Differential Box-counting Method Research Articles

Investigating the fractal dimension of flame fronts of the biodiesel-diesel blends combustion in atmospheric conditions and engine cylinders: An experimental study

The fractal dimension of the flame front is an important parameter that quantifies the degree of flame folding and twisting as well as the instantaneous mass burning rate. Due to the geometric nature of flame surface folding, the fractal approach is also considered one of the most appropriate methods in power law modelling. Although biodiesel fuel is an important renewable fuel, studies on the fractal dimension of the flame during the combustion of biodiesel and its blends are limited. To address this research gap, combustion assays of biodiesel-diesel mixtures were systematically conducted within a coaxial diffusion burner and a diesel engine. High-resolution images of the evolving combustion flames were captured using a high-speed video camera. Digital image processing was applied to facilitate the extraction of key features from these images. Through the differential box counting (DBC) method, the fractal dimensions of the flame fronts were derived. A comprehensive analysis was carried out, revealing the influence of various parameters on the flame's fractal dimensions, including the biodiesel-diesel blend proportion, blend temperature, engine operational speed, and air flow rate. The results highlight that the fractal dimension undergoes a non-monotonic variation as biodiesel content increases. Specifically, the combustion of B25 blend exhibited the minimum fractal dimension, while the B100 blend showed the maximum. Further, in engine environments, peak fractal dimensions were observed to occur earlier at higher speeds (2200 r/min) with an increase in the biodiesel ratio. These insights not only bridge a critical knowledge gap in power-law modeling for biodiesel combustion but also offer a quantitative understanding of how blend ratios and engine conditions affect flame fractal properties.

A new quasigroup isomorphism invariant arising from fractal image patterns

The analysis and recognition of fractal image patterns derived from Cayley tables has turned out to play a relevant role for distributing distinct types of algebraic and combinatorial structures into isomorphism classes. In this regard, Dimitrova and Markovski described in 2007 a graphical representation of quasigroups by means of fractal image patterns. It is based on the construction of pseudo-random sequences arising from a given quasigroup. In particular, isomorphic quasigroups give rise to the same fractal image pattern, up to permutation of underlying colors. This possible difference may be avoided by homogenizing the standard sets related to these patterns. Based on the differential box-counting method, the mean fractal dimension of homogenized standard sets constitutes a quasigroup isomorphism invariant which is analyzed in this paper in order to distribute quasigroups of the same order into isomorphism classes.

Fractal Analysis of Tunnel Structural Damage Caused by High-Temperature and Explosion Impact

The tunnel is one of the most important components in modern underground engineering. Due to long and narrow shape constraints, it very easily results in large-scale fire and explosion when deflagration is caused by the accidents of vehicles that transport dangerous goods in the tunnel. Previously, the studies on the damage to tunnel lining caused by high-temperature impacts in these kinds of disasters were often limited to a discussion of only one influencing factor, either fire or explosion, but they rarely considered the two factors simultaneously. In this work, the damage properties of full-size tunnel lining induced by high temperature and impact were evaluated, and the concrete samples from the whole lining arch were selected for CT scanning. The improved differential box-counting method was used for the fractal analysis of the CT images to obtain the damage-distribution properties of the tunnel lining structure under the two coupled influencing factors: the high temperature caused by fire, and the impact caused by deflagration.

Problems of the Grid Size Selection in Differential Box-Counting (DBC) Methods and an Improvement Strategy.

The differential box-counting (DBC) method is useful for determining the fractal dimension of grayscale images. It is simple to learn and implement and has been extensively utilized. However, this approach has several problems, such as over- or undercounting the number of boxes due to inappropriate parameter choices, limiting the calculation accuracy. Many studies have been conducted to increase the algorithm’s computational accuracy by improving the calculating parameters of the differential box-counting method. The grid size is a crucial parameter for the DBC method. Generally, there are two typical ways for selecting the grid size in relevant studies: consecutive integer and divisors of image size. However, both methods for grid size selection are problematic. The consecutive integer method cannot partition the image entirely and will result in the undercounting of boxes; the divisors of image size can partition the image completely. However, this method uses fewer grid sizes to compute fractal dimensions and has a relatively huge distance error (DE). To address the shortcomings of the above-mentioned two approaches, this research presents an improved grid size selection strategy. The improved method enhances computational accuracy by computing the discarded image edge areas in the consecutive integer method, allowing the original image information to be used as thoroughly as the divisor strategy. Based on fractional Brownian motion (FBM), Brodatz, and Aerials image sets, the accuracy of the three grid size selection techniques (consecutive integer method, divisors of image size method, and the improved algorithm) to compute the fractal dimension is then compared. The results reveal that, compared to the two prior techniques, the revised algorithm described in this study minimizes the distance error and increases the accuracy of the fractal dimension computation.

Heat transfer analysis of irradiation-induced gas bubble in tungsten from a fractal dimension perspective

The mechanism of heat transfer profoundly depends on the microscopic topological structure. In reality, the intrinsic nature of (un)-irradiated metals possesses the fractal characteristic that repeats a pattern at different scales and sizes regardless of the complexity. Herein we combine the conventional thermal conduction model with the analytical fractal solution in order to understand the effect of fractal parameter on estimating the proportion of each phase in dual- and triple-phase microstructures, the role of gaseous pressure in conducting heat, and the variation of effective thermal conductivity of the bubble-containing tungsten. The fractal dimension obtained from the differential box-counting (DBC) method is quantitatively compared with that from the classical box-counting (BC) method, and the relevance between two approaches in assessing the fractal dimension is analyzed and discussed. Importantly, the basic porosity-fractal dimension formula is qualitatively modified by introducing a regulatory factor to identify the microstructure with striking characteristic. This study provides evidence from the analytical fractal model that the effective thermal conductivity changes with grain size and bubble density. Meanwhile, the predicted results from the proposed model match well with the previous experimental data and the corresponding results predicted from the numerical finite element simulation. Focusing on the heat performance of (un)-irradiated material, the specific procedure allows us to extract the valuable data effectively and efficiently.

The Fractal Dimension Suggests Two Chromatin Configurations in Small Cell Neuroendocrine Lung Cancer and Is an Independent Unfavorable Prognostic Factor for Overall Survival.

Experimental studies have shown that in small cell neuroendocrine lung carcinomas (SCLC) global opening of the chromatin structure is associated with a higher transcription activity and increase of tumor aggressiveness and metastasis. The study of the fractal characteristics (FD) of nuclear chromatin has been widely used to describe the cell nuclear texture and its changes correspond to changes in nuclear metabolic and transcription activity. Hence, we investigated whether the nuclear fractal dimension could be a prognostic factor in SCLC. Hematoxylin-eosin stained brush cytology slides from 49 patients with SCLC were retrieved from our files. The chromatin (FD) was calculated in digitalized and interactively segmented nuclei using a differential box-counting method. The 3,575 nuclei studied showed a bimodal distribution (peaks at FD1 = 2.115 and FD2 = 2.180). The 75 percentile of the FD was an independent unfavorable prognostic factor for overall survival when tested together with ECOG (Eastern Cooperative Oncology Group) performance status, tumor extension, and therapy in a multivariate Cox regression. Our study corroborates the concept of two main chromatin configurations in small cell neuroendocrine carcinomas and that globally more open chromatin indicates a higher risk of metastasis and therefore a shorter survival of the patient.

The microscopic characteristics of particle matter and image algorithm based on fractal theory

The effects of ash and sulfur content on the morphology of particulate matter (PM) in diesel particle filter (DPF) were investigated with five different components of lubricants. The aggregate morphology of primary particles in diesel were analyzed using transmission electron microscopy (TEM). The fractal dimensions of carbon particles were calculated by box-counting method (BCM), differential box-counting method (DBC), relative differential box-counting method (RDBC) and MAD-based box counting method (MAD-DBC), and the results were compared. The results showed that the microstructure of PM developed from chain-like structure to agglomerate structure with the increase of sulfur and ash content in lubricating oil. The fractal dimension of carbon particles increased with the increase of sulfur and ash content. The SSE of RDBC fitting results was smaller, and the R-square is larger. MAD-DBC fitting results had stronger anti-noise interference performance.

Improved Image Analysis Method to Evaluate Tracking Property under Successive Flashover Based on Fractal Theory

Successive flashover would result in carbonized tracking on insulator surface and cause deterioration to the insulation. Thus, investigation of the tracking can be beneficial in understanding flashover characteristics during long-term operation. In this paper, DC flashover was operated on the insulator, and the image of tracking after successive discharge were captured. Improved differential box-counting method (IDBM) was applied to analyze these images based on fractal theory. Weighted item was suggested during the counting procedure for rectangle image with margin covered by cut-size box. Fractal dimension of the tracking was calculated according to the suggested method. It is claimed that the suggested method could estimate the discharge propagation property and deterioration characteristics on the insulator surface. Moreover, IDBM showed advantages in image pre-processing and deterioration property revealed compared to traditional box-counting method attributing to the consideration of color depth. This image analysis method shows universality in dealing with tracking image and could provide additional information to flashover voltage. This paper suggested a potential approach for the investigation of discharge mechanism and corresponding deterioration in future research.

An improved differential box counting method to measure fractal dimensions for pavement surface skid resistance evaluation

Fractal dimension had been recognized as a texture indicator to evaluate skid resistance performance of pavement surface. One of the most frequently used methodologies to measure fractal dimension is differential box counting (DBC). This paper proposes an improved DBC (IMDBC) to estimate fractal dimension of three-dimensional (3D) pavement texture image based on grid displacement mechanism. Considering the contact characteristics between pavement surface and tire, the most suitable fractal texture surface for skid resistance is found based on the fractal and stratification method. The IMDBC improves the fitting accuracy by 18.8% (complete texture) and 900% (partial texture) compared to the traditional DBC. The correlation coefficient between the fractal dimension and friction reaches up to 0.6 when vertical height ratio is 0.5 ~ 0.6 h, which means that the proposed fractal dimension indicator of the upper layer texture could be used as a potential parameter to evaluate pavement surface skid resistance performance.

Comparative study on damage process of concrete subjected to uniaxial tensile and compression loads based on CT test and improved differential box counting method

Concrete is prone to be damaged due to external forces. Traditional concrete failure analysis and research mostly focus on the characterization of macro – mechanical properties, which cannot describe the damage of internal structure and the process of concrete failure after being subjected to a load on a mesoscopic level. Thus, a series of CT tests were carried out on the damage process of concrete under tensile and compression loads, and CT images of the whole process of concrete failure were obtained. On this basis, the fractal dimension calculated by the Improved Differential Box Counting (IDBC) method was used to quantitatively study the change of the internal structure of concrete during the loading process. Then the mean values of the CT number of concrete CT images were statistically analyzed. The results revealed that the fractal dimension of different stress stages in each section increased with the increase of scan times under tensile load, while the fractal dimension increasing first, then decreasing and finally increasing with the increase of scan times under compression load. Correspondingly, with the increase of scan times, the mean values of CT number under tensile load showed a trend of decreasing, while a trend of decreasing first, then increasing and finally decreasing of the mean values of CT number was exhibited when the concrete specimen was subjected to compression load, which indicated that the concrete specimen undergone the process of expansion, crack propagation, and failure under tensile load, whereas the concrete experienced the process of compaction, crack propagation, and failure under a compression load. Furthermore, research results have shown that the fractal dimension and mean value of CT number can be used to quantitatively describe the damage process of concrete, whether it is subjected to tensile or compression load.

Determination and evolution of fractal property of n-heptane pool fires caused by depressurization process in an aircraft cargo compartment

Fractality is one of the inherent characteristics to present the irregularity and self-similarity of an object. In this study, the fractal theory was applied to the fire scene under decreasing pressure in an aircraft cargo compartment, considering that the compartment is usually inaccessible in flight and flame images are a kind of convenient monitoring information compared to other detection methods. A series of n-heptane pool fires were performed under decreasing pressure with four depressurized rates. The flame image was captured by a video camera with fixed aperture and exposure time. The fractal dimension (FD) of flame images was computed via two approaches, i.e., the box-counting (BC) and differential box-counting (DBC) methods. The results show that n-heptane pool fires have a good fractal property for both binary and grayscale images. The FD via BC method shows little sensitivity with decreasing pressure. However, the FD via DBC method presents a significant linear relationship with a slope of 0.0016 in the range from 2.85 to 2.75 during the depressurization. The quantitative correlation was brought into the classical radiation modeling, and the exponential function between FD and MLR of n-heptane was established.

Is SARS CoV-2 a Multifractal?—Unveiling the Fractality and Fractal Structure

A first report of unveiling the fractality and fractal nature of severe acute respiratory syndrome coronavirus (SARS CoV-2) responsible for the pandemic disease widely known as coronavirus disease 2019 (COVID 19) is presented. The fractal analysis of the electron microscopic and atomic force microscopic images of 40 coronaviruses (CoV), by the normal and differential box-counting method, reveals its fractal structure. The generalised dimension indicates the multifractal nature of the CoV. The higher value of fractal dimension and lower value of Hurst exponent (H) suggest higher complexity and greater roughness. The statistical analysis of generalised dimension and H is understood through the notched box plot. The study on CoV clusters also confirms its fractal nature. The scale-invariant value of the box-counting fractal dimension of CoV yields a value of 1.820. The study opens the possibility of exploring the potential of fractal analysis in the medical diagnosis of SARS CoV-2.

Bone microarchitecture characterization based on fractal analysis in spatial frequency domain imaging

ABSTRACTThis paper suggests a new technique for trabecular bone characterization using fractal analysis of X‐Ray and MRI texture images for osteoporosis diagnosis. Osteoporosis is a chronic disease characterized by a decrease in bone density that can lead to fracture and disability. In essence, the proposed fractal model makes use of the differential box‐counting method (DBCM) to estimate the fractal dimension (FD) after an appropriate image preprocessing stage that ensures a robust estimation process. In this study, we showed that within the frequency domain generated through discrete cosine transform (DCT), only a quarter of DCT coefficients are enough to characterize osteoporotic tissues. The algorithmic complexity of the developed approach is of the order of where N stands for the size of the image, which, in turn, likely yields important gain in terms of medication cost. We report a successful separation of healthy and pathological cases in term of both P − value (using statistical Wilcoxon rank sum test) and margin difference. A comparative statistical analysis has been performed using a publicly available database that contains a set of MRI and X‐Ray texture images of both healthy and osteoporotic bone tissues. The statistical results demonstrated the feasibility and accepted performance level of our fractal model‐based diagnosis to discriminate healthy and unhealthy trabecular bone tissues. The developed approach has been implemented on a medical device prototype.

Detection of abnormality in wireless capsule endoscopy images using fractal features

One of the most recent non-invasive technologies to examine the gastrointestinal tract is wireless capsule endoscopy (WCE). As there are thousands of endoscopic images in an 8–15 h long video, an evaluator has to pay constant attention for a relatively long time (60–120 min). Therefore the possibility of the presence of pathological findings in a few images (displayed for evaluation for a few seconds only) brings a significant risk of missing the pathology with all negative consequences for the patient. Hence, manually reviewing a video to identify abnormal images is not only a tedious and time consuming task that overwhelms human attention but also is error prone. In this paper, a method is proposed for the automatic detection of abnormal WCE images. The differential box counting method is used for the extraction of fractal dimension (FD) of WCE images and the random forest based ensemble classifier is used for the identification of abnormal frames. The FD is a well-known technique for extraction of features related to texture, smoothness, and roughness. In this paper, FDs are extracted from pixel-blocks of WCE images and are fed to the classifier for identification of images with abnormalities. To determine a suitable pixel block size for FD feature extraction, various sizes of blocks are considered and are fed into six frequently used classifiers separately, and the block size of 7×7 giving the best performance is empirically determined. Further, the selection of the random forest ensemble classifier is also done using the same empirical study. Performance of the proposed method is evaluated on two datasets containing WCE frames. Results demonstrate that the proposed method outperforms some of the state-of-the-art methods with AUC of 85% and 99% on Dataset-I and Dataset-II respectively.

Quantification of Structural Heterogeneity Using Fractal Analysis of Contrast-Enhanced CT Image to Predict Survival in Gastric Cancer Patients.

Malignant tumor essentially implies structural heterogeneity. Fractal analysis of medical imaging has a potential to quantify this structural heterogeneity in the tumor AIMS: The purpose of this study is to quantify this structural abnormality in the tumor applying fractal analysis to contrast-enhanced computed tomography (CE-CT) image and to evaluate its biomarker value for predicting survival of surgically treated gastric cancer patients. A total of 108 gastric cancer patients (77 men and 31 women; mean age: 69.1years), who received curative surgery without any neoadjuvant therapy, were retrospectively investigated. Portal-phase CE-CT images were analyzed with use of a plug-in tool for ImageJ (NIH, Bethesda, USA), and the fractal dimension (FD) in the tumor was calculated using a differential box-counting method to quantify structural heterogeneity in the tumor. Tumor FD was compared with clinicopathologic features and disease-specific survival (DSS). High FD value of the tumor significantly associated with high T stage and high pathological stage (P = 0.009, 0.007, respectively). In Kaplan-Meier analysis, patients with higher FD tumors (FD > 0.9746) showed a significantly worse DSS (P = 0.009, log rank). Multivariate analysis demonstrated that tumor FD, T stage, and N stage were independent prognostic factors for DSS. In subset analysis of lymph-node positive gastric cancers, only tumor FD was an independent prognostic factor for DSS. CT fractal analysis can be a useful biomarker for gastric cancer patients, reflecting survival and clinicopathologic features.

Quantitative study of meso-damage process on concrete by CT technology and improved differential box counting method

Abstract To study the change of concrete internal structure before and after stress, the meso-mechanical experiment of concrete was conducted using CT technology and a portable dynamic loading device. And CT images of the concrete internal structure change under uniaxial compression condition were obtained. For establishing the relationship between the micro-structure and macro-mechanics of concrete, an improved differential box counting (IDBC) method was proposed to calculate the fractal dimension of the CT images, which could describe the entire damage process of concrete. By comparing the fractal dimension calculated by the two methods, it can be seen that the fitting error of the fractal dimension obtained by the IDBC method is reduced by nearly 50% compared with that obtained by the DBC method. Afterward, the damage variable expression was established on the basis of the fractal dimension. The research results indicated that the damage variable established in this research can describe the damage process of concrete well. That is, the damage variable equaled to 0 during the first scan, and before reaching the peak stress, the damage variable showed a trend of increasing first and then decreasing. After the concrete specimen has completely fractured, the damage variable reached a maximum of one. Furthermore, through extracting the area of CT image cracks region at different stress stages, the relationship between the area of cracks region and the stress was studied. It appeared that the CT technology and fractal theory are promising methods to analyze the evolution of the concrete damage process under static uniaxial compression.

Image texture surface analysis using an improved differential box counting based fractal dimension

Surface texture of an image plays an imperative role in understanding objects such as aggregates, woods, grains, soils, trees, and houses. Fractal dimension (FD) helps to characterise these objects by quantifying their complex texture patterns. Differential box counting (DBC) is one of the popularly used methods to measure FD of a gray-scale image. However, it suffers from several limitations. So, this study introduces three improved DBC methods using three box heights based on eigenvalue, kurtosis, and skewness of an image respectively. These methods also use a new xy-plane shifting mechanism and a modified formula for computing nr. Moreover, weighted least squares (WLS) regression is adopted, where a trapezoidal membership function (TMF) based rule is proposed for assigning weights to each grid size. The image surface exhibits multifractal nature, hence two multifractal analysis methods are also analysed and included in the experiments. All the experiments are performed on three gray-scale image databases viz., the synthesized Fractal Brownian motion (FBM) images, natural texture images of Brodatz database, and images of Outex database that are captured at different rotations. Experimental results show that the proposed methods outperform some of the state-of-the-art methods.

Fractal Dimension Estimation for Color Texture Images

As an important feature of image texture, fractal dimension is widely used to index, segment or classify texture images. Previously, many methods have been proposed to estimate fractal dimension. However, most of them are for binary and gray-scale images. Only a few of them are for color images. In this paper, by extending the differential box-counting approach to five-dimensional Euclidean hyper-space, we present a new robust and simple algorithm to estimate fractal dimension for color texture images. By the differential box-counting method, a real empty box not covering any pixel even with ideal resolution is also counted as a box having pixels, resulting in errors of computations. This problem is eliminated by our method based on fractal Brownian surface model. Our experiments demonstrate that our algorithm is able to capture the complexity of color natures, and outperforms other methods in terms of color texture complexity ranking and discrimination, robustness and computational complexity.

Quantitative texture measurement of gray-scale images: Fractal dimension using an improved differential box counting method

Texture analysis methods have been used in various image processing applications, such as segmentation, classification, and recognition, shape analysis. The performance of computer vision and image processing algorithms depend on visual texture patterns. However, it is difficult to define. Fractal Dimension (FD) helps to measure the level of roughness present in an image. A significant amount of work is available in literature to measure FD. Differential Box Counting (DBC) is one such successful method exploited in image texture analysis study, because it is simple and easy to understand. DBC is modified several times to yield a better FD value. However, most of the state-of-the-art methods suffer from the over-counting and under-counting of boxes, larger box-height, less number of grid sizes, inappropriate line fitting method. This work introduces a gray-level shift-invariant DBC method, which uses a new formula for counting boxes along z-direction to solve over-counting of boxes, a partitioning-shifting-partitioning mechanism to fix under-counting of boxes along xy-direction, a smaller box-height to enhance the FD value and robust least squares regression to estimate a line accurately. All the experiments are performed on synthesized Fractal Brownian Motion image dataset and four real datasets. The obtained results illustrate that the proposed DBC method outperforms state-of-the-art DBC methods.

Differential box counting methods for estimating fractal dimension of gray-scale images: A survey

Abstract Fractal dimension is extensively in use as features in computer vision applications to characterize roughness and self-similarity of objects in an image for many years. These features have been adopted successfully mainly in texture segmentation and classification. Differential box counting method is one of the widely accepted approaches, those exist in literature to estimate fractal dimension of an image. In this work, we comprehensively reviewed the available differential box counting methods. First, the differential box counting method is discussed in detail along with its computer vision applications and drawbacks. Second, various variants of differential box counting method are thoroughly studied and grouped using different parameters of differential box counting method. Third, the synthetic and real-world databases, considered for demonstrating experimental results by the state-of-the-art methods have been presented. Fourth, some of the state-of-the-art methods have been implemented and corresponding results obtained in this study are reported. Fifth, three evaluation metrics have also been reviewed. However, these metrics work only for synthetic fractal Brownian motion images because the theoretical fractal dimension values for these images are known and have been used as a set of ground truths. Finally, we concluded the status of differential box counting methods and explored the possible future directions.