Fractal Maps Research Articles

Nonexpansiveness and Fractal Maps in Hilbert Spaces

Nonexpansiveness and Fractal Maps in Hilbert Spaces

FractalRG: Advanced fractal region growing using Gaussian mixture models for left atrium segmentation

This paper presents an advanced region growing method for precise left atrium (LA) segmentation and estimation of atrial wall thickness in CT/MRI scans. The method leverages a Gaussian mixture model (GMM) and fractal dimension (FD) analysis in a three-step procedure to enhance segmentation accuracy. The first step employs GMM for seed initialization based on the probability distribution of image intensities. The second step utilizes fractal-based texture analysis to capture image self-similarity and texture complexity. An enhanced approach for generating 3D fractal maps is proposed, providing valuable texture information for region growing. In the last step, fractal-guided 3D region growing is applied for segmentation. This process expands seed points iteratively by adding neighboring voxels meeting specific similarity criteria. GMM estimations and fractal maps are used to restrict the region growing process, reducing the search space for global segmentation and enhancing computational efficiency. Experiments on a dataset of 10 CT scans with 3,947 images resulted in a Dice score of 0.85, demonstrating superiority over traditional techniques. In a dataset of 30 MRI scans with 3,600 images, the proposed method achieved a competitive Dice score of 0.89±0.02, comparable to Deep Learning-based models. These results highlight the effectiveness of our approach in accurately delineating the LA region across diverse imaging modalities.

Porphyry and skarn Cu prospectivity of Sumatra with GIS spatial modelling techniques

Although Sumatra Island hosts only one mid-sized, economically viable porphyry Cu deposit and a few exploited Cu-rich polymetallic skarns, studies indicate that more Cu resources remain to be discovered. This contribution integrated geochemical, structural, and relief evidential layers with the fuzzy logic- and weights-of-evidence (WoE) methods to produce mineral prospectivity maps that show a strong correlation between the generated favourable areas and the existing porphyry and skarn Cu prospects, as well as potentially new exploration districts throughout Sumatra. Moreover, a local singularity fractal map of Cu-enriched zones reduces the proposed target areas and cross-validates the results of the fuzzy model. The inclusion of relief data in the modelling filters out signal noise arising from other evidential layers and overcomes the over-fitting issue encountered in the model that excluded a relief map. The resulting Cu potential maps can thus be used as base maps for regional porphyry and skarn Cu exploration.

KASP-IEva: an intelligent typing evaluation model for KASP primers.

KASP marker technology has been used in molecular marker-assisted breeding because of its high efficiency and flexibility, and an intelligent evaluation model of KASP marker primer typing results is essential to improve the efficiency of marker development on a large scale. To this end, this paper proposes a gene population delineation method based on NTC identification module and data distribution judgment module to improve the accuracy of K-Means clustering, and introduces a decision tree to construct the KASP-IEva primer typing evaluation model. The model firstly designs the NTC identification module and data distribution judgment module to extract four types of data, grouping and categorizing to achieve the improvement of the distinguishability of amplification product signals; secondly, the K-Means algorithm is used to aggregate and classify the data, to visualize the five aggregated clusters and to obtain the morphology location eigenvalues; lastly, the evaluation criteria for the typing effect level are constructed, and the logical decision tree is used to make conditional discrimination on the eigenvalues in order to realize the score prediction. The performance of the model was tested by the KASP marker typing test results of 2519 groups of cotton varieties, and the following conclusions were obtained: the model is able to visualize the aggregation and classification effects of the amplification products of NTC, pure genotypes, heterozygous genotypes, and untyped genotypes, enabling rapid and accurate KASP marker typing evaluation. Comparing and analyzing the model evaluation results with the expert evaluation results, the average accuracy rate of the four grades evaluated by the model was 87%, and the overall evaluation results showed an uneven distribution of the grades with significant differential characteristics. When evaluating 2519 KASP fractal maps, the expert evaluation consumes 15 hours, and the model evaluation only uses 8min27.45s, which makes the model intelligent evaluation significantly better than the expert evaluation from the perspective of time. The establishment of the model will further enhance the application of KASP markers in molecular marker-assisted breeding and provide technical support for the large-scale screening and identification of excellent genotypes.

Parameters estimation of three-diode photovoltaic model using fractional-order Harris Hawks optimization algorithm

Effective and efficient systems that calculate the parameters of PV models to be optimal have attracted great interest from many researchers. In this study, a novel algorithm approach is proposed to improve the search performance of the Harris Hawks optimization (HHO) algorithm based on fractal maps. The random parameter that is effective in besiege of the prey in the exploration and exploitation stages of the HHO algorithm is adjusted using the fractal henon chaotic map. Unimodal, multimodal and fixed-dimension benchmark functions were used to verify the quality and efficiency of the proposed Fractional Henon Chaotic Harris Hawks Optimization (FCHHHO) algorithms and variants in providing optimal solutions. Moreover, RTC France photovoltaic cell and Photowatt-PWP 201 photovoltaic models with single, double and three-diodes have been determined accurately, stably and reliably. Comparative and statistical results obtained from the developed algorithms showed that the algorithms have superior performance with regard to accuracy and robustness in comparison with their variants and compared with other metaheuristic algorithms.

Multi-directional annular multi-wing chaotic system based on Julia fractals

Fractals and chaos are inextricably linked, and relevant study has been done on both. However, the studies that combine fractals and chaotic systems to produce multi- scroll or multi-wing chaotic attractors are all extend multi-scroll or multi-wing in the even direction. To this end, by improving the traditional fractal mapping, a chaotic system that can extend multiple wings in any direction of odd and even is designed and analyzed. The main point of this method is as follows: Firstly, a new family of even-symmetric segmented composite functions is proposed and applied to a family of generalized Lorenz systems, which can generate multi-wing chaotic systems in two directions and the number of wings in each direction can also be increased with the parameter N. Secondly, the improved Julia fractal is applied to two-direction multi-wing chaotic systems to generate a class of single-direction multi-wing chaotic systems. Finally, the combination of this class of single-direction multi-wing chaotic systems and higher-order Julia fractals is used to realize chaotic systems with extended multi-wing in odd and even arbitrary directions. Computer simulations confirm the method's viability. In addition, the maximum Lyapunov exponent, bifurcation diagram, Poincare diagram, and PE complexity chaos diagram are also analyzed as examples of multi-directional multi-wing Lü chaotic systems, demonstrating the system's superiority in odd directions over even directions and the existence and effective scalability of multi-directional multi-wing chaotic systems.

Correction to: Prospecting for polymetallic mineralization in Middle Morocco using fractal mapping

Correction to: Prospecting for polymetallic mineralization in Middle Morocco using fractal mapping

Prospecting for polymetallic mineralization in Middle Morocco using fractal mapping

Prospecting for polymetallic mineralization in Middle Morocco using fractal mapping

MULTI-DIRECTION CHAIN AND GRID CHAOTIC SYSTEM BASED ON JULIA FRACTAL

In this paper, based on the multi-scroll chaotic system, multi-direction chain and grid chaotic attractors are generated by a new Julia fractal mapping process. The feasibility and effectiveness of the proposed method are verified by numerical simulation. This scheme not only realizes the combination of unidirectional and bidirectional distributed multi-scroll chaotic system and Julia fractal, but also applies to three-directional distributed 3D grid-like multi-scroll generalized Jerk system. This paper takes unidirectionally distributed multi-scroll chaos as an example. It discusses the influence of Julia fractals with coefficients and complex constants on the system and generalizes them to the higher-order Julia fractal mapping process. Then, three types of chaotic systems with controllable scroll numbers distributed in multiple directions are obtained. The results of the dynamic analysis method show that the post-fractal chaotic system not only increases the bifurcation interval of its parameters compared with the original chaotic system, but also increases the complexity of its sequence and the maximum Lyapunov exponent, and its attraction domain has a very complex fractal boundary. A kind of multi-directional chain chaotic attractor is realized by the Digital Signal Processors (DSP). The phase diagram of the oscilloscope is consistent with the result of numerical simulation, which verifies the possibility of this method in the digital circuit.

Some stochastic chaotic attractors with global derivative and stochastic fractal mapping: Existence, uniqueness and applications

One of the great ability of humans is to use mathematical formulas to replicate what they see in their daily activities, they start with simple models, when these models are unable to achieve their goals and they modify or replace them with more complex mathematical formulas. Chaos has been modelled using classical differential operators, classical with fractional orders. On the other hand, many mathematical mapping have been used to simulate some fractals behaviours. In this paper, in order to capture more fractal and chaos behaviours, we introduce the notion of chaos stochastic models and fractal stochastic mapping. We additionally present the conditions of existence and uniqueness of chaos‐stochastic models. Some numerical simulations are presented to help see the effect of stochastic functions.

Fractal signature of coronaviruses related to severe acute respiratory syndrome

PurposeThe aim of this work was to investigate the genome of SARS-CoV, MERS-CoV, and SARS-CoV-2 by the paradigm of chaos theory and fractal geometry. Coronavirus is the agent that causes the severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS) and emergent infections such as the present pandemic of COVID-19. Understanding its genome pattern is important for developing new and faster ways of testing for identifying the genome of the virus and also for better understanding of its origin and evolution.MethodsFor that, it was calculated the alpha coefficient by detrended fluctuation analysis (DFA) for the sequences of these genomes converted to binary numbers in order to determine if it is a chaotic or a random series of data. Also, it applied the random walking for obtaining a fractal map of the whole genome and calculated the fractal dimension (FD) by box-counting of this map by two different software.ResultsWith this, it was found that the alpha coefficient of the first SARS viruses was > 0.5, indicating that the series is chaotic or fractal, and has a persistent long-range memory or self-similarity along its sequence. This is not the case for MERS virus, which showed to have a completely random sequence (α < 0.5). For the fractal dimension, SARS viruses presented a FD around 1.5, and for MERS the fractal dimension decreases (FD < 1.5).ConclusionThe images generated by random walking of the entire RNA genome are by itself a fractal signature of the virus, which may be applied for studying its origin and for faster diagnostic of COVID-19.

Uranium, thorium and potassium insights on Campania region (Italy) soils: Sources patterns based on compositional data analysis and fractal model

Uranium, thorium and potassium are distinguished in different compartments of the environmental media according to the inputs, the fate and the distribution patterns that occur in the ecosystems. Thus, an understanding of the sources patterns and behaviours of these elements in soils, their rock-soil relationships and potential toxicity or deficiency problems associated with them is nowadays a relevant concern for environmental protection and human health. A robust compositional computation analysis coupled with frequency spatial-method (Fractal model) is applied on 3 selected elements (U, Th and K), based on a large dataset of 7134 topsoil samples over the whole Campania region, and investigated elements concentrations were analysed (after aqua regia extraction) by inductively coupled plasma mass spectrometry (ICP-MS). In this survey, U concentrations ranged from 0.05 to 43.2 mg kg−1 with a mean value of 3.35 mg kg−1, Th concentrations ranged from 0.3 to 64.30 mg kg−1 with a mean values of 11.95 mg kg−1 and K concentrations ranged from 0.03 to 5.97% with a mean value of 0.68%. The mean values for U and Th are slightly above the mean crustal abundances, while K is slightly below them. Fractal mapping of Centred-log transformed (clr) data of investigated elements show that the U, Th and K geochemical anomalies can be ascribed to geology-related volcanic sources. Indeed, the highest values in soils were assessed in the surveyed alkaline magmatic areas in the central-western part of the region characterized by volcanic rocks, whereas lowest values are found in areas characterized by silico-clastic and carbonate deposits, occurring mostly in the southern and eastern part of the region. Evidence from this study showed that compositional data transformations such as clr transformation could help to avoid artefacts, prior to statistical computations.

A REVISIT TO α-FRACTAL FUNCTION AND BOX DIMENSION OF ITS GRAPH

One of the tools offered by fractal geometry is fractal interpolation, which forms a basis for the constructive approximation theory for nondifferentiable functions. The notion of fractal interpolation function can be used to obtain a wide spectrum of self-referential functions associated to a prescribed continuous function on a compact interval in [Formula: see text]. These fractal maps, the so-called [Formula: see text]-fractal functions, are defined by means of suitable iterated function system which involves some parameters. Building on the literature related to the notion of [Formula: see text]-fractal functions, the current study targets to record the continuous dependence of the [Formula: see text]-fractal function on parameters involved in its definition. Furthermore, the paper attempts to study the box dimension of the graph of the [Formula: see text]-fractal function.

Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus

Abstract Human being live in three-dimensional space; they can accurately visualize processes taking place in one, two and three dimensions. Although the set of bi-complex numbers and quaternion have attracted attention of many researchers in physics and related branches, they do not really represent processes taking place in the space where human being are located. We suggested a new set of complex number called “the Trinition”. The new set is comprised between complex number with one imaginary part and complex number with three imaginary parts called quaternion/bi-complex numbers. We established a bijection between the new set and the three-dimensional space. We presented some important properties of the new set. We showed that all chaotic attractors in three dimension are simply three-dimensional mapping in the new set. Fewer examples of mapping in such set were presented. A new methodology that can be used to obtain more strange attractors are equally suggested. The methodology combines fractional chaotic models and some fractal mapping within the new set. Some illustrative figures are presented.

Wavelet-based cross-correlation analysis of structure scaling in turbulent clouds

We propose a statistical tool to compare the scaling behaviour of turbulence in pairs of molecular cloud maps. Using artificial maps with well defined spatial properties, we calibrate the method and test its limitations to ultimately apply it to a set of observed maps. We develop the wavelet-based weighted cross-correlation (WWCC) method to study the relative contribution of structures of different sizes and their degree of correlation in two maps as a function of spatial scale, and the mutual displacement of structures in the molecular cloud maps. We test the WWCC for circular structures having a single prominent scale and fractal structures showing a self-similar behavior without prominent scales. Observational noise and a finite map size limit the scales where the cross-correlation coefficients and displacement vectors can be reliably measured. For fractal maps containing many structures on all scales, the limitation from the observational noise is negligible for signal-to-noise ratios >5. (abbrev). Application of the WWCC to the observed line maps of the giant molecular cloud G333 allows to add specific scale information to the results obtained earlier using the principle component analysis. It confirms the chemical and excitation similarity of $^{13}$CO and C$^{18}$O on all scales, but shows a deviation of HCN at scales of up to 7' (~7 pc). This can be interpreted as a chemical transition scale. The largest structures also show a systematic offset along the filament, probably due to a large-scale density gradient. The WWCC can compare correlated structures in different maps of molecular clouds identifying scales that represent structural changes such as chemical and phase transitions and prominent or enhanced dimensions.

Fractal analysis of geochemical landscapes using scaling noise model

Fractal/multifractal concepts have facilitated the description and analysis of complex geochemical data in both mineral exploration and environmental studies. Scaling (1/fβ) noise has been ubiquitously found in geosciences but lack in-depth studies for geochemical distribution pattern in mineral exploration. In the present paper, the 1/f scaling natures of geochemical landscapes are investigated using spectral method through geochemical samples in stream sediments from Nanling Range, China. The results show explicit differences in scaling exponent (β) between major elements and trace elements, between highly enriched elements (e.g., W, Sn, Mo and Bi) and relatively low enriched elements (e.g., Au, Ag and Cu), where β measures the strength of persistence (or the degree of roughness) of geochemical landscapes. Furthermore, fractal mapping of geochemical patterns (W and Sn element) is undertaken to reveal the spatial association between local fractal dimension and mineralization. The finding is that most of the W and Sn deposits in study area exhibit rough geochemical patterns with fractal dimension ranging from 2.3 and 2.7 (2.6≤β≤3.4). We proposed to recognize the complex geochemical anomalies containing both stochastic (irregular) and deterministic (regular) components from fractal noise perspective.

A multigrid-based preconditioned solver for the Helmholtz equation with a discretization by 25-point difference scheme

In this paper, a preconditioned iterative method is developed to solve the Helmholtz equation with perfectly matched layer (Helmholtz-PML equation). The complex shifted-Laplacian is generalized to precondition the Helmholtz-PML equation, which is discretized by an optimal 25-point finite difference scheme that we presented in Chen et al. (2011). A spectral analysis is given for the discrete preconditioned system from the perspective of linear fractal mapping, and Bi-CGSTAB is used to solve it. The multigrid method is employed to invert the preconditioner approximately, and a new matrix-based prolongation operator is constructed in the multigrid cycle. Numerical experiments are presented to illustrate the efficiency of the multigrid-based preconditioned Bi-CGSTAB method with the new prolongation operator. Numerical results are also given to compare the performance of the new prolongation operator with that of the prolongation operator based on the algebraic multigrid (AMG) principle.

Fractal Depth Map Sequence Coding Algorithm with Motion-vector-field-based Motion Estimation

Three-dimensional video coding is one of the main challenges restricting the widespread applications of 3D video and free viewpoint video. In this paper, a novel fractal coding algorithm with motion-vector-field-based motion estimation for depth map sequence is proposed. We firstly add pre-search restriction to rule the improper domain blocks out of the matching search process so that the number of blocks involved in the search process can be restricted to a smaller size. Some improvements for motion estimation including initial search point prediction, threshold transition condition and early termination condition are made based on the feature of fractal coding. The motion-vector-field-based adaptive hexagon search algorithm on the basis of center-biased distribution characteristics of depth motion vector is proposed to accelerate the search. Experimental results show that the proposed algorithm can reach optimum levels of quality and save the coding time. The PSNR of synthesized view is increased by 0.56 dB with 36.97% bit rate decrease on average compared with H.264 Full Search. And the depth encoding time is saved by up to 66.47%. Moreover, the proposed fractal depth map sequence codec outperforms the recent alternative codecs by improving the H.264/AVC, especially in much bitrate saving and encoding time reduction.

Sudy on Determination of Cadmium Content in Rice

Wavelet function using a variety of experimental spectral decomposition of the image area, the results show that the spectral fractal dimension db series db5 wavelet decomposition of the third layer is the most stable and reliable signal. And cadmium contamination levels in two different areas of Changchun area, for example, on the Hyperion image wavelet decomposition to obtain details of the signal spectrum fractal dimension map, identifying the extent of a wide range of rice cadmium pollution stress. This method is rapid, macro, accurate monitoring of cadmium contamination of rice provides a convenient way.

Angle Independence Properties of Fractal Dimension Maps Estimated From SAR Data

The extremely remarkable properties of angle independence exhibited by an innovative SAR product, the fractal dimension map estimated from a single SAR image, are discussed. The theoretical analysis is supported by a noticeable data set of actual SAR images acquired, with look angles varying from 20° to 45°, in the stripmap operational mode by the COSMO-SkyMed constellation. The behavior of the fractal dimension maps at different look angles is discussed for both natural and urban scenarios and emphasis is also posed on areas within the same image that, according to the scene macroscopic topography, are characterized by different incidence angles. The whole analysis is aimed at highlighting, on the one hand, the specific independencies of natural surface fractal dimension maps from the look angle and from the local incidence angle, which can be very useful in information extraction and SAR post-processing techniques and, on the other hand, the different fractal dimension maps behavior whereas urban areas are analyzed.