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Nonstationary particle processes
Jirsák, Čeněk ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
Title: Nonstacionary particle processes Author: Čeněk Jirsák Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Jan Rataj, CSc., Mathematical Institute, Charles University Supervisor's e-mail address: rataj@karlin.mff.cuni.cz Abstract: Many real phenomena can be modeled as random closed sets of different Hausdorff dimension in Rd . One of the main characteristics of such random set is its expected Hausdorff measure. In case that this measure has a density, the density is called intensity function. In present paper we define a nonparametric kernel estimation of the intensity function. The concept of Hk -rectifiable set has a key role here. Properties of kernel estimation such as unbiasness or convergence behavior are studied. As the esti- mation may be difficult to compute precisely numerical approximations are derived for practical use. Parametric models are also briefly mentioned and the kernel estimation is used with the minimum contrast method to estimate the parameters of the model. At last the suggested methods are tested on simulated data. Keywords: stochastic geometry, intensity measure, random closed set, kernel estimation 1
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Nonstationary particle processes
Jirsák, Čeněk ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
Title: Nonstacionary particle processes Author: Čeněk Jirsák Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Jan Rataj, CSc., Mathematical Institute, Charles University Supervisor's e-mail address: rataj@karlin.mff.cuni.cz Abstract: Many real phenomena can be modeled as random closed sets of different Hausdorff dimension in Rd . One of the main characteristics of such random set is its expected Hausdorff measure. In case that this measure has a density, the density is called intensity function. In present paper we define a nonparametric kernel estimation of the intensity function. The concept of Hk -rectifiable set has a key role here. Properties of kernel estimation such as unbiasness or convergence behavior are studied. As the esti- mation may be difficult to compute precisely numerical approximations are derived for practical use. Parametric models are also briefly mentioned and the kernel estimation is used with the minimum contrast method to estimate the parameters of the model. At last the suggested methods are tested on simulated data. Keywords: stochastic geometry, intensity measure, random closed set, kernel estimation 1
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Obrazová analýza řezů pórovitého materiálu
Hejtmánek, Vladimír ; Čapek, P.
Two methods of image analysis of cross-sections through two-phase porous media are compared. Linear filtering uses the recursive (IIR) Gaussian filter with the same vertical and horizontal blur radii. The images are also treated by removing small clusters of pixels in the void and solid phases. After linear filtering and before removing small clusters, the grey-level threshold is determined in order to partition all grey pixels into pore and solid pixels. Linear filtering reduces the specific surface very efficiently creating ``smooth'' pore walls and leaves small clusters of pixels in the images. Removing preserves ``rough'' pore walls and removes simultaneously small clusters from both the phases.
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