量化三维网格模型复杂度的Fractal Dimension^(+)方法

对三维网格模型复杂度的评估及应用进行研究,提出基于Fractal Dimension^(+)方法量化三维网格模型复杂度。该方法利用分形几何学的分形维度来确定三维网格模型的不规则性;为了全面量化复杂度并且提高量化的准确性和可信度,在分形维度算法中添加孔隙比和亏格数的概念,全面考虑介质内部空间分布和几何结构的连接性以及孔洞分布的信息。对大量三维网格模型进行实验,结果表明Fractal Dimension^(+)方法可以有效计算出准确的复杂度。

Computational Quantification of Map Projection Distortion by Fractal Dimension of Coastlines

Maps, essential tools for portraying the Earth’s surface, inherently introduce distortions to geographical features. While various quantification methods exist for assessing these distortions, they often fall short when evaluating actual geographic features. In our study, we took a novel approach by analyzing map projection distortion from a geometric perspective. We computed the fractal dimensions of different stretches of coastline before and after projection using the divide-and-conquer algorithm and image processing. Our findings revealed that map projections, even when preserving basic shapes, inevitably stretch and compress coastlines in diverse directions. This analysis method provides a more realistic and practical way to measure map-induced distortions, with significant implications for cartography, geographic information systems (GIS), and geomorphology. By bridging the gap between theoretical analysis and real-world features, this method greatly enhances accuracy and practicality when evaluating map projections.

Fractal analysis of major faults and fractal dimension of lineaments in the Indo-Gangetic Plain on a regional scale

The Indo-Gangetic Plain(IGP)is one of the most seismically vulnerable areas due to its proximity to the Himalayas.Geographic information system(GIS)-based seismic characterization of the IGP was performed based on the degree of deformation and fractal dimension.The zone between the Main Boundary Thrust(MBT)and the Main Central Thrust(MCT)in the Himalayan Mountain Range(HMR)experienced large variations in earthquake magnitude,which were identified by Number-Size(NS)fractal modeling.The central IGP zone experienced only moderate to low mainshock levels.Fractal analysis of earthquake epicenters reveals a large scattering of earthquake epicenters in the HMR and central IGP zones.Similarly,the fault fractal analysis identifies the HMR,central IGP,and south-western IGP zones as having more faults.Overall,the seismicity of the study region is strong in the central IGP,south-western IGP,and HMR zones,moderate in the western and southern IGP,and low in the northern,eastern,and south-eastern IGP zones.

Novel Path Counting-Based Method for Fractal Dimension Estimation of the Ultra-Dense Networks

Next-generation networks,including the Internet of Things(IoT),fifth-generation cellular systems(5G),and sixth-generation cellular systems(6G),suf-fer from the dramatic increase of the number of deployed devices.This puts high constraints and challenges on the design of such networks.Structural changing of the network is one of such challenges that affect the network performance,includ-ing the required quality of service(QoS).The fractal dimension(FD)is consid-ered one of the main indicators used to represent the structure of the communication network.To this end,this work analyzes the FD of the network and its use for telecommunication networks investigation and planning.The clus-ter growing method for assessing the FD is introduced and analyzed.The article proposes a novel method for estimating the FD of a communication network,based on assessing the network’s connectivity,by searching for the shortest routes.Unlike the cluster growing method,the proposed method does not require multiple iterations,which reduces the number of calculations,and increases the stability of the results obtained.Thus,the proposed method requires less compu-tational cost than the cluster growing method and achieves higher stability.The method is quite simple to implement and can be used in the tasks of research and planning of modern and promising communication networks.The developed method is evaluated for two different network structures and compared with the cluster growing method.Results validate the developed method.

Hausdorff Dimension and Fractal Dimension of the Global Attractor for the Higher-Order Coupled Kirchhoff-Type Equations

This paper mainly deals with the higher-order coupled Kirchhoff-type equations with nonlinear strong damped and source terms in a bounded domain. We obtain some results that are estimation of the upper bounds of Hausdorff dimension and Fractal dimension of the global attractor.

Cloud detection from visual band of satellite image based on variance of fractal dimension

Cover ratio of cloud is a very important factor which affects the quality of a satellite image, therefore cloud detection from satellite images is a necessary step in assessing the image quality. The study on cloud detection from the visual band of a satellite image is developed. Firstly, we consider the differences between the cloud and ground including high grey level, good continuity of grey level, area of cloud region, and the variance of local fractal dimension (VLFD) of the cloud region. A single cloud region detection method is proposed. Secondly, by introducing a reference satellite image and by comparing the variance in the dimensions corresponding to the reference and the tested images, a method that detects multiple cloud regions and determines whether or not the cloud exists in an image is described. By using several Ikonos images, the performance of the proposed method is demonstrated.

Numerical Approximation of Fractal Dimension of Gaussian Stochastic Processes

In this paper we propose a numerical method to estimate the fractal dimension of stationary Gaussian stochastic processes using the random Euler numerical scheme and based on an analytical formulation of the fractal dimension for filtered stochastic signals. The discretization of continuous time processes through this random scheme allows us to find, numerically, the expected value, variance and correlation functions at any point of time. This alternative method for estimating the fractal dimension is easy to implement and requires no sophisticated routines. We use simulated data sets for stationary processes of the type Random Ornstein Uhlenbeck to graphically illustrate the results and compare them with those obtained whit the box counting theorem.

The Fracture Density and Fractal Dimension Prediction Based on Support Vector Machine

The key of effective development for the fractured reservoir is to describe the distribution of the fracture and build the fracture geological model. To acquire more optimal exploration and development of the oilfield, objective geologic model of reservoir fractures is needed for further knowledge of the spatial distribution condition of fractures. Adopting well-logging and seismic techniques can be expensive and usually yield multiple solutions, yet resolution will disturb the seismic method, making it difficult to acquire accurate and sound fracture parameters from seismic data. In this paper, the theoretical foundation for support vector machine, fractal geometry, combined the various information in geology, logging, well core, seism and field outcrop about the fracture and calculate the parameters of the fracture (fracture density and fractal dimension), and the good foundation is established for fractured reservoir description of discrete fracture network model. Based on analysis of conventional prediction methods of development indices and factor influencing the parameters of the fracture, a support vector machine method is established to predict the parameters of the fracture. The new support vector machine method is based on time series analysis to select the kernel function. Trains and tests the support vector machine network with historical data to construct the support vector regression prediction model. A case was fit into the model to test and analyse its reliability, the results suggested that the model had a high prediction performance, and could be well applied to the prediction of fracture parameters.

A Fractal Dimension Based Framework for Night Vision Fusion

In this paper, a novel fusion framework is proposed for night-vision applications such as pedestrian recognition,vehicle navigation and surveillance. The underlying concept is to combine low-light visible and infrared imagery into a single output to enhance visual perception. The proposed framework is computationally simple since it is only realized in the spatial domain. The core idea is to obtain an initial fused image by averaging all the source images. The initial fused image is then enhanced by selecting the most salient features guided from the root mean square error(RMSE) and fractal dimension of the visual and infrared images to obtain the final fused image.Extensive experiments on different scene imaginary demonstrate that it is consistently superior to the conventional image fusion methods in terms of visual and quantitative evaluations.

Comparison of Correlation Dimension and Fractal Dimension in Estimating BIS index

This paper compares the correlation dimension (D2) and Higuchi fractal dimension (HFD) approaches in estimating BIS index based on of electroencephalogram (EEG). The single-channel EEG data was captured in both ICU and operating room and different anesthetic drugs, including propofol and isoflurane were used. For better analysis, application of adaptive segmentation on EEG signal for estimating BIS index is evaluated and compared to fixed segmentation. Prediction probability (PK) is used as a measure of correlation between the predictors and BIS index to evaluate the proposed methods. The results show the ability of these algorithms (specifically HFD algorithm) in predicting BIS index. Also, evolving fixed and adaptive windowing methods for segmentation of EEG reveals no meaningful difference in estimating BIS index.

Study on the Spatial Pattern Changes of Land Use Based on Fractal Dimensions in Tianjin New Coastal Area

[Objective] The aim was to discuss the spatial pattern changes of land use in Tianjin new coastal area based on fractal dimensions.[Method] By dint of remote and geographic information system technology to obtain the data of urban land use in new coastal area from 1993 to 2008,the boundary dimension,radius dimension and information dimension of each land use type were calculated based on fractal dimension.In addition,the revealed land use spatial dimension changes characteristics were analyzed.[Result] The spatial distribution of each land use type in new costal area had distinct fractal characteristics.And,the amount and changes of three types of dimension values effectively revealed the changes of complicatedness,centeredness and evenness of spatial pattern of land use in the study area.The boundary dimension of unused land and salty earth increased incessantly,which suggested its increasing complicatedness.The boundary of the port and wharf and shoal land was getting simpler.The radius dimension of the cultivated land was larger than 2,which suggested that its area spread from center to the surroundings;the one in salty land and waters distributed evenly within different radius space to the center of the city;the one in other land use types reduced gradually from center to the surroundings.The information dimension value in the woodland and orchard land,unused land and shoal land was small,and was in obvious concentrated distribution;the spatial distribution of cultivated and salty land concentrated in the outside area;the construction area in the port and wharf spread gradually on the basis of original state;the spatial distribution of waters and residents and mines were even.[Conclusion] Applying fractal dimensions to the study of spatial pattern changes of urban land use can make up for some disadvantages in classical urban spatial pattern quantitative research,which has favorable practical value.

A novel box-counting method for quantitative fractal analysis of threedimensional pore characteristics in sandstone

Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.

Particle Swarm Optimized Optimal Threshold Value Selection for Clustering based on Correlation Fractal Dimension

The work on the paper is focused on the use of Fractal Dimension in clustering for evolving data streams. Recently Anuradha et al. proposed a new approach based on Relative Change in Fractal Dimension (RCFD) and damped window model for clustering evolving data streams. Through observations on the aforementioned referred paper, this paper reveals that the formation of quality cluster is heavily predominant on the suitable selection of threshold value. In the above-mentionedpaper Anuradha et al. have used a heuristic approach for fixing the threshold value. Although the outcome of the approach is acceptable, however, the approach is purely based on random selection and has no basis to claim the acceptability in general. In this paper a novel method is proposed to optimally compute threshold value using a population based randomized approach known as particle swarm optimization (PSO). Simulations are done on two huge data sets KDD Cup 1999 data set and the Forest Covertype data set and the results of the cluster quality are compared with the fixed approach. The comparison reveals that the chosen value of threshold by Anuradha et al., is robust and can be used with confidence.

Fractal Dimension Based Shot Transition Detection in Sport Videos

Increase in application fields of video has boosted the demand to analyze and organize video libraries for efficient scene analysis and information retrieval. This paper addresses the detection of shot transitions, which plays a crucial role in scene analysis, using a novel method based on fractal dimension (FD) that carries information on roughness of image intensity surface and textural structure. The proposed method is tested on sport videos including soccer and tennis matches that contain considerable amount of abrupt and gradual shot transitions. Experimental results indicate that the FD based shot transition detection method offers promising performance with respect to pixel and histogram based methods available in the literature.

Texture Filters and Fractal Dimension on Image Segmentation

Texture analysis is important in several image segmentation and classification problems. Different image textures manifest themselves by dissimilarity in both the property values and the spatial interrelationships of their component texture primitives. We use this fact in a texture discrimination system. This paper focuses on how to apply texture operators based on co-occurrence matrix, texture filters and fractal dimension to the problem of object recognition and image segmentation.

A Statistical Method for Determining the Fractal Dimension of Time Series

In this paper,we present a new method for determining the fractal dimension of time series and the algorithm of H index(Hurst index).

Fractal dimension for porous metal materials of FeCrAl fiber

The FeCrA1 fiber was used to prepare porous metal materials with air-laid technology, and then followed by sintering at 1300 ℃ for a holding period of 2 h in the vacuum. In addition, a novel fractal soft, which was developed based on the fractal theory and the computer image processing technology, was explored to describe the pore structure of porous metal materials. Furthermore, the fractal dimension of pore structure was calculated by the soft and the effects of magnification and porosity on ffactal dimension were also discussed. The results show that the fractal dimension decreases with increase in the magnification, while it increases continuously with the porosity enhancing. The interrelationship between the fractal dimension and the magnification or porosity can be presented by the equation of D=α_0exp（-x/α_1）＋α_2和D=k_2-（k_1-k_2）/[1＋exp（（θ-k_0）/k_3）], respectively.

Quantum Gravity and Dark Energy Using Fractal Planck Scaling

Following an inspiring idea due to D. Gross, we arrive at a topological Planck energy Ep and a corresponding topological Planck length effectively scaling the Planck scale from esoterically large and equally esoterically small numbers to a manageably where P(H) is the famous Hardy’s probability for quantum entanglement which amounts to almost 9 percent and Based on these results, we conclude the equivalence of Einstein-Rosen “wormhole” bridges and Einstein’s Podolsky-Rosen’s spooky action at a distance. In turn these results are shown to be consistent with distinguishing two energy components which results in , namely the quantum zero set particle component which we can measure and the quantum empty set wave component which we cannot measure , i.e. the missing dark energy. Together the two components add to where E is the total energy, m is the mass and c is the speed of light. In other words, the present new derivation of the world’s most celebrated formula explains in one stroke the two most puzzling problems of quantum physics and relativistic cosmology, namely the physicomathematical meaning of the wave function and the nature of dark energy. In essence they are one and the same when looked upon from the view point of quantum-fractal geometry.

Effect of pouring temperature on fractal dimension of primary phase morphology in semi-solid A356 alloy

The fractal dimensions of primary phase morphology in semi-solid A356 alloy prepared by low superheat pouring and slightly electromagnetic stirring were calculated,and the effect of pouring temperature on fractal dimension of primary phase morphology in semi-solid A356 alloy was researched.The results indicate that it is feasible to prepare semisolid A356 alloy slurry by low superheat pouring and slightly electromagnetic stirring,and there is an important effect of pouring temperature on the morphology and the grain size of the primary phase in semi-solid A356 alloy,in which the reduction of pouring temperature can obviously improve grain size and shape factor of primary phase in semi-solid A356 alloy under the condition of a certain stirring power.The primary phase morphology of semi-solid A356 alloy prepared by low superheat pouring and slightly electromagnetic stirring can be characterized by fractal dimension,and the primary phase morphology obtained by the different processing parameters has the different fractal dimensions.Solidification of semi-solid alloy is a course of change in fractal dimension.

FRACTAL DIMENSION AS MORPHOLOGY AND SIZE PARAMETERS OF FRACTURED PARTICLES OF ROCK

Semi-automatic image analysis system was used for the surface fractal dimension measurement of 15 kinds of rock-fractured particles. The results indicate that the surfaces of the particles that have a size of about 200 mesh are fractal, and that the fractal dimension is in the range of 2.07～2.11. Therefore, some definitions of particle size such as surface diameter, equivalent volumetricly surface diameter are equivocal without giving the step length and the drag diameter, free-fall diameter, and stokes’diameter may be modified by the fractal dimension.