Fractal analysis of images in medicine and morphology: basic principles and methodologies

Background. Fractal analysis is an informative and objective method of mathematical analysis that can complement existing methods of morphometry and provides a comprehensive quantitative assessment of the spatial configuration of irregular anatomical structures. Objective: a comparative analysis of fractal analysis methods used for morphometry in biomedical research. Methods. A comprehensive analysis of morphological studies, based on fractal analysis. Results. Different types of medical images with different preprocessing algorithms can be used for fractal analysis. The parameter determined by fractal analysis is the fractal dimension, which is a measure of the complexity of the spatial configuration and the degree of filling of space with a certain geometric object. The most known methods of fractal analysis are the following: box counting, caliper, pixel dilation, "mass-radius", cumulative intersection, grid intercept. The box counting method and its modifications is the most commonly used method due to the simplicity and versatility. Different methods of fractal analysis have a similar principle: fractal measures (different geometric figures) of a certain size completely cover the structure in the image, size of fractal measure is iteratively changed, and the minimum number of fractal measures covering the structure is calculated. Methods of fractal analysis differ in the type of fractal measure, which can be a linear segment, a square of a fractal grid, a cube, a circle, a sphere etc. Conclusion. The choice of the method of fractal analysis and image preprocessing method depends on the studied structure, features of its spatial configuration, the type of image used for the analysis, and the aim of the study.

Multifractal analysis of ultrasonically machined surfaces of cylindrical quartz crystals: the effect of the abrasive grits

Since synthetic quartz is essential to produce 3-D resonators for numerous applications in precision electronics, in this work the surface topography of cylindrical quartz bars is investigated using the multifractal technique. The cylindrical bars were manufactured with ultrasonic machining using with five SiC grits ranging from 6 to 50 µm. The machined surfaces were initially characterized by contact profilometry and scanning electron microscopy (SEM). The multifractality of the machined surfaces was scrutinized using a box-counting method applied to the images obtained with 500X magnification. The multifractal spectrum indicated that the fractal dimension f(α) and the width of the fractal spectrum Δα are dependent on the grit size, but this dependence is not monotonic. The lowest (negative) value for Δf(α) was found for 25 µm grits indicating that for these grits the lower frequency events (grooves with tens µm width occurring along the USM direction) controls the surface topography much more than high frequency events related to brittle microcracking. The abrasive wear due to the continuous slurry recycling in lateral tool-workpiece interfaces contributed to smooth the groove texture as well as the sharpness of microscopic indentations, which remained observed on the surfaces machined with 50 µm grits. The opposite paths observed for the arithmetical mean deviation of the measured profile (Ra) and Δf(α) parameters with the cutting rate measured for each grit size were valuable to differentiate flat-rough and unlevelled-rough topographies in quartz bars.

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.

Evaluation of the Trabecular Structure of Mandibular Condyles in Children Using Fractal Analysis

Objective: To evaluate the trabecular internal structure of the mandibular condyle with fractal analysis on panoramic radiography in children. Study Design: 159 panoramic radiographs were separated into 8 groups according to age and gender. The radiographs were standardized as 8-bit images. Regions of interest, located on both mandibular condyles, were selected as 64x64 pixel squares. Image J v1.50i software was used to obtain the fractal dimension (FD) values by the box-counting method. Results: The data obtained from the right and left condyles were analyzed in terms of gender and age groups. No statistically significant difference was observed between the genders in respect of the mean FD values for both condyles (p>0.05). Mean, standard deviations and the 95% confidence intervals for the FD values of the left and right condyles were obtained according to age. A statistically significant difference was observed in the mean FD values for both left (p= 0.019) and right (p= 0.000) condyles when all groups were compared and no statistically significant difference was found between all groups except the 6-year-old group for both condyles. In both condyles, the significantly lowest mean FD values were determined in the 6 years age group. Conclusions: The FD values of the mandibular condyle trabecular structure changed with age. It will be possible to evaluate these changes from panoramic radiographs by making calculations using the fractal analysis method.

Multilayered Complexity Analysis in Architectural Design: Two Measurement Methods Evaluating Self-Similarity and Complexity

This article contributes to clarifying the questions of whether and how fractal geometry, i.e., some of its main properties, are suitable to characterize architectural designs. This is done in reference to complexity-related aesthetic qualities in architecture, taking advantage of the measurability of one of them; the fractal dimension. Research in this area so far, has focused on 2-dimensional elevation plans. The authors present several methods to be used on a variety of source formats, among them a recent method to analyze pictures taken from buildings, i.e., 2.5-dimensional representations, to discuss the potential that lies within their combination. Color analysis methods will provide further information on the significance of a multilayered production and observation of results in this realm. In this publication results from the box-counting method are combined with a coordinate-based method for analyzing redundancy of proportions and their interrelations as well as the potential to include further layers of comparison are discussed. It presents a new area of box-counting implementation, a methodologically redesigned gradient analysis and its new algorithm as well as the combination of both. This research shows that in future systems it will be crucial to integrate several strategies to measure balanced aesthetic complexity in architecture.

Between 2D and 3D: Studying Structural Complexity of Urban Fabric Using Voxels and LiDAR-Derived DSMs

Cities are complex systems and their physical forms are the manifestation of cultural, social and economic processes shaped by the geometry of natural and man-made elements. Digital Surface Models (DSM) using LiDAR provide an efficient volumetric transformation of urban fabric including all built and natural elements which allows the study of urban complexity through the lens of fractal dimension (D). Founded on the “box-counting” method, we reveal a voxelization technique developed in GIS (Geographic Information System) to estimate D values of ten DSM samples across central Melbourne. Estimated D values of surface models (between 2 and 3) provide a measure to interpret the structural complexity of different urban characters defined by the pattern of developments and densities. The correlations between D values with other DSM properties such as elevation, volume, solar radiation and surface roughness, showed a strong relationship between DSM volume and mean elevation. Lower strength correlations were recorded with solar radiation and surface roughness. The proposed method provides opportunities for fractal research to study pressing issues in complex urban environments such as declining physical fitness, mental health and urban biodiversity.

Influence of the Fractal Geometry on the Mechanical Resistance of Cantilever Beams Designed through Topology Optimization

In this work, the complex geometry of beams obtained from topology optimization is characterized through the fractal dimension (FD). The fractal dimension is employed as an efficiency measure of the mass distribution in the beams, that is, the capacity of the optimized solutions to be efficiently distributed in the design space. Furthermore, the possible relationships between the fractal dimension and beams’ mechanical properties are explored. First, a set of theoretical beams are studied based on their well-known fractal dimension. A 3D fractal called Menger sponge is reproduced on a Michell’s beam (cantilever with a single force applied at the end). The programming codes that generate those beams are created in Matlab software, as are the algorithms for estimating the fractal dimension (box-counting method). Subsequently, identical beams are modelled in the software Inspire in order to apply the topology optimization and determine the mechanical parameters from the static analysis. Results indicate that the fractal dimension is affected by the design geometry and proposed optimized solutions. In addition, several relationships among fractal dimension and some mechanical resistance parameters could be established. The obtained relations depended on the objectives that were initially defined in the topology optimization.

Application of An Exponential Mathematical Law of Cardiac Dynamics In 16 Hours

Introduction and objectives: nonlinear dynamics and fractal geometry have allowed the advent of an exponential mathematical law applicable to diagnose cardiac dynamics in 21 hours, however, it would be beneficial to reduce the time required to diagnose cardiac dynamics with this method in critical scenarios, in order to detect earlier complications that may require medical attention. The objective of this research is to confirm the clinical applicability of the mathematical law in 16 hours, with a comparative study against the Gold Standard. Methods: There were taken 450 electrocardiographic records of healthy patients and with cardiac diseases. A physical-mathematical diagnosis was applied to study cardiac dynamics, which consists of generating cardiac chaotic attractors based on the sequence of heart rate values during 16 hours, which were then measured with two overlapping grids according to the Box-Counting method to quantify the spatial occupation and the fractal dimension of each cardiac dynamic, with its respective statistical validation. Results: The occupation spaces of normal dynamics calculated in 16 hours were compatible with previous parameters established, evidencing the precision of the methodology to differentiate normality from abnormality. Sensitivity and specificity values of 100% were found, as well as a Kappa coefficient of 1. Conclusions: it was possible to establish differences between cardiac dynamics for 16 hours, suggesting that this method could be clinically applicable to analyze and diagnose cardiac dynamics in real time.

Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning

Atmospheric turbulence studies indicate the presence of self-similar scaling structures over a range of scales from the inertial outer scale to the dissipative inner scale. A measure of this self-similar structure has been obtained by computing the fractal dimension of images visualizing the turbulence using the widely used box-counting method. If applied blindly, the box-counting method can lead to misleading results in which the edges of the scaling range, corresponding to the upper and lower length scales referred to above are incorporated in an incorrect way. Furthermore, certain structures arising in turbulent flows that are not self-similar can deliver spurious contributions to the box-counting dimension. An appropriately trained Convolutional Neural Network can take account of both the above features in an appropriate way, using as inputs more detailed information than just the number of boxes covering the putative fractal set. To give a particular example, how the shape of clusters of covering boxes covering the object changes with box size could be analyzed. We will create a data set of decaying isotropic turbulence scenarios for atmospheric turbulence using Large-Eddy Simulations (LES) and analyze characteristic structures arising from these. These could include contours of velocity magnitude, as well as of levels of a passive scalar introduced into the simulated flows. We will then identify features of the structures that can be used to train the networks to obtain the most appropriate fractal dimension describing the scaling range, even when this range is of limited extent, down to a minimum of one order of magnitude.

Experimental Study of Crushed Granular Materials by the Notion of Fractal Dimension in 2D and 3D.

The micro-texture of the aggregates of a pavement layer has a direct influence on their resistance. Whatever the position of these aggregates in a pavement structure, they must withstand, during construction or during life, the stresses of attrition and impact. In this study, a series of mechanical tests (Proctor, Los-Angeles and Micro-Deval) are carried out on grains of local materials (limestone and shale), the degree of crushing of the grains has been quantified using the concept of fractal dimension. The fractal dimension was calculated for the different grains constituting the samples before and after each test, with the use of two two-dimensional 2D methods (Masses Method at the scale of a sample and the Box Counting Method at the scale of a grain) and a three-dimensional 3D method (Blanket on a grain scale) which is based on the use of the difference between erosion and dilation. We seek to determine from these methods the correlation between the two fractal dimensions, namely 2D and 3D and study the influence of different parameters on the mechanical characteristics of the materials chosen: the shape and size of the grains, the presence or absence of water, the stress intensity as well as the nature of the material. The results obtained show that the three-dimensional method has a positive effect on the description of the 3D microstructure of the surface of the grains subjected to the various mechanical tests.