A Three-Dimensional Fractal Analysis Method for Ground Monocrystal Sapphire Surface

2014 ◽  
Vol 1017 ◽  
pp. 187-192
Author(s):  
Qiu Yan Wang ◽  
Zhi Qiang Liang ◽  
Xi Bin Wang ◽  
Wen Xiang Zhao ◽  
Yong Bo Wu ◽  
...  
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Fractal Analysis ◽  
Three Dimensional ◽  
Ground Surface ◽  
Surface Characteristics ◽  
Surface Phenomenon ◽  
Analysis Method ◽  
Characterization Methods ◽  
Box Counting

Conventional characterization methods of grinding surface using surface roughness parameters, e.g., Ra, depend on either the resolution of the measuring instrument or the length of the sample. But fractal dimension (FD) as a scale-independent fractal parameter is effective to evaluate the ground surface at any length scale and represent lots of surface phenomenon at its relevant length scales. In this paper, a three-dimensional (3D) box-counting fractal analysis method is used to investigate ground surface morphology of monocrystal sapphire by calculating 3D fractal dimension of the ground surface. The results obtained show that fractal dimension decreases with the increasing surface roughness. For the ground surface with higher fractal dimension, its microtopography is more exquisite with minor defects. Once the fractal dimension become smaller, deep cracks and pronounced defects are exhibited in ground surface. Moreover, the ground surface obtained in ductile mode has much higher fractal dimension than that in brittle mode. Therefore, the fractal analysis method has the potential to reveal the ground surface characteristics of monocrystal sapphire.

Fractal analysis with scanning probe microscopy

1992 ◽  
Vol 50 (2) ◽  
pp. 1046-1047
Author(s):  
Paul E. West ◽  
Sid Marchesse-Rugona ◽  
Zhuoning Li
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Physical Properties ◽  
Fractal Analysis ◽  
Three Dimensional ◽  
Optical Microscope ◽  
Two Dimensional ◽  
Atomic Force ◽  
Dimensional Measurements ◽  
Surface Profiles

Surface roughness determined qualitatively by direct visualization can be correlated to several physical properties. However, finding a suitable method of quantifying surface roughness, until recently, has been difficult. The concept of Fractal Dimension, recently popularized by Mandelbrot(1982) has been extremely successful in quantifying surface roughness and relating it to such measurable physical properties such as; cleanability, catalytic activity, rate of corrosion, and even flavor.Atomic Force Microscopes permit direct three dimensional measurements of surface microstructure. AFM images are obtained by measuring the motion of a sharp stylus as it is scanned across a surface. Because the AFM directly measures three dimensional topograms, it is ideally suited for two dimensional and three dimensional fractal analysis. Other microscope techniques such as the scanning electron or optical microscope give only two dimensional magnification and fractal measurements are not easily made.The Atomic Force Microscope enables us to obtain the fractal dimension of surface profiles as well as surface areas. For surface profiles we use a box counting method (Mandelbrot 1986, Chesters et al. 1989).

COMPARISON OF SEVERAL METHODS FOR CALCULATING FRACTAL-BASED TOPOGRAPHIC CHARACTERIZATION PARAMETERS

Fractals ◽  
1994 ◽  
Vol 02 (03) ◽  
pp. 437-440 ◽  
Author(s):  
WILLIAM A. JOHNSEN ◽  
CHRISTOPHER A. BROWN
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Data Sets ◽  
Characterization Methods ◽  
Topographic Data ◽  
Box Counting ◽  
Correlation Methods ◽  
Analysis Methods ◽  
Wide Range ◽  
Point Correlation

The objective of this work is to compare fractal-based, topographic characterization parameters calculated by several different fractal analysis methods. Four fractal characterization methods (compass, patchwork, box counting, and 2-point correlation) are systematically applied to five topographic data sets, which encompass a wide range of scale, and the results are compared. The compass and patchwork methods calculate similar values for the fractal dimension and smooth/rough crossover. The box and 2-point correlation methods calculate similar values for the fractal dimension. The compass and patchwork methods are capable of calculating the smooth/rough crossover.

Optimal Density Determination of Bouguer Anomaly Using Fractal Analysis (Case of study: Charak area,IRAN)

2005 ◽  
Vol 1 (1) ◽  
pp. 21-24
Author(s):  
Hamid Reza Samadi
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Host Rock ◽  
Bouguer Anomaly ◽  
Research Area ◽  
Density Difference ◽  
Optimal Density

In exploration geophysics the main and initial aim is to determine density of under-research goals which have certain density difference with the host rock. Therefore, we state a method in this paper to determine the density of bouguer plate, the so-called variogram method based on fractal geometry. This method is based on minimizing surface roughness of bouguer anomaly. The fractal dimension of surface has been used as surface roughness of bouguer anomaly. Using this method, the optimal density of Charak area insouth of Hormozgan province can be determined which is 2/7 g/cfor the under-research area. This determined density has been used to correct and investigate its results about the isostasy of the studied area and results well-coincided with the geology of the area and dug exploratory holes in the text area

scholarly journals ASSESSMENT OF URBAN SPRAWL USING SHANNON’S ENTROPY AND FRACTAL ANALYSIS: A CASE STUDY OF ATAKUM, ILKADIM AND CANIK (SAMSUN, TURKEY)

2017 ◽  
Vol 25 (3) ◽  
pp. 264-276 ◽  
Author(s):  
Derya OZTURK
Keyword(s):  
Urban Sprawl ◽  
Fractal Analysis ◽  
Urban Form ◽  
Entropy Method ◽  
Shannon’S Entropy ◽  
Analysis Method ◽  
Maximum Likelihood Classification ◽  
Shannon's Entropy ◽  
Box Counting

Urban sprawl is one of the most important problems in urban development due to its negative environmental and societal impacts. Therefore, the spatial pattern of urban growth should be accurately analyzed and well understood for effective urban planning. This paper focuses on urban sprawl analysis in the Atakum, Ilkadim and Canik districts of Samsun, Turkey. In this study, urban sprawl was examined over a period of 24 years using Shannon's entropy and fractal analysis based on remote sensing and Geographic Information System (GIS). The built-up areas in 1989, 2000 and 2013 were extracted from Landsat TM/ETM+/OLI images using the maximum likelihood classification method, and urban form changes in the 1989–2013 period were investigated. The Shannon's entropy method was used to determine the degree of urban sprawl, and a fractal analysis method based on box counting was used to characterize the urban sprawl. The results show that Atakum, Ilkadim and Canik experienced important changes and have considerable sprawl and complex characteristics now. The study also revealed that there is no monotonic relationship between Shannon's entropy and fractal dimension.

Outer Cutoff Value for the Box-Counting Method for Fractal Analysis of the Nucleus Using Kirsch Edge Detection

2020 ◽  
pp. 1-8
Author(s):  
Haruhiko Yoshioka ◽  
Kouki Minami ◽  
Hirokazu Odashima ◽  
Keita Miyakawa ◽  
Kayo Horie ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Edge Detection ◽  
Fractal Analysis ◽  
Nuclear Size ◽  
Original Position ◽  
Cytological Diagnosis ◽  
Cutoff Value ◽  
Position Bias ◽  
Box Counting ◽  
Nuclear Rotation

<b><i>Objective:</i></b> The complexity of chromatin (i.e., irregular geometry and distribution) is one of the important factors considered in the cytological diagnosis of cancer. Fractal analysis with Kirsch edge detection is a known technique to detect irregular geometry and distribution in an image. We examined the outer cutoff value for the box-counting (BC) method for fractal analysis of the complexity of chromatin using Kirsch edge detection. <b><i>Materials:</i></b> The following images were used for the analysis: (1) image of the nucleus for Kirsch edge detection measuring 97 × 122 pix (10.7 × 13.4 μm) with a Feret diameter of chromatin mesh (<i>n</i> = 50) measuring 17.3 ± 1.8 pix (1.9 ± 0.5 μm) and chromatin network distance (<i>n</i> = 50) measuring 4.4 ± 1.6 pix (0.49 ± 0.18 μm), and (2) sample images for Kirsch edge detection with varying diameters (10.4, 15.9, and 18.1 μm) and network width of 0.4 μm. <b><i>Methods:</i></b> Three types of bias that can affect the outcomes of fractal analysis in cytological diagnosis were defined. (1) Nuclear position bias: images of 9 different positions generated by shifting the original position of the nucleus in the middle of a 256 × 256 pix (28.1 μm) square frame in 8 compass directions. (2) Nuclear rotation bias: images of 8 different rotations obtained by rotating the original position of the nucleus in 45° increments (0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°). (3) Nuclear size bias: images of varying size (diameter: 190 pix [10.4 μm], 290 pix [15.9 μm], and 330 pix [18.1 μm]) with the same mesh pattern (network width: 8 pix [0.4 μm]) within a 512 × 512 pix square. Different outer cutoff values for the BC method (256, 128, 64, 32, 16, and 8 pix) were applied for each bias to assess the fractal dimension and to compare the coefficient of variation (CV). <b><i>Results:</i></b> The BC method with the outer cutoff value of 32 pix resulted in the least variation of fractal dimension. Specifically, with the cutoff value of 32 pix, the CV of nuclear position bias, nuclear rotation bias, and nuclear size bias were &#x3c;1% (0.1, 0.4, and 0.3%, respectively), with no significant difference between the position and rotation bias (<i>p</i> = 0.19). Our study suggests that the BC method with the outer cutoff value of 32 pix is suitable for the analysis of the complexity of chromatin with chromatin mesh.

Fractal analysis of a new generation of HVOF sprayed coatings based on optical surface metrology

2016 ◽  
Vol 83 (6) ◽  
Author(s):  
Yibo Zou ◽  
Markus Kästner ◽  
Eduard Reithmeier
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Laser Scanning ◽  
Three Dimensional ◽  
Confocal Laser Scanning Microscope ◽  
Confocal Laser ◽  
Surface Metrology ◽  
Roughness Parameters ◽  
Scale Invariant ◽  
Sprayed Coatings

AbstractIn this article, fractal analysis combined with roughness measurement is proposed to characterize the new generations of HVOF sprayed coatings' surface textures. Two-dimensional and three-dimensional box counting algorithms are introduced to determine the fractal dimension, which is considered as a scale-invariant parameter and is able to describe chaos and complexity of the surface. For surface roughness metrology, a confocal laser scanning microscope with different lenses is used to acquire the areal topography, providing a sequence of height maps with different image resolutions. Typical areal roughness parameters are assessed based on the international standard ISO-25178. The results show that the fractal dimension is a powerful tool to depict the nature of the surface texture of the investigated coatings. Moreover, it is found that the traditional amplitude roughness parameters depend strongly on the range of the measurement field as well as the datasets' resolution, whereas the fractal dimension is rather invariant to the scales of the measured datasets. Finally, the correlation between the fractal dimension and roughness parameters is given at the end of this paper.

scholarly journals Review about the Application of Fractal Theory in the Research of Packaging Materials

Materials ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 860
Author(s):  
Qingshan Duan ◽  
Jiejie An ◽  
Hanling Mao ◽  
Dongwu Liang ◽  
Hao Li ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Fractal Theory ◽  
Inorganic Materials ◽  
Packaging Materials ◽  
Analysis Method ◽  
Characteristic Parameters ◽  
Fractal Characteristic ◽  
Fractal Characteristics ◽  
The Relationship

The work is intended to summarize the recent progress in the work of fractal theory in packaging material to provide important insights into applied research on fractal in packaging materials. The fractal analysis methods employed for inorganic materials such as metal alloys and ceramics, polymers, and their composites are reviewed from the aspects of fractal feature extraction and fractal dimension calculation methods. Through the fractal dimension of packaging materials and the fractal in their preparation process, the relationship between the fractal characteristic parameters and the properties of packaging materials is discussed. The fractal analysis method can qualitatively and quantitatively characterize the fractal characteristics, microstructure, and properties of a large number of various types of packaging materials. The method of using fractal theory to probe the preparation and properties of packaging materials is universal; the relationship between the properties of packaging materials and fractal dimension will be a critical trend of fractal theory in the research on properties of packaging materials.

scholarly journals FRACTAL ANALYSIS OF TRABECULAR BONE: A STANDARDISED METHODOLOGY

2011 ◽  
Vol 19 (1) ◽  
pp. 45 ◽  
Author(s):  
Ian Parkinson ◽  
Nick Fazzalari
Keyword(s):  
Fractal Dimension ◽  
Trabecular Bone ◽  
Fractal Analysis ◽  
Fractal Dimensions ◽  
Box Counting ◽  
Cell Tissue ◽  
Image Analyser ◽  
Model Independent ◽  
Box Counting Method ◽  
Image Orientation

A standardised methodology for the fractal analysis of histological sections of trabecular bone has been established. A modified box counting method has been developed for use on a PC based image analyser (Quantimet 500MC, Leica Cambridge). The effect of image analyser settings, magnification, image orientation and threshold levels, was determined. Also, the range of scale over which trabecular bone is effectively fractal was determined and a method formulated to objectively calculate more than one fractal dimension from the modified Richardson plot. The results show that magnification, image orientation and threshold settings have little effect on the estimate of fractal dimension. Trabecular bone has a lower limit below which it is not fractal (λ<25 μm) and the upper limit is 4250 μm. There are three distinct fractal dimensions for trabecular bone (sectional fractals), with magnitudes greater than 1.0 and less than 2.0. It has been shown that trabecular bone is effectively fractal over a defined range of scale. Also, within this range, there is more than 1 fractal dimension, describing spatial structural entities. Fractal analysis is a model independent method for describing a complex multifaceted structure, which can be adapted for the study of other biological systems. This may be at the cell, tissue or organ level and compliments conventional histomorphometric and stereological techniques.

scholarly journals Binarization methods and investigation of their influence on the fractal dimension of functional coatings

2020 ◽  
pp. 30-42
Author(s):  
Anna Zhurba ◽  
Michail Gasik
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Quantitative Characteristic ◽  
Image Binarization ◽  
Counting Method ◽  
Functional Coatings ◽  
Binary Images ◽  
Box Counting ◽  
Box Counting Method ◽  
Binarization Threshold

An essential element of fractal analysis of functional coatings is the fractal dimension, which is an important quantitative characteristic. Typically, coating images are represented as colored or halftone, and most fractal dimension algorithms are for binary images. Therefore, an important step in fractal analysis is binarization, which is a threshold separation operation and the result of which is a binary image.The purpose of the study is to study and program the methods of image binarization and to study the influence of these methods on the value of fractal dimension of functional coatings.As a result of the binarization threshold, the image is split into two regions, one containing all pixels with values below a certain threshold and the other containing all pixels with values above that threshold. Of great importance is the determination of the binarization threshold.The study analyzed a number of functional coating images, determined the fractal dimension of the image by the Box Counting method at different binarization thresholds and when applying different binarization methods (binarization with lower and upper threshold, with double restriction, and the average method for determining the optimal binarization threshold) images. The Box Counting method is used to depict any structure on a plane. This method allows us to determine the fractal dimension of not strictly self-similar objects. Each image binarization method is used for different types of images and for solving different problems.As a result, the methods of image binarization were developed and implemented, the fractal dimension of binary images was calculated, and the influence of these methods on the value of fractal dimension of functional coatings was investigated.The surfaces of composite steel structure, metallic porous materials, and natural cave structures are analyzed.

Magnetorheological finishing of copper cylindrical roller for its improved performance in printing machine

2020 ◽  
pp. 095440892094523
Author(s):  
Manpreet Singh ◽  
Anant Kumar Singh
Keyword(s):  
Surface Roughness ◽  
Surface Hardness ◽  
Ground Surface ◽  
Surface Characteristics ◽  
Fluid Composition ◽  
Finishing Process ◽  
Finished Surface ◽  
Measuring Machine ◽  
Mr Polishing Fluid ◽  
Cylindrical Roller

The copper cylindrical roller plays an important role in the printing operation. The copper roller requires fine and uniform finishing to uniformly distribute the colours and ingot material. Fine and uniform finishing of copper cylindrical rollers get difficulty using the traditional finishing processes due to their ductility and low hardness. Therefore, to achieve this fine finishing requirement, the rotary rectangular tool core-based magnetorheological (MR) finishing process is employed. Initially, the suitable MR polishing fluid composition is selected for the effective fine finishing of the surface of the copper cylindrical rollers. Furthermore, the central composite design is used to optimize the MR finishing process parameters. The surface roughness profiles, surface texture, and reflection tests are performed on the initial ground surface and the MR finished surface of the copper roller. The surface roughness value gets reduced from 190 nm to 25 nm after 4 hrs MR finishing with the optimum parametric conditions over the copper cylindrical roller surface having a dimension of 120 mm in length and 25 mm in diameter. The present MR finishing process found effective to significantly reduce the surface roughness value and enhance the surface characteristics of the copper cylindrical rollers. The geometrical dimensions in terms of circularity and straightness are also checked on the initial ground surface and finished surface of the copper cylindrical roller using the coordinate measuring machine and waviness test. The enhancement in surface characteristics, dimensional accuracy, and surface hardness after the present MR finishing process is found to be beneficial for improving the functional performance of the copper cylindrical rollers in the printing processing machine.

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