### Abstract

A novel estimator for estimating the mean length of
fibres is proposed for censored data observed in square
shaped windows. Instead of observing the fibre lengths,
we observe the ratio between the intensity estimates of
minus-sampling and plus-sampling. It is well-known that
both intensity estimators are biased. In the current
work, we derive the ratio of these biases as a function
of the mean length assuming a Boolean line segment model
with exponentially distributed lengths and uniformly
distributed directions. Having the observed ratio of the
intensity estimators, the inverse of the derived function
is suggested as a new estimator for the mean length. For
this estimator, an approximation of its variance is
derived. The accuracies of the approximations are
evaluated by means of simulation experiments. The novel
method is compared to other methods and applied to
real-world industrial data from nanocellulose
crystalline.

Original language | English |
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Pages (from-to) | 147-155 |

Number of pages | 9 |

Journal | Image Analysis and Stereology |

Volume | 33 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- Boolean model
- nanocellulose
- fibres
- length distribution
- mean length
- simulation

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## Cite this

Niilo-Rämä, M., Kärkkäinen, S., Gasbarra, D., & Lappalainen, T. (2014). Inclusion ratio based estimator for the mean length of the boolean line segment model with an application to nanocrystalline cellulose.

*Image Analysis and Stereology*,*33*(2), 147-155. https://doi.org/10.5566/ias.v33.p147-155