Variation Of Fractal Dimension Research Articles

Changes in Pore Structure and Fractal Characteristics of Solvents Pretreated High-Rank Coal under Supercritical CO2

CO2 injection in coal seams, which is a significant initiative to mitigate environmental problems caused by greenhouse gases, often leads a sequence of changes in the physical properties of coal reservoirs. To look into how the pore structure changes in the process of CO2 sequestration, we selected fresh coal from Huoerxinhe coal mine in China as the object. Then, acid treatment and SC-CO2 extraction were used to dissolve Organic and inorganic components in coal. Thus, by using SEM, LTGA-N2 apparatus and XRD, the characteristics of pore parameter and fractal dimension variation were discussed. The research results show that, the APS of samples THF, HCL-HF and Y-C increase, while the total PV decreases and the pore connectivity deteriorates. The pore connectivity of Samples THF and HCL-HF is improved (THF-C, HCL-HF-C), but the total pore volume continuously reduces. In addition, solvents treatment and SC-CO2 extraction mainly act on the microporous fraction. After solvents pretreatment, the changes in the pore size distribution curves are mainly manifested in the reduction of number of micropores, especially in the micropores around 3–4 nm. There is a small increase in micropores for samples Y-C and HCL-HF-C, with the pore size mainly concentrated around 4 nm, while the pores of the sample THF-C mainly show an increase within the scope of 3–16 nm. Generally, solvent pretreatment and SC-CO2 extraction help to simplify pore structure. However, the sample HCL-HF-C shows opposite change characteristics. In a short period of time, the larger pore fractal dimension, the less beneficial it is to the flow of CO2, while pore fractal dimension becomes progressively less useful in assessing pore connectivity with increasing time.

The applicability of stripe light projection data to the determination of fractal dimensions of blast-cleaned steel surfaces

This paper is concerned with the assessment of fractal dimensions of blast-cleaned steel substrates using stripe line projection on 12 surface configurations. By means of the box-counting method and a Design of Experiment (DoE) approach, the factors abrasive type, surface preparation grade, surface roughness, and measurement direction were analyzed. The results revealed that the factor surface roughness contributed 52.5 % to the fractal dimensions. The factor measurement direction contributed 14.7 % (even up to 20 % once interactions were included) to the fractal dimension. Particularly the profile “formation” values were significantly influenced by the factor measurement direction, with contributions ranging up to 17.2 %. The potential limitations on the optical stripe light projection in capturing variations in fractal dimensions, especially in the abscissa-direction (formation), are highlighted. Whereas the box-counting method offers valuable insights, the stripe line projection introduces variations in the derived fractal dimensions. Stripe line projection shall be carefully considered when applied to fractal analysis.

FRACTAL DIMENSIONS OF THE LOGARITHM OF CONTINUOUS FUNCTIONS

This paper investigates the variation of fractal dimensions of continuous functions under operations. It is found that the Box dimension of the logarithm of positive continuous functions remains constant, and any nonzero real power of a positive continuous function can keep fractal dimensions variation closed. The study also discusses the fractal dimension of logarithmic function with bases of fractal functions, and the corresponding fractal dimension is determined by the function with the larger fractal dimension.

Differential diagnosis of Lewy body dementias using multivariate EEG classifiers

AbstractBackgroundLewy body dementia (LBD) results from pathology in ɑ‐synuclein and is the second most common cause of dementia. LBD has two manifestations: Parkinson’s disease dementia (PDD) and dementia with Lewy bodies (DLB). There is currently no non‐invasive, fast, reliable way to distinguish between early‐stage PDD, DLB, and Alzheimer’s disease (AD). Although previous traditional EEG visual rating methods have not been useful in delineating dementia subtypes, resting‐state electroencephalography (rsEEG) is a promising method for differential diagnosis of dementia, because it is inexpensive, non‐invasive, and widely available for clinical use. Here, we demonstrate how a multivariate classifier using rsEEG as input can differentiate PDD, DLB, and Alzheimer’s disease.MethodWe analyzed rsEEG data from older adults (N = 133, age 59‐89) with AD (n = 33), DLB (n = 27) or PDD (n = 45). We computed the aperiodic and periodic parameterization of brain activity, along with metrics of variation in fractal dimension within specific frequency bands. We then used these features, along with age and sex, to develop XGBoost machine‐learning models to predict patient diagnoses. Models were evaluated using cross‐validation with balanced accuracy, weighted sensitivity, and weighted specificity.ResultThe first task was a two‐way classification: Control vs Dementia (accuracy = 86%, sensitivity = 93%, specificity = 79%, p < .001). The second task was a three‐way classification: AD vs DLB vs PDD (accuracy = 66%, sensitivity = 70%, specificity = 82%, p < .001). We also evaluated two‐way models distinguishing between AD vs DLB (accuracy = 86%, sensitivity = 87%, specificity = 86%, p < .001), AD vs PDD (accuracy = 84%, sensitivity = 85%, specificity = 83%, p < .001), AD vs LBD (combined DLB and PDD; accuracy = 82%, sensitivity = 85%, specificity = 78%, p < .01), and DLB vs PDD (accuracy = 68%, sensitivity = 71%, specificity = 65%, p < .05).ConclusionMachine‐learning and rsEEG are capable of distinguishing between AD and LBD. Moreover, machine‐learning combined with spectral and fractal features can distinguish between DLB and PDD, despite both forms of dementia arising from ɑ‐synuclein pathology.

CHANGES IN FRACTAL DIMENSION OF THIN AND THICK BLOOD VESSELS FROM RETINAL FUNDUS IMAGES FOR DIFFERENT STAGES IN DIABETIC RETINOPATHY

Retinal vasculature feature extraction plays a critical role in the diagnosis and treatment of systemic conditions, particularly in the cases of diabetic retinopathy (DR). This research introduces an algorithm that utilizes segmented blood vessels in retinal images to identify and differentiate five stages of DR, including mild, moderate, severe and proliferative. The algorithm effectively extracts retinal blood vessels by integrating morphological operators and matched filters, yielding a more precise output. The algorithm’s performance is evaluated using the database IDRiD, demonstrating precision and sensitivity scores comparable to those of a trained observer. A box-counting method was incorporated to measure the fractal dimension (FD) of DR-segmented vessel images at various stages to enhance the accuracy of DR staging. The FD analysis was applied to both thick and thin segments of the blood vessels, enabling the assessment of accuracy, sensitivity and specificity. The results indicate that the algorithm successfully identifies the different stages of DR with an accuracy of 93.65% for the mild stage, of 93.33% for the moderate and severe stages and of 92.71% for proliferative DR compared to the images without DR. The study reveals that the variation in FD between the thick and thin vessel components can be an effective biomarker for identifying the different stages of DR, contributing to a better understanding of disease progression. By combining morphological operators, matched filters and fractal dimension analysis, this research presents a promising approach for specialists involved in diagnosing and treating DR, eventually leading to improved patient care and consequences.

Evaluation method of rock damage under uniaxial compression based on unit series-parallel electrical conductive model

The evaluation of rock damage behaviour is an important requirement for ensuring stability control and safety prediction in rock engineering. However, they have not been able to obtain sufficiently accurate and dynamic results due to the insufficient evaluation method. In this study, by means of fractals and unit series division, a unit series-parallel conductive model of damaged rock is derived, and a new evaluation method of rock damage under uniaxial compression was proposed. Rock was damaged by uniaxial compression, while electrical measurements and X-ray microscopy tests were performed to obtain the damaged rock resistivity, porosity, and fractal dimension variation. By establishing the relationship between defined meso-damage factor and resistivity, rock damage evolution law under axial compression was obtained. The results indicate that the growth trend was agree with the classical statistical damage model, which verified the accuracy of the results obtained by the proposed method. Moreover, as the strain increased, the damage factor determined by resistivity gradually decreased to −0.06 firstly and then increased rapidly to 0.79. Different from previous damage evolution law, brittle failure was observed and the cracks development in each stage was considered, including the closure (negative damage) and expansion (positive damage) of cracks.

Pore Structure and Fractal Characteristics of Tight Sandstone: A Case Study for Huagang Formation in the Xihu Sag, East China Sea Basin, China

Various experiments, including routine petrophysical measurements, thin section and scanning electronic microscope (SEM), high-pressure mercury intrusion (HPMI), and nuclear magnetic resonance (NMR), were performed to characterize the microscopic pore structure of tight sandstone in the Huagang Formation (E3h), Xihu Sag, East China Sea Basin, China. Specifically, NMR was used to investigate the dynamic variation of fractal dimensions during centrifugation, and the comparison of HPMI and NMR were used to clarify the difference of fractal dimensions. The results showed that there were four types of pores observed in thin section and SEM images: primary intergranular pores, intergranular dissolution pores, intragranular dissolution pores, and micropores within clay aggregates. The geometric shape and pore size of different pore types showed huge differences, indicating the formation of complex and diverse pore structures in the E3h formation. The flow capability of the reservoir was dominated by large pores, while the storage capacity was determined by small pores. The dynamic variation of fractal dimensions calculated by NMR data showed the water residing in the pore structure with low fractal dimensions was removed preferentially, and the pore structure of the resided water was always more complicated than the pore structure of removed water, which indicated the flow capability of the reservoir was affected by the complexity of the pore structure. Based on the comparison of the fractal dimension data from HPMI and NMR, it was found that the variation trends of the fractal dimensions were consistent, as the radius of the pore throat increased, the fractal dimensions increased, and the pore structure became more complicated. Both fractal dimensions of macropores (Dmac) and movable-fluid pores (Dmov) can reflect the flow capability of reservoir effectively, but the correlations between fractal dimensions from HMPI and NMR were poor, which could be due to the different working mechanism in these methods.

FRACTAL DIMENSION VARIATION OF CONTINUOUS FUNCTIONS UNDER CERTAIN OPERATIONS

In this paper, we make research on the fractal structure of the space of continuous functions and explore how the fractal dimension of continuous functions under certain operations changes. We prove that any nonzero real power and the logarithm of a positive continuous function can keep the fractal dimensions of bi-Lipschitz invariance closed. For continuous functions having finite zero points, the relationship between its global behavior and the local behavior of its square on zero points has been given. Further, we discuss the fractal dimension of the product of continuous functions and provide the product decomposition of a continuous function in terms of the lower and upper box dimensions. Some special properties of the space of one-dimensional continuous functions have also been shown.

Research on macroscopic mechanical properties and microscopic evolution characteristic of sandstone in thermal environment

To study macro and micro characteristics of rock and their correlation in high temperature environment, uniaxial compression experiment is conducted to analyze macroscopic mechanical properties of sandstone samples after different temperature treatment, and AE source location technology is used to study corresponding microscopic evolution characteristics. Then, the correlation between macroscopic mechanical properties and microscopic evolution characteristics is discussed qualitatively, and their quantitative relationship is revealed further. The research results show that the brittleness of the sandstone sample experiences transform process of “first increasing and then decreasing” as treatment temperature increases from room temperature to 1000 °C and turning point is 400 °C, according to the change trend in peak stress, peak strain and fracture characteristics. Based on the variations of fractal dimension and microfracture type of AE events, it is evident that the microcrack system of the sandstone sample experiences the change process of “first simple and then complex” with an increase of treatment temperature. The reason for these variations is that treatment temperature can change microcrack system through the game process of thermal healing effect and thermal fracture effect, resulting in the difference of macroscopic mechanical properties. As treatment temperature increases from room temperature to 400 °C, thermal healing effect is always stronger than thermal fracture effect and the difference enlarges continuously. The microcrack system shrinks gradually, and the sandstone sample presents less properties of the microcrack system and its brittleness increase. As treatment temperature is above 400 °C, a continued increase in treatment temperature makes the difference between the two effects start to decreases until thermal fracture effect exceeds thermal healing effect, then their difference enlarges again. The microcrack system develops gradually, and the sandstone sample presents more properties of the microcrack system and its brittleness decrease. Further, according to data fitting analysis, it can be considered that thermal healing effect presents a linear correlation with treatment temperature during the game process and thermal fracture effect shows an nonlinear correlation with treatment temperature. The difference in the increase rate of these two effects results in nonlinear relationship between the microcrack system and treatment temperature. Then, the linear relationship between the microcrack system and macroscopic mechanical properties is reveal and quantitative relationship model between macroscopic mechanical properties and treatment temperature is established, which has good agreement with the experimental data. This research is helpful to know the correlation of macro and micro variations of the sandstone sample in high temperature environment. It possesses important guiding significance and reference value for ensuring stability of the surrounding rock in deep underground engineering.

Pore‐Fracture Distribution Heterogeneity of Shale Reservoirs Determined by using HPMI and LPN2 GA Tests

AbstractThe compressibility of shale matrix reflects the effects of reservoir lithology, material composition, pore structure and tectonic deformation. It is important to understand the factors that influence shale matrix compressibility (SMC) and their effects on pore size distribution (PSD) heterogeneity in order to evaluate the properties of unconventional reservoirs. In this study, the volumes of pores whose diameters were in the range 6–100 nm were corrected for SMC for 17 shale samples from basins in China using high‐pressure mercury intrusion and low‐temperature nitrogen gas adsorption analyses, in order to investigate the factors influencing the SMC values. In addition, the variations in fractal dimensions before and after pore volume correction were determined, using single and multifractal models to explain the effects of SMC on PSD heterogeneity. In this process, the applicability of each fractal model for characterizing PSD heterogeneity was determined using statistical analyses. The Menger and Sierpinski single fractal models, the thermodynamic fractal model and a multi‐fractal model were all used in this study. The results showed the following. The matrix compression restricts the segmentation of the fractal dimension curves for the single fractal Menger and Sierpinski models, which leads to a uniformity of PSD heterogeneity for different pore diameters. However, matrix compression has only a weak influence on the results calculated using a thermodynamic model. The SMC clearly affects the multifractal value variations, showing that the fractal dimension values of shale samples under matrix compression are small. Overall PSD heterogeneity becomes small for pores with diameters below 100 nm and the SMC primarily affects the PSD heterogeneity of higher pore volume areas. The comparison of fractal curves before and after correction and the variance analysis indicate that the thermodynamic model is applicable to quantitatively characterize PSD heterogeneity of shale collected from this sampling area. The results show that PSD heterogeneity increases gradually as micro‐pore volumes increase.

Study on Fractal Characteristics of Evolution of Mining-Induced Fissures in Karst Landform

The karst landscape is widespread in the southern region of China. As a result of underground mining activities, the original stress equilibrium is disrupted, causing the redistribution of stress in the overlying rock layer, inducing the longitudinal fracture of mining to expand and penetrate upwards, resulting in the rupture and destabilization of the karst cave roof, thus triggering a series of engineering problems such as karst cave collapse, landslide, the discontinuous deformation of the ground surface, and soil erosion. In order to study the evolutionary characteristics of buried rock fissures in shallow coal seam mining under the karst landform, taking the shallow coal seam with the typical karst cave development landform in Guizhou as the engineering background, based on the similarity simulation experiment and fractal theory, the evolution law of buried rock fissures and network fractal characteristics under the disturbance of the karst landform mining are analyzed. The research shows that the mining-induced fracture reaches the maximum development height of 61 m on the left side of the cave, and the two sides of the cave produce uncoordinated deformation. The separation fracture below the cave is relatively developed, and the overall distribution pattern of the cave rock fracture network presents a “ladder” shape. The correlation coefficient of the fractal dimension of the rock fractures under different advancing distances is more than 0.90, and the rock fracture network under the karst landform has high self-similarity. The variation of fractal dimension with the advancing degree of the working face can be divided into four stages. The first and second stages show an exponential growth trend, and the third and fourth stages show linear changes with slopes of 0.0007 and 0.0014, respectively. The fluctuation of the fractal dimension is small. The periodic weighting of the upper roof in the cave-affected zone is frequent, the fragmentation of the fractured rock mass becomes larger, and the fractures of the upper rock mass are relatively developed. The research results can provide a reference for the study on the evolution law of mining-induced rock fissures under similar karst landforms.

Fractal characteristics of fractures in crystalline basement rocks: Insights from depth-dependent correlation analyses to 5 km depth

The scaling laws describing the spatial arrangement of fractures along six deep boreholes penetrating the crystalline rocks in the Rhine Graben were derived using a correlation analysis. Five of the wells, two to 5 km depth, were located at the Soultz geothermal site and one well to 5 km depth was located at Basel, some 150 km from Soultz. Five datasets were derived from borehole imaging logs, whilst one stemmed from the analysis of 810 m of continuous core at Soultz. The two differed inasmuch as the core dataset included essentially all fractures, whereas the image log dataset had few fractures narrower than 1–3 mm. The results of the analysis for all image datasets showed that the spatial arrangement of fractures followed fractal behavior at all scales from meters to several hundred meters, the largest scale amenable to assessment, and that the fractal dimensions were confined to the narrow range 0.85–0.9. However, the core dataset showed significant deviation from fractal behavior, the best-fit fractal dimension of 0.8 being somewhat lower than values obtained from imaging logs in neighboring wells. Eliminating fractures with apertures less than 1 mm from the core dataset to improve comparability led to even lower fractal dimension estimates, indicating the discrepancy was not due to imaging log resolution. Analysis of successive depth sections of the core log suggested the discrepancy was due to the presence of a localized zone between 1750 and 2070 m where the fractal organization is disturbed or takes a lower dimension than elsewhere. Aside from this zone, no systematic variation of fractal dimension with depth was observed in any dataset, implying that a single exponent together with intensity adequately describes the arrangement of fractures along the entire length of the boreholes. The results are relevant to the parameterization of DFN models of deep rock masses. • The fracture population derived from image logs followed fractal statistic with very similiar fractal dimensions. • The core-derived fracture dataset showed significant deviation from fractal behavior. • No systematic variations of fractal dimension with depth were observed in any of the datasets.

DECODING THE CORRELATION AMONG LEG MUSCLE AND BRAIN ACTIVATIONS IN DIFFERENT BODY MOVEMENTS

We investigated the correlation among brain and leg muscle activations by analyzing Electroencephalogram (EEG) and Electromyogram (EMG) signals in different conditions. Twelve subjects performed four tasks, including (1) quarter turns, (2) U-turns, (3) bypass obstacles, and (4) repeating quarter turns and U-turns two times. Then, we quantified the alterations of the complexity of these signals by computing the fractal dimension and sample entropy. The results showed that EEG and EMG signals in the case of the first task are more complex than the second task, in which they are more complex than the third task. Furthermore, the brain and muscle signals show the least complexity in the case of the fourth task. Moreover, we found strong correlations in the variations of fractal dimension ([Formula: see text]) and sample entropy ([Formula: see text]) between EEG and EMG signals in various tasks. Therefore, brain and muscle activations are strongly correlated in different tasks. Similar analyses can be conducted in the case of other organs to decode their correlations.

Quantitative evaluation of impact cracks near the borehole based on 2D image analysis and fractal theory

Impact cracks near the borehole are beneficial to geothermal energy development in many aspects, such as hydraulic fracturing, perforation and so on. However, the quantitative evaluation of the impact cracks near the borehole has not been studied. In the present paper, the fractal dimension of the impact crack has been calculated by the box-counting method to capture the complexity and irregularity of the impact crack pattern. The results show that the impact crack has fractal characteristics, which proves that it is feasible to use the fractal dimension method to quantitatively evaluate the impact cracks. The fractal dimension of the impact cracks is between 1.27 and 1.64. Moreover, the variation of fractal dimension with the increase of the dimensionless distance shows a remarkable regionalization. Simultaneously, the impact crack patterns are classified and divided into seven kinds according to their shape characteristics, and T-shaped, Y-shaped, and wave-shaped cracks are the main kinds of impact cracks. Finally, the surface crack porosity parameter is introduced to characterize the actual content of impact cracks, and the maximum surface crack porosity is 15.9%. Through the comprehensive evaluation of fractal dimension and surface crack porosity, it can be estimated that the influence of percussion drilling on rocks near the borehole ranges from about 1.5 times to 1.7 times the diameter of the borehole. The concept of fractal dimension provides a new perspective for a quantitative understanding of impact cracks, and the presented experimental results can be a useful reference for building numerical models.

BrainFD: Measuring the Intracranial Brain Volume With Fractal Dimension.

A few methods and tools are available for the quantitative measurement of the brain volume targeting mainly brain volume loss. However, several factors, such as the clinical conditions, the time of the day, the type of MRI machine, the brain volume artifacts, the pseudoatrophy, and the variations among the protocols, produce extreme variations leading to misdiagnosis of brain atrophy. While brain white matter loss is a characteristic lesion during neurodegeneration, the main objective of this study was to create a computational tool for high precision measuring structural brain changes using the fractal dimension (FD) definition. The validation of the BrainFD software is based on T1-weighted MRI images from the Open Access Series of Imaging Studies (OASIS)-3 brain database, where each participant has multiple MRI scan sessions. The software is based on the Python and JAVA programming languages with the main functionality of the FD calculation using the box-counting algorithm, for different subjects on the same brain regions, with high accuracy and resolution, offering the ability to compare brain data regions from different subjects and on multiple sessions, creating different imaging profiles based on the Clinical Dementia Rating (CDR) scores of the participants. Two experiments were executed. The first was a cross-sectional study where the data were separated into two CDR classes. In the second experiment, a model on multiple heterogeneous data was trained, and the FD calculation for each participant of the OASIS-3 database through multiple sessions was evaluated. The results suggest that the FD variation efficiently describes the structural complexity of the brain and the related cognitive decline. Additionally, the FD efficiently discriminates the two classes achieving 100% accuracy. It is shown that this classification outperforms the currently existing methods in terms of accuracy and the size of the dataset. Therefore, the FD calculation for identifying intracranial brain volume loss could be applied as a potential low-cost personalized imaging biomarker. Furthermore, the possibilities measuring different brain areas and subregions could give robust evidence of the slightest variations to imaging data obtained from repetitive measurements to Physicians and Radiologists.

Surface Evaluation of Orthodontic Wires Using Texture and Fractal Dimension Analysis.

Mechanical properties of orthodontic wires can have a very significant impact both on the resistance of the entire appliance to the oral cavity conditions and directly on the effectiveness of the therapy. Striving to achieve repeatability of mechanical characteristics of orthodontic wires of a given type should be an obligatory condition in their production. To achieve it, these components should be thoroughly analyzed using various mechanical tests. Twenty-four steel and nickel-titanium orthodontic wires from four different manufacturers were examined. Each wire was subjected to fractal dimension analysis and texture analysis. The two sides of each wire were compared against each other, as well as in terms of variation in the surface area for each wire type made by different manufacturers. Most wires showed significant variation in fractal dimension and texture, both when comparing two sides of the same wire and between individual wires of a given type made by a single manufacturer. When conducting research and clinically using orthodontic wires made of Ni-Ti alloys and stainless steel, it should be assumed that the surface of orthodontic wires shows a significant degree of variation, and wires of the same type from the same manufacturer may differ significantly in this respect.

Fracture Performance of a Large-Stone Asphalt Mixture Based on a Monotonic Tensile Overlay Test

A large-stone asphalt mixture (LSAM) is usually used as the base course of asphalt pavement to prevent the generation of cracks. However, there are few studies on the fracture performance and crack resistance of LSAMs. Under the monotonic tensile loading mode, the overlay test (OT) was explored to investigate the influence of different test factors on the cracking resistance and fracture performance. The study results indicated that a change in temperature and aging results in a variation in fractal dimension, the fracture energy of crack growth is higher than that of crack initiation, and fracture energy is increased to a certain extent by decreased temperature, an increased loading rate or increased aging. Finally, a constitutive model is established based on the disturbed state concept (DSC), and the proposed constitutive model is consistent with the test results.

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

Objective: 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. Materials: 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 (n = 50) measuring 17.3 ± 1.8 pix (1.9 ± 0.5 μm) and chromatin network distance (n = 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. Methods: 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). Results: 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 <1% (0.1, 0.4, and 0.3%, respectively), with no significant difference between the position and rotation bias (p = 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.

Pores Evolution of Soft Clay under Loading/Unloading Process

Loading/unloading tests and field emission scanning electron microscope (FESEM) tests were performed on undisturbed soft clay specimens to study the pore evolution under the loading/unloading process. The results showed that small pores (&lt;0.2 μm) had intrinsic characteristics, and the distribution and the fragmentation fractal dimension of small pores were basically unchanged with pressure, while large pores (&gt;0.6 μm) changed greatly under loading/unloading. The pore-size distribution was mainly influenced by large pores. The microstructure of soft clay before unloading has an influence on the change of the swelling index (Cs) and the pores evolution under unloading. Cs increased as the surface fractal dimension of the pores and the area of large pores decreased, and the fragmentation fractal dimension of the pores increased under the loading process. The variations in fractal dimensions and large pore area increased under unloading. Moreover, the compression index (Cc) changed nonlinearly with the pore evolution under loading. Below 100 kPa, Cc increased slightly with a small increase of the fractal dimensions and large pores area under loading. From 100 kPa to 400 kPa, Cc increased to a peak value of 0.484, and the fractal dimensions and large pore area were the greatest under loading. Above 400 kPa, all of them changed slowly. Based on the evolution of pore fractal characteristics, the loading/unloading process could be divided into three stages: the natural structural stage, the structural adjustment stage, and the new equilibrium stage, which was important to study the loading/unloading properties of soft clay.

FRACTAL-BASED ANALYSIS OF THE RELATION BETWEEN TOOL WEAR AND MACHINE VIBRATION IN MILLING OPERATION

Tool wear is one of the unwanted phenomena in machining operations where tool has direct contact with the workpiece. Tool wear is an important issue in milling operation that is caused due to different parameters such as machine vibration. Tool wear shows complex structure, and machine vibration is a chaotic signal that also is complex. In this research, we analyze the correlation between tool wear and machine vibration using fractal theory. We run the experiments in which machining parameters, namely depth of cut, feed rate and spindle speed change, and accordingly analyze the variations of fractal dimension of tool wear versus the fractal dimension of machine vibration signal. Based on the obtained results, variations of complexity of tool wear are reversely correlated with the variations of complexity of vibration signal. Fractal analysis could potentially be applied to other machining operations in order to investigate the relation between tool wear and machine vibration.