• Journal of Korean Society of Industrial and Systems Engineering
• /
• v.19 no.37
• /
• pp.233-240
• /
• 1996

Julia set, Fatou set와 Mandelbrot set 가 컴퓨터에 의하여 도형화된 후부터 혼돈 역학체계 (chaotic dynamical system)에 대한 연구가 모든 학계에 비상한 관심을 모으고 있으며 특히 수학자들에 의하여 많은 연구가 이루어지고 있다. 또한 혼돈 역학체계를 기초로 하여 컴퓨터 그래픽스를 이용한 후랙탈(fractal)들의 매혹적인 시각적 표현으로 인하여 최근들어 과학자들 뿐 아니라 일반대중의 후랙탈에 대한 관심이 매우 높아지고 있다. 후랙탈이란 말은 라틴어 fractus(부서진 상태를 뜻함)에서 유래되었으며 1975년 Mandelbrot가 수학 및 자연계의 비정규적 패턴들에 대한 체계적 고찰을 담은 자신의 에세이의 표제를 주기 위해서 만들었다(〔6〕). 후랙탈을 기술하는데 있어서 가장 중요한 양은 차원(dimension)으로, 예컨데 Cantor 1/3 집합은 길이 1인 선분으로부터 시작하야 매 단계마다 모든 선분들의 가운데 1/3을 잘라내는 것을 무한히 반복함으로써 얻어지는데 이 집합의 Lebesgue measure는 0이지만 후랙탈 차원은 log2/log3 로 정수차원이 아닌 실수차원을 갖으며 또한 Cantor 1/3집합은 연속이 아니면서 점도 선도 아닌 집합인 것이다. 이 논문에서는 Box counting dimension 과 Hausdorff dimension에 대한 몇 가지 정의를 하고 정리 2.6, 정리2.7 및 정리 3.3을 증명함으로써 어떤 성질을 갖는 후랙탈의 가장 중요한 양인 후랙탈 차원에 대하여 논의 하고자 한다.

• Journal of Navigation and Port Research
• /
• v.40 no.4
• /
• pp.207-212
• /
• 2016

The box-counting(BC) method is one of the most commonly used methods for fractal dimension calculation of binary images in the fields of Engineering, Science, Medical Science, Geology, etc due to its simplicity and reliability. It deals with only square images with each size equal to the power of 2 to prevent it from discarding unused pixels for images of arbitrary size. In this paper, we presents a more efficient BC method based on the original one, which is applicable to images of arbitrary size. The proposed approach allows the number of the counting boxes to be real to improve the estimation accuracy. The mean absolute error performance is computed on two deterministic fractal images whose theoretical dimensions are well known to compare with those of the existing BC method and triangular BC method. The experimental results show that the proposed method can outperform the two methods and assess the complexity of coastline images of Korea and Chodo island taken from the Google map.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.26 no.4
• /
• pp.609-616
• /
• 2002

The characteristics of mechanical behavior are investigated for Ti-6Al-4V alloy. Four kinds of the specimens are prepared under different heat treatments in order to produce different microstructures. In the present investigations, impact, tensile and fatigue crack growth tests are performed for each test specimen. The results obtained through the investigations are compared. Additionally fr actal dimensions of crack pass are obtained using the box counting method. The results are, 1) the microstructures shows as equiaxed, bimodal and Widmanstatten microstructures respectively, 2) the impact energy and elongation are superior fur the bimodal microstructure, and the hardness and tensile strength are superior fur the Widmanstatten microstructure, 3) the fatigue crack growth rate is similar to all microstructures in low ΔK region while that of equiaxed microstructure is the largest, and that of Widmanstatten microstructure is the lowest in high ΔK region respectively, 4) the fractal dimension D of Widmanstatten microstructure shows higher value than that of the equiaxed and bimodal microstructures under 200 magnification view of the SEM micrographs.

• Economic and Environmental Geology
• /
• v.29 no.4
• /
• pp.523-528
• /
• 1996

In Korean topography the behaviors of random fractals and multifractality are analytically and numerically studied on the mountain heights shown between $128{\sim}129^{\circ}E$ and $37{\sim}38^{\circ}N$. The phase transitions on the fractal structure are approximately found at the critical length $N_c=2000m$ from the values of standard deviations that it varies with both the longitudinal and latitudinal lengths. In the multifractal structure we assume that the mountain heights divided by the intervals of 20 m are located on the horizontal plane in two dimensional square lattice, and estimate the values of the generalized dimension and the scaling exponents by using the the box counting method for the three cases of square area ($1{\times}1km^2$, $2{\times}2km^2$, $4{\times}4km^2$).

• Geomechanics and Engineering
• /
• v.9 no.5
• /
• pp.619-629
• /
• 2015

Influenced by various mining activities, fractures in rock masses have different densities, set numbers and lengths, which induce different mechanical properties and failure modes of rock masses. Therefore, precisely expressing the failure criterion of the fractured rock influenced by coal mining is significant for the support design, safety assessment and disaster prevention of underground mining engineering subjected to multiple mining activities. By adopting PFC2D particle flow simulation software, this study investigated the propagation and fractal evolution laws of the micro cracks occurring in two typical kinds of rocks under uniaxial compressive condition. Furthermore, it calculated compressive strengths of the rocks with different confining pressures and box-counting dimensions. Moreover, the quantitative relation between the box-counting dimension of the rocks and the empirical parameters m and s in Hoek-Brown strength criterion was established. Results showed that with the increase of the strain, the box-counting dimension of the rocks first increased slowly at the beginning and then exhibited an exponential increase approximately. In the case of small strains of same value, the box-counting dimensions of hard rocks were smaller than those of weak rocks, while the former increased rapidly and were larger than the latter under large strain. The results also presented that there was a negative correlation between the parameters m and s in Hoek-Brown strength criterion and the box-counting dimension of the rocks suffering from variable mining activities. In other words, as the box-counting dimensions increased, the parameters m and s decreased linearly, and their relationship could be described using first order polynomial function.

• The Journal of Engineering Geology
• /
• v.24 no.2
• /
• pp.161-169
• /
• 2014

We present the results of an experimental physical weathering study that focuses on fresh and slightly weathered gneiss samples from the Wonju area of Korea. The study investigated changes in the physico-mechanical properties of these samples during accelerated laboratory-based weathering, including analyses of microfracture formation. The deteriorated samples used in the study were subjected to 100-150 freeze-thaw cycles, with index properties and microfracture geometries measured between each cycle. Each complete freeze-thaw cycle lasted 24 hours, and consisted of 2 hours of saturation in a vacuum chamber, 8 hours of freezing at $-21^{\circ}C{\pm}1^{\circ}C$, and 14 hours of thawing at room temperature. Specific gravity and seismic velocity values were negatively correlated with the number of freeze-thaw cycles, whereas absorption values tended to increase. The amount of deterioration of the rock samples was dependent on the degree of weathering of the rock prior to the start of the analysis. Absorption, specific gravity, and seismic velocity values can be used to infer the amount of physical weathering experienced by a gneiss in the study area. The sizes and density of microfracture in the rock specimens varied with the number of freeze-thaw cycles. We found that box fractal dimensions can be used to quantify the formation and propagation of microfracture in the samples. In addition, these box fractal dimensions can be used as a weathering index for the mid-and long-term prediction of rock weathering. The present results indicate that accelerated-weathering analysis can provide a detailed overview of the weathering characteristics of deteriorated rocks.

• Journal of Korea Multimedia Society
• /
• v.7 no.2
• /
• pp.173-186
• /
• 2004

This paper proposed the content-based retrieval system as a method for performing image retrieval through the effective feature extraction of the object of significant meaning based on the characteristics of man's visual system. To allow the object region of interest to be primarily detected, the region, being comparatively large size, greatly different from the background color and located in the middle of the image, was judged as the major object with a meaning. To get the original features of the image, the cumulative sum of tile declination difference vector the segment of the object contour had and the signature of the bipartite object were extracted and used in the form of being applied to the rotation of the object and the change of the size after partition of the total length of the object contour of the image into the normalized segment. Starting with this form feature, it was possible to make a retrieval robust to any change in translation, rotation and scaling by combining information on the texture sample, color and eccentricity and measuring the degree of similarity. It responded less sensitively to the phenomenon of distortion of the object feature due to the partial change or damage of the region. Also, the method of imposing a different weight of similarity on the image feature based on the relationship of complexity between measured objects using the fractal dimension by the Boxing-Counting Dimension minimized the wrong retrieval and showed more efficient retrieval rate.

• Economic and Environmental Geology
• /
• v.45 no.5
• /
• pp.487-502
• /
• 2012

This paper is focused on mineral compositions, microstructures and distributional characters of remained grains in the fault rocks collected from a fault developed in Yongdang-ri, Yangbuk-myeon, Gyeongju City, Korea, using X-ray diffraction (XRD), optical microscope, laser grain size analysis and fractal dimension analysis methods. The exposed fault core zone is about 1.5 meter thick. On the average, the breccia zone is 1.2 meter and the gouge zone is 20cm thick, respectively. XRD results show that the breccia zone consists predominantly of rock-forming minerals including quartz and feldspar, but the gouge zone consists of abundant clay minerals such as chlorite, illite and kaolinite. Mineral vein, pyrite and altered minerals commonly observed in the fault rock support evidence of fault activity associated with hydrothermal alteration. Fractal dimensions based on box counting, image analysis and laser particle analysis suggest that mineral grains in the fault rock underwent fracturing process as well as abrasion that gave rise to diminution of grains during the fault activity. Fractal dimensions(D-values) calculated by three methods gradually increase from the breccia zone to the gouge zone which has commonly high D-values. There are no noticeable changes in D-values in the gouge zone with trend being constant. It means that the bulk-crushing process of mineral grains in the breccia zone was predominant, whereas abrasion of mineral grains in the gouge zone took place by continuous fault activity. It means that the bulk-crushing process of mineral grains in the breccia zone was predominant, whereas abrasion of mineral grains in the gouge zone took place by continuous fault activity. Mineral compositions in the fault zone and peculiar trends in grain distribution indicate that multiple fault activity had a considerable influence on the evolution of fault zones, together with hydrothermal alteration. Meanwhile, fractal dimension values(D) in the fault rock should be used with caution because there is possibility that different values are unexpectedly obtained depending on the measurement methods available even in the same sample.

• Tunnel and Underground Space
• /
• v.16 no.1 s.60
• /
• pp.26-37
• /
• 2006

This study addresses the applicability of box fractal dimension, $D_B$, as an index of statistical homogeneity of fractured rock mass. The box-count method's capability in quantifying the combined effect of fracture density and size distribution is examined systematically. Total of 129 two-dimensional fracture configurations were generated based on different combinations of fracture size distribution and fracture density. $D_B$was calculated for the generated fracture network systems using the box-counting method. It was found that was standard deviation of trace length and fracture orientation have no effect on calculated $D_B$. The estimated $D_B$ was found to increase with increasing total density and/or mean trace length. To explore the field applicability of this study, the statistical homogeneity of fractured rock mass was investigated at the rock slope and the underground facility using the box-counting method as well as conventional contingency table analysis. The results obtained in this study clearly show that the methodologies given in this paper have the capability of determining the statistical homogeneity of fractured rock mass.

• Imaging Science in Dentistry
• /
• v.42 no.1
• /
• pp.5-12
• /
• 2012

Purpose : This study was performed to evaluate possible variations in maxillary and mandibular bone texture of normal population using the fractal analysis, particles count, and area fraction in intraoral radiographs. Materials and Methods : Periapical radiographs of patients who had full mouth intraoral radiographs were collected. Regions of interest ($100{\times}100$ pixels) were located between the teeth of the maxillary anterior, premolar, and molar area, as well as the mandibular anterior, premolar, and molar areas. The fractal dimension (FD) was calculated by using the box counting method. The particle count (PC) and area fraction (AF) analyses were also performed. Results : There was no significant difference in the FD values among the different groups of age, gender, upper, and lower jaws. The mean FD value was $1.49{\pm}0.01$. The mean PC ranged from 44 to 54, and the mean AF ranged from 10.92 to 11.85. The values of FD, PC, and AF were significantly correlated with each other except for the upper molar area. Conclusion : According to the results, patients with normal trabecular pattern showed a FD of approximately 1.5. Based on these results, further investigation would be recommended if the FD value of patient significantly differenct from this number, since the alteration of this value indicates microstructural modification of trabecular pattern of the jaws. Additionally, with periapical radiographs, simple and cost-effective, PC and AF could be used to assess the deviation from the normal.