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Orientational analysis of planar fibre systems observed as a Poisson shot‐noise process
Author(s) -
KÄRKKÄINEN SALME,
LANTUÉJOUL CHRISTIAN
Publication year - 2007
Publication title -
journal of microscopy
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.569
H-Index - 111
eISSN - 1365-2818
pISSN - 0022-2720
DOI - 10.1111/j.1365-2818.2007.01826.x
Subject(s) - shot noise , boolean model , grayscale , poisson distribution , mathematics , planar , sampling (signal processing) , point process , noise (video) , observable , mathematical analysis , pixel , statistical physics , image (mathematics) , optics , statistics , physics , computer science , artificial intelligence , computer graphics (images) , quantum mechanics , detector
Summary We consider two‐dimensional fibrous materials observed as a digital greyscale image. The problem addressed is to estimate the orientation distribution of unobservable thin fibres from a greyscale image modelled by a planar Poisson shot‐noise process. The classical stereological approach is not straightforward, because the point intensities of thin fibres along sampling lines may not be observable. For such cases, Kärkkäinen et al. (2001) suggested the use of scaled variograms determined from grey values along sampling lines in several directions. Their method is based on the assumption that the proportion between the scaled variograms and point intensities in all directions of sampling lines is constant. This assumption is proved to be valid asymptotically for Boolean models and dead leaves models, under some regularity conditions. In this work, we derive the scaled variogram and its approximations for a planar Poisson shot‐noise process using the modified Bessel function. In the case of reasonable high resolution of the observed image, the scaled variogram has an approximate functional relation to the point intensity, and in the case of high resolution the relation is proportional. As the obtained relations are approximative, they are tested on simulations. The existing orientation analysis method based on the proportional relation is further experimented on images with different resolutions. The new result, the asymptotic proportionality between the scaled variograms and the point intensities for a Poisson shot‐noise process, completes the earlier results for the Boolean models and for the dead leaves models.