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1

On the Carrying Dimension of Occupation Measures for Self-Affine Random Fields

Kern Peter, Sönmez Ercan

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 2

459--479

EN

Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. The aim is to demonstrate the following interesting relation to a series of articles by U. Zähle (1984, 1988, 1990, 1991). Under natural regularity assumptions, we prove that the Hausdorff dimension of the graph of self-affine fields coincides with the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Zähle. As a remarkable consequence we obtain a general formula for the Hausdorff dimension given by means of the singular value function.

2

Zastosowanie mikroskopii sił atomowych (AFM) w diagnostyce warstwy wierzchniej

Bramowicz M., Kłysz S.

Prace Naukowe Instytutu Technicznego Wojsk Lotniczych

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2007

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nr 22

159-166

PL

W niniejszej pracy przedstawiono mikroskopię sił atomowych jako zaawansowaną metodę inżynierską, dającą szerokie możliwości w nowoczesnej kontroli jakości wysokiej klasy elementów optyki, fotoniki czy półprzewodników. Ponadto silnie zaakcentowano konieczność opisu morfologii warstwy wierzchniej poprzez podanie wymiaru fraktalnego oraz długości korelacji jako zasadniczych parametrów opisujących samopodobieństwo oraz samoafiniczność powierzchni elementów technicznych. Przytaczaną metodykę poparto badaniami eksperymentalnymi, w ramach których przeprowadzono analizę fraktalną topografii powierzchni cienkiej folii stopu Heuslera typu Ni-Mn-Ga z zastosowaniem metody RMS (Root Mean Square).

EN

This paper introduces the Atomic Force Microscopy as an advanced engineering method that offers great potential in the modern, high-class quality control of optic, photonic, and semiconductor elements. Furthermore, what has been strongly emphasized is a real need to describe the morphology of a surface layer using the fractal parameter and the correlation length as fundamental parameters that describe self-similarity and self-affinity of surfaces of technical elements. The methodology shown in the paper is supported with some effects of experimental work, including fractal analysis of the topography of the surface of thin Ni-Mn-Ga type Heusler alloy with the Root Mean Square (RMS) method.

3

Feature generation from digital images using pseudo-fractal algorithm and its four modifications

Janaszewski M., Kącki E.

Machine Graphics and Vision

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2007

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Vol. 16, No. 1/2

117-138

EN

The main aim of the paper is to present the authors' original method of feature generation from digital images and to report on a comparison of five various algorithms, which implemented that method. The algorithms are based on an idea by the same authors', which consists in producing a quantitative description of similarity intensity between various parts of an image in various scales. To develop it the algorithms take advantage of fractal coding based on an Iterated Function System. Therefore, the generated features can rightly be called similarity features. In this paper we show that similarity features, when combined with other well known ones, can improve recognition results in some image classification tasks. After presenting how the algorithm works, we compare their properties and report the classification results obtained in two different pattern recognition experiments. Moreover, the paper contains a discussion of the obtained results, and of possible future applications of the similarity features.