Fractals Dotaz Zobrazit nápovědu
We introduce the spherical wedge billiard, a dynamical system consisting of a particle moving along a geodesic on a closed non-Euclidean surface of a spherical wedge. We derive the analytic form of the corresponding Poincaré map and find very complex dynamics, ranging from completely chaotic to very regular, exhibiting fractal features. Further, we show that upon changing the billiard parameter, the fixed points of the Poincaré map merge in complex ways, which has origin in the spherical aberration of the billiard mapping. We also analyze in detail the regular regime when phase space diagram is closely related to Talbot carpets.
- Publikační typ
- časopisecké články MeSH
Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.
- Klíčová slova
- Hausdorff measure, Quermass-interaction process, box-counting dimension, breast cancer, pathology,
- MeSH
- algoritmy MeSH
- diagnóza počítačová metody MeSH
- duktální karcinom prsu MeSH
- fraktály MeSH
- hodnocení rizik metody MeSH
- lidé MeSH
- nádory prsu diagnóza patologie MeSH
- patologie metody MeSH
- počítačová simulace MeSH
- stochastické procesy MeSH
- Check Tag
- lidé MeSH
- ženské pohlaví MeSH
- Publikační typ
- časopisecké články MeSH
- práce podpořená grantem MeSH
- srovnávací studie MeSH
Structured and complex data can be found in many applications in research and development, and also in industrial practice. We developed a methodology for describing the structured data complexity and applied it in development and industrial practice. The methodology uses fractal dimension together with statistical tools and with software modification is able to analyse data in a form of sequence (signals, surface roughness), 2D images, and dividing lines. The methodology had not been tested for a relatively large collection of data. For this reason, samples with structured surfaces produced with different technologies and properties were measured and evaluated with many types of parameters. The paper intends to analyse data measured by a surface roughness tester. The methodology shown compares standard and nonstandard parameters, searches the optimal parameters for a complete analysis, and specifies the sensitivity to directionality of samples for these types of surfaces. The text presents application of fractal geometry (fractal dimension) for complex surface analysis in combination with standard roughness parameters (statistical tool).
The fractal character of exponential particle size distribution makes it possible to use fractal dimension for the evaluation of the results of granulometric analysis. The conditions of sedimentation analysis of talc in aqueous solution were optimized by adding nonionogenic tenside and deflocculation electrolyte. Dispersion analysis demonstrated the suitability of minimally 0.2 g/l of polysorbate 80 for talc soaking. In the evaluation of pharmaceutical granulometry, it is advisable to supplement the usually reported specific surface area with fractal dimension, which primarily contributes to a more general interpretation of variability of particle size.
- MeSH
- chemie farmaceutická * MeSH
- fraktály * MeSH
- mastek MeSH
- roztoky MeSH
- velikost částic * MeSH
- Publikační typ
- anglický abstrakt MeSH
- časopisecké články MeSH
- práce podpořená grantem MeSH
- Názvy látek
- mastek MeSH
- roztoky MeSH
Flowability of powder excipients is directly influenced by their size and shape although the granulometric influence of the flow and shear behaviour of particulate matter is not studied frequently. In this work, the influence of particle size on the mass flow rate through the orifice of a conical hopper, and the cohesion and flow function was studied for four free-flowable size fractions of sorbitol for direct compression in the range of 0.080-0.400 mm. The particles were granulometricaly characterized using an optical microscopy; a boundary fractal dimension of 1.066 was estimated for regular sorbitol particles. In the particle size range studied, a non-linear relationship between the mean particle size and the mass flow rate Q10 (g/s) was detected having amaximum at the 0.245mm fraction. The best flow properties of this fraction were verified with aJenike shear tester due to the highest value of flow function and the lowest value of the cohesion. The results of this work show the importance of the right choice of the excipient particle size to achieve the best flow behaviour of particulate material.Key words: flowability size fraction sorbitol for direct compaction Jenike shear tester fractal dimension.
- MeSH
- fraktály * MeSH
- pevnost ve smyku MeSH
- pomocné látky * MeSH
- prášky, zásypy, pudry MeSH
- sorbitol * MeSH
- velikost částic MeSH
- Publikační typ
- časopisecké články MeSH
- Názvy látek
- pomocné látky * MeSH
- prášky, zásypy, pudry MeSH
- sorbitol * MeSH
One of the most recent non-invasive technologies to examine the gastrointestinal tract is wireless capsule endoscopy (WCE). As there are thousands of endoscopic images in an 8-15 h long video, an evaluator has to pay constant attention for a relatively long time (60-120 min). Therefore the possibility of the presence of pathological findings in a few images (displayed for evaluation for a few seconds only) brings a significant risk of missing the pathology with all negative consequences for the patient. Hence, manually reviewing a video to identify abnormal images is not only a tedious and time consuming task that overwhelms human attention but also is error prone. In this paper, a method is proposed for the automatic detection of abnormal WCE images. The differential box counting method is used for the extraction of fractal dimension (FD) of WCE images and the random forest based ensemble classifier is used for the identification of abnormal frames. The FD is a well-known technique for extraction of features related to texture, smoothness, and roughness. In this paper, FDs are extracted from pixel-blocks of WCE images and are fed to the classifier for identification of images with abnormalities. To determine a suitable pixel block size for FD feature extraction, various sizes of blocks are considered and are fed into six frequently used classifiers separately, and the block size of 7×7 giving the best performance is empirically determined. Further, the selection of the random forest ensemble classifier is also done using the same empirical study. Performance of the proposed method is evaluated on two datasets containing WCE frames. Results demonstrate that the proposed method outperforms some of the state-of-the-art methods with AUC of 85% and 99% on Dataset-I and Dataset-II respectively.
- Klíčová slova
- Anomaly detection, Differential box-counting, Fractal dimensions, Wireless capsule endoscopy,
- MeSH
- fraktály MeSH
- gastrointestinální trakt MeSH
- kapslová endoskopie * MeSH
- lidé MeSH
- Check Tag
- lidé MeSH
- Publikační typ
- časopisecké články MeSH
- práce podpořená grantem MeSH
The crystallization in mixture of physiological solution (0.9% NaCl) with human albumin (model system) or blood serum leads to characteristic dendritic crystals. The composition of blood serum influences crystallization results in dependence on simultaneous pathological process of the organism. The crystallographic method is based on comparison and evaluation of dendritic patterns created by crystallization of NaCl in blood serum. The methods of image analysis are applied to classify the characteristic dendritic crystals in blood serum. This contribution describes possibility of fractal geometry appliance.
- MeSH
- biologické modely * MeSH
- fraktály * MeSH
- krevní plazma chemie MeSH
- krystalografie * MeSH
- lidé MeSH
- sérový albumin chemie MeSH
- Check Tag
- lidé MeSH
- Publikační typ
- anglický abstrakt MeSH
- časopisecké články MeSH
- práce podpořená grantem MeSH
- Názvy látek
- sérový albumin MeSH
In recent years, alcohol addiction has become a major public health concern and a global threat due to its potential negative health and social impacts. Beyond the health consequences, the detrimental consumption of alcohol results in substantial social and economic burdens on both individuals and society as a whole. Therefore, a proper understanding and effective control of the spread of alcohol addictive behavior has become an appealing global issue to be solved. In this study, we develop a new mathematical model of alcohol addiction with treatment class. We analyze the dynamics of the alcohol addiction model for the first time using advanced operators known as fractal-fractional operators, which incorporate two distinct fractal and fractional orders with the well-known Caputo derivative based on power law kernels. The existence and uniqueness of the newly developed fractal-fractional alcohol addiction model are shown using the Picard-Lindelöf and fixed point theories. Initially, a comprehensive qualitative analysis of the alcohol addiction fractional model is presented. The possible equilibria of the model and the threshold parameter called the reproduction number are evaluated theoretically and numerically. The boundedness and biologically feasible region for the model are derived. To assess the stability of the proposed model, the Ulam-Hyers coupled with the Ulam-Hyers-Rassias stability criteria are employed. Moreover, utilizing effecting numerical schemes, the models are solved numerically and a detailed simulation and discussion are presented. The model global dynamics are shown graphically for various values of fractional and fractal dimensions. The present study aims to provide valuable insights for the understanding the dynamics and control of alcohol addiction within a community.
- Klíčová slova
- Alcohol addiction model, Existence and uniqueness, Fractal–fractional Caputo operator, Simulation, Ulam–Hyers stability,
- MeSH
- alkoholismus * MeSH
- ethanol MeSH
- fraktály MeSH
- lidé MeSH
- návykové chování * MeSH
- počítačová simulace MeSH
- Check Tag
- lidé MeSH
- Publikační typ
- časopisecké články MeSH
- Názvy látek
- ethanol MeSH
Fractal dimension of the particles of powdered auxiliary substances expresses the raggedness of their surface, which can influence both the bulk density and their flow rate through the opening of the powder funnel. In the size fractions of 0.20-0.25 mm of six powdered auxiliary substances, their linear fractal dimensions within a range of 1.031 to 1.109 and the corresponding flow rates through the circular openings of the powder funnel of diameters of 6; 8; 10, and 12 mm were found. A significant correlation has been demonstrated between the flow rate expressed in volume (cm3/s) and the fractal dimension, which after an increase by one unit characterizes the surface of the particles. With increasing fractal dimension, the porosity of freely poured layer of powdered auxiliary substances is increased and simultaneously also their flow rate expressed in volume. For the usually reported flow rate expressed in weight (g/s), such influence is not significant.
- MeSH
- chemické jevy MeSH
- chemie farmaceutická MeSH
- chemie fyzikální MeSH
- farmaceutické pomocné látky chemie MeSH
- fraktály MeSH
- prášky, zásypy, pudry * MeSH
- velikost částic MeSH
- Publikační typ
- časopisecké články MeSH
- práce podpořená grantem MeSH
- Názvy látek
- farmaceutické pomocné látky MeSH
- prášky, zásypy, pudry * MeSH
Compression of ECG signal is essential especially in the area of signal transmission in telemedicine. There exist many compression algorithms which are described in various details, tested on various datasets and their performance is expressed by different ways. There is a lack of standardization in this area. This study points out these drawbacks and presents new compression algorithm which is properly described, tested and objectively compared with other authors. This study serves as an example how the standardization should look like. Single-cycle fractal-based (SCyF) compression algorithm is introduced and tested on 4 different databases-CSE database, MIT-BIH arrhythmia database, High-frequency signal and Brno University of Technology ECG quality database (BUT QDB). SCyF algorithm is always compared with well-known algorithm based on wavelet transform and set partitioning in hierarchical trees in terms of efficiency (2 methods) and quality/distortion of the signal after compression (12 methods). Detail analysis of the results is provided. The results of SCyF compression algorithm reach up to avL = 0.4460 bps and PRDN = 2.8236%.
