruzin SE (1999) Plant microtechnique and microscopy. Oxford University Press, OxfordGoogle Scholar
  17. Sabo K (2014) Center-based \(l_1\)-clustering method. Int J Appl Math Comput Sci 24:151–163CrossRefGoogle Scholar
  18. Sabo K, Benšić M (2009) Border estimation of a disc observed with random errors solved in two steps. J Comput Appl Math 229:16–26CrossRefGoogle Scholar
  19. Schneeweiss H (2004) Estimating the endpoint of a uniform distribution under measurement errors. CEJOR 12:221–231Google Scholar
  20. Sen PB, Singer JM, de Lima ACP (2010) From finite sample to asymptotic methods in statistics. Cambridge University Press, New YorkGoogle Scholar
  21. Stefanski L, Carroll RJ (1990) Deconvoluting kernel density estimators. Statistics 21:169–184CrossRefGoogle Scholar
  22. Turkalj Ž, Markulak D, Singer S, Scitovski R (2016) Research project grouping and ranking by using adaptive Mahalanobis clustering. Croat Oper Res Rev 7:81–96CrossRefGoogle Scholar
  23. Zhang CH (1990) Fourier methods for estimating mixing densities and distributions. Ann Stat 18:806–830CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Faculty of Metallurgy and MaterialsUniversity of ZenicaZenicaBosnia and Herzegovina
  2. 2.Department of MathematicsJosip Juraj Strossmayer University OsijekOsijekCroatia

Personalised recommendations