scholarly journals Investigation on Non-Segmentation Based Algorithms for Microvasculature Quantification in OCTA Images

2021 ◽  
Vol 7 (2) ◽  
pp. 247-250
Author(s):  
Amr Abuzer ◽  
Ady Naber ◽  
Simon Hoffmann ◽  
Lucy Kessler ◽  
Ramin Khoramnia ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Imaging Modality ◽  
Ground Truth ◽  
Added Value ◽  
Data Set ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Retinal Microvasculature ◽  
Higher Sensitivity

Abstract Optical Coherence Tomography Angiography (OCTA) is an imaging modality that provides threedimensional information of the retinal microvasculature and therefore promises early diagnosis and sufficient monitoring in ophthalmology. However, there is considerable variability between experts analysing this data. Measures for quantitative assessment of the vasculature need to be developed and established, such as fractal dimension. Fractal dimension can be used to assess the complexity of vessels and has been shown to be independently associated with neovascularization, a symptom of diseases such as diabetic retinopathy. This investigation assessed the performance of three fractal dimension algorithms: Box Counting Dimension (BCD), Information Dimension (ID), and Differential Box Counting (DBC). Two of those, BCD and ID, rely on previous vessel segmentation. Assessment of the added value or disturbance regarding the segmentation step is a second aim of this study. The investigation was performed on a data set composed of 9 in vivo human eyes. Since there is no ground truth available, the performance of the methods in differentiating the Superficial Vascular Complex (SVC) and Deep Vascular Complex (DVC) layers apart and the consistency of measurements of the same layer at different time-points were tested. The performance parameters were the ICC and the Mann- Whitney U tests. The three applied methods were suitable to tell the different layers apart and showed consistent values applied in the same slab. Within the consistency test, the non-segmentation-based method, DBC, was found to be less accurate, expressed in a lower ICC value, compared to its segmentation-based counterparts. This result is thought to be due to the DBC’s higher sensitivity when compared to the other methods. This higher sensitivity might help detect changes in the microvasculature, like neovascularization, but is also more likely prone to noise and artefacts.

Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning

2021 ◽  
Author(s):  
Nicholas Dudu ◽  
Arturo Rodriguez ◽  
Gael Moran ◽  
Jose Terrazas ◽  
Richard Adansi ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Atmospheric Turbulence ◽  
Isotropic Turbulence ◽  
Passive Scalar ◽  
Counting Method ◽  
Data Set ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Box Counting Method

Abstract 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.

ON THE MENDÈS FRANCE FRACTAL DIMENSION OF STRANGE ATTRACTORS: THE HÉNON CASE

2002 ◽  
Vol 12 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
M. PIACQUADIO ◽  
R. HANSEN ◽  
F. PONTA
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Strange Attractors ◽  
Planar Curves ◽  
Box Counting ◽  
Box Counting Dimension

We consider the Hénon attractor ℋ as a curve limit of continuous planar curves H(n), as n → ∞. We describe a set of tools for studying the Hausdorff dimension dim H of a certain family of such curves, and we adapt these tools to the particular case of the Hénon attractor, estimating its Hausdorff dimension dim H (ℋ) to be about 1.258, a number smaller than the usual estimates for the box-counting dimension of the attractor. We interpret this discrepancy.

scholarly journals Sparse Representation of Deformable 3D Organs with Spherical Harmonics and Structured Dictionary

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Dan Wang ◽  
Ahmed H. Tewfik ◽  
Yingchun Zhang ◽  
Yunhe Shen
Keyword(s):  
Sparse Representation ◽  
Spherical Harmonic ◽  
Ex Vivo ◽  
Ground Truth ◽  
Maximum Error ◽  
Training Data ◽  
Data Set ◽  
Low Dimensionality ◽  
Reconstructed Surface

This paper proposed a novel algorithm to sparsely represent a deformable surface (SRDS) with low dimensionality based on spherical harmonic decomposition (SHD) and orthogonal subspace pursuit (OSP). The key idea in SRDS method is to identify the subspaces from a training data set in the transformed spherical harmonic domain and then cluster each deformation into the best-fit subspace for fast and accurate representation. This algorithm is also generalized into applications of organs with both interior and exterior surfaces. To test the feasibility, we first use the computer models to demonstrate that the proposed approach matches the accuracy of complex mathematical modeling techniques and then both ex vivo and in vivo experiments are conducted using 3D magnetic resonance imaging (MRI) scans for verification in practical settings. All results demonstrated that the proposed algorithm features sparse representation of deformable surfaces with low dimensionality and high accuracy. Specifically, the precision evaluated as maximum error distance between the reconstructed surface and the MRI ground truth is better than 3 mm in real MRI experiments.

scholarly journals An Analysis of COVID-19 in Europe Based on Fractal Dimension and Meteorological Data

2021 ◽  
Author(s):  
Cristina Maria Pacurar ◽  
Victor Dan Păcurar ◽  
Marius Paun
Keyword(s):  
Fractal Dimension ◽  
Meteorological Parameters ◽  
Prediction Models ◽  
Meteorological Data ◽  
Global Radiation ◽  
Reproduction Rate ◽  
Box Counting ◽  
Box Counting Dimension ◽  
First Choice ◽  
Research Efficiency

The present paper proposes a fractal analysis of the Covid-19 dynamics in 45 European countries. We introduce a new idea of using the box-counting dimension of the epidemiologic curves as a means of classifying the Covid-19 pandemic in the countries taken into consideration. The classification can be a useful tool in deciding upon the quality and accuracy of the data available. We also investigated the reproduction rate, which proves to have significant fractal features, thus enabling another perspective on this epidemic characteristic. Moreover, we studied the correlation between two meteorological parameters: global radiation and daily mean temperature and two Covid-19 indicators: daily new cases and reproduction rate. The fractal dimension differences between the analysed time series graphs could represent a preliminary analysis criterion, increasing research efficiency. Daily global radiation was found to be stronger linked with Covid-19 new cases than air temperature (with a greater correlation coefficient -0.386, as compared with -0.318), and consequently it is recommended as the first-choice meteorological variable for prediction models.

scholarly journals The linkage between box-counting and geomorphic fractal dimensions in the fractal structure of river networks: the junction angle

2020 ◽  
Vol 51 (6) ◽  
pp. 1397-1408
Author(s):  
Xianmeng Meng ◽  
Pengju Zhang ◽  
Jing Li ◽  
Chuanming Ma ◽  
Dengfeng Liu
Keyword(s):  
Fractal Dimension ◽  
Fractal Structure ◽  
Fractal Dimensions ◽  
River Networks ◽  
Stream Length ◽  
Box Counting ◽  
Linear Relationships ◽  
Box Counting Dimension ◽  
Junction Angle

Abstract In the past, a great deal of research has been conducted to determine the fractal properties of river networks, and there are many kinds of methods calculating their fractal dimensions. In this paper, we compare two most common methods: one is geomorphic fractal dimension obtained from the bifurcation ratio and the stream length ratio, and the other is box-counting method. Firstly, synthetic fractal trees are used to explain the role of the junction angle on the relation between two kinds of fractal dimensions. The obtained relationship curves indicate that box-counting dimension is decreasing with the increase of the junction angle when geomorphic fractal dimension keeps constant. This relationship presents continuous and smooth convex curves with junction angle from 60° to 120° and concave curves from 30° to 45°. Then 70 river networks in China are investigated in terms of their two kinds of fractal dimensions. The results confirm the fractal structure of river networks. Geomorphic fractal dimensions of river networks are larger than box-counting dimensions and there is no obvious relationship between these two kinds of fractal dimensions. Relatively good non-linear relationships between geomorphic fractal dimensions and box-counting dimensions are obtained by considering the role of the junction angle.

scholarly journals Application of Fractal Dimension of Terrestrial Laser Point Cloud in Classification of Independent Trees

2021 ◽  
Vol 5 (1) ◽  
pp. 14
Author(s):  
Ju Zhang ◽  
Qingwu Hu ◽  
Hongyu Wu ◽  
Junying Su ◽  
Pengcheng Zhao
Keyword(s):  
Fractal Dimension ◽  
Tree Species ◽  
Point Cloud ◽  
Point Clouds ◽  
Tree Canopy ◽  
Ecological Effect ◽  
Forest Species ◽  
Calculation Algorithm ◽  
Box Counting ◽  
Box Counting Dimension

Tree precise classification and identification of forest species is a core issue of forestry resource monitoring and ecological effect assessment. In this paper, an independent tree species classification method based on fractal features of terrestrial laser point cloud is proposed. Firstly, the terrestrial laser point cloud data of an independent tree is preprocessed to obtain terrestrial point clouds of independent tree canopy. Secondly, the multi-scale box-counting dimension calculation algorithm of independent tree canopy dense terrestrial laser point cloud is proposed. Furthermore, a robust box-counting algorithm is proposed to improve the stability and accuracy of fractal dimension expression of independent tree point cloud, which implementing gross error elimination based on Random Sample Consensus. Finally, the fractal dimension of a dense terrestrial laser point cloud of independent trees is used to classify different types of independent tree species. Experiments on nine independent trees of three types show that the fractal dimension can be stabilized under large density variations, proving that the fractal features of terrestrial laser point cloud can stably express tree species characteristics, and can be used for accurate classification and recognition of forest species.

scholarly journals Comparison of box counting and correlation dimension methods in well logging data analysis associate with the texture of volcanic rocks

2016 ◽  
Author(s):  
D. Mou ◽  
Z. W. Wang
Keyword(s):  
Fractal Dimension ◽  
Correlation Dimension ◽  
Gamma Ray ◽  
Volcanic Rocks ◽  
Well Logging ◽  
Oil Field ◽  
Pyroclastic Rock ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Volcanic Lava

Abstract. We have developed a fractal analysis method to estimate the dimension of well logging curves in Liaohe oil field, China. The box counting and correlation dimension are methods that can be applied to predict the texture of volcanic rocks with calculation the fractal dimension of well logging curves. The well logging curves are composed of gamma ray (GR), compensated neutron logs (CNL), acoustic (AC), density (DEN), Resistivity lateral log deep (RLLD), every curve contains a total of 6000 logging data. The dimension of well logging curves are calculated using box counting and correlation algorithms respectively. It is shown that two types of dimension of CNL, DEN and AC have the same average value. The box counting dimension of volcanic lava is lower than the pyroclastic rock obviously. The majority of correlation dimension of volcanic lava is lower than the pyroclastic rock, but a small amount of correlation dimension of volcanic lava is equal to the pyroclastic rock. It is demonstrated that the box counting dimension is more suitable for predicting the texture of volcanic rocks. Applications to logging data, A well show the relationship between the fractal dimension and the texture of volcanic rock in certain depth.

Box-Counting Dimension of Fractal Urban Form

2012 ◽  
Vol 3 (3) ◽  
pp. 41-63 ◽  
Author(s):  
Shiguo Jiang ◽  
Desheng Liu
Keyword(s):  
Fractal Dimension ◽  
Urban Form ◽  
Fractal Dimensions ◽  
Reliable Estimate ◽  
Box Dimension ◽  
Beijing City ◽  
Box Counting ◽  
Window Method ◽  
Box Counting Dimension ◽  
Data Points

The difficulty to obtain a stable estimate of fractal dimension for stochastic fractal (e.g., urban form) is an unsolved issue in fractal analysis. The widely used box-counting method has three main issues: 1) ambiguities in setting up a proper box cover of the object of interest; 2) problems of limited data points for box sizes; 3) difficulty in determining the scaling range. These issues lead to unreliable estimates of fractal dimensions for urban forms, and thus cast doubt on further analysis. This paper presents a detailed discussion of these issues in the case of Beijing City. The authors propose corresponding improved techniques with modified measurement design to address these issues: 1) rectangular grids and boxes setting up a proper box cover of the object; 2) pseudo-geometric sequence of box sizes providing adequate data points to study the properties of the dimension profile; 3) generalized sliding window method helping to determine the scaling range. The authors’ method is tested on a fractal image (the Vicsek prefractal) with known fractal dimension and then applied to real city data. The results show that a reliable estimate of box dimension for urban form can be obtained using their method.

scholarly journals Reconstruction for Time-Domain In Vivo EPR 3D Multigradient Oximetric Imaging—A Parallel Processing Perspective

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Christopher D. Dharmaraj ◽  
Kishan Thadikonda ◽  
Anthony R. Fletcher ◽  
Phuc N. Doan ◽  
Nallathamby Devasahayam ◽  
...  
Keyword(s):  
Imaging Modality ◽  
Low Cost ◽  
Single Point ◽  
Biophysical Parameters ◽  
Data Set ◽  
Reconstruction Process ◽  
Matlab Code ◽  
Fourier Reconstruction ◽  
Parallel Matlab

Three-dimensional Oximetric Electron Paramagnetic Resonance Imaging using the Single Point Imaging modality generates unpaired spin density and oxygen images that can readily distinguish between normal and tumor tissues in small animals. It is also possible with fast imaging to track the changes in tissue oxygenation in response to the oxygen content in the breathing air. However, this involves dealing with gigabytes of data for each 3D oximetric imaging experiment involving digital band pass filtering and background noise subtraction, followed by 3D Fourier reconstruction. This process is rather slow in a conventional uniprocessor system. This paper presents a parallelization framework using OpenMP runtime support and parallel MATLAB to execute such computationally intensive programs. The Intel compiler is used to develop a parallel C++ code based on OpenMP. The code is executed on four Dual-Core AMD Opteron shared memory processors, to reduce the computational burden of the filtration task significantly. The results show that the parallel code for filtration has achieved a speed up factor of 46.66 as against the equivalent serial MATLAB code. In addition, a parallel MATLAB code has been developed to perform 3D Fourier reconstruction. Speedup factors of 4.57 and 4.25 have been achieved during the reconstruction process and oximetry computation, for a data set with23×23×23gradient steps. The execution time has been computed for both the serial and parallel implementations using different dimensions of the data and presented for comparison. The reported system has been designed to be easily accessible even from low-cost personal computers through local internet (NIHnet). The experimental results demonstrate that the parallel computing provides a source of high computational power to obtain biophysical parameters from 3D EPR oximetric imaging, almost in real-time.

The Blemish Identify in Ultrasonic Testing of Friction Welded Joints Based on Fractal Theory

2015 ◽  
Vol 799-800 ◽  
pp. 942-946
Author(s):  
Yuan Peng Liu ◽  
Xin Yin ◽  
Min Wang
Keyword(s):  
Fractal Dimension ◽  
Welded Joints ◽  
Ultrasonic Testing ◽  
Fractal Theory ◽  
Metal Alloys ◽  
Box Dimension ◽  
Box Counting ◽  
Signal Characteristics ◽  
Box Counting Dimension ◽  
Dimension Number

The friction welded joints made by GH4169 heat metal alloys are detected by U1traPAC system of the ultrasonic wave explore instrument. Aimed at the blemish signal characteristics, a method is put forward to identify the blemish signal in ultrasonic testing of friction welded joints based on fractal theory., A new kind of fractal dimension number calculation method—normalized scale box-counting dimension method is used to calculate the box-dimension of the blemish signal, and statistic and analysis the scopes of the box-dimension of the blemish signal and the square differ of the result. The preliminary experimental results show that the blemish signal and have each fractal dimension zones, and don’t hand over to fold. So it can use to judge whether the blemish exist or not. The method is identified to have better covariance characteristics about the blemish identify of the friction welded joints.

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