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  1. Fractal characterization of bpn weights evolution
    Gunasekaran S, Venkatesh B, Sagar BS
    Int J Neural Syst, 2004 Apr;14(2):139-45.
    PMID: 15112371
    Training methodology of the Back Propagation Network (BPN) is well documented. One aspect of BPN that requires investigation is whether or not the BPN would get trained for a given training data set and architecture. In this paper the behavior of the BPN is analyzed during its training phase considering convergent and divergent training data sets. Evolution of the weights during the training phase was monitored for the purpose of analysis. The evolution of weights was plotted as return map and was characterized by means of fractal dimension. This fractal dimensional analysis of the weight evolution trajectories is used to provide a new insight to understand the behavior of BPN and dynamics in the evolution of weights.
    Matched MeSH terms: Fractals*
  2. Decoding of the relationship between human brain activity and walking paths
    Kamal SM, Sim S, Tee R, Nathan V, Aghasian E, Namazi H
    Technol Health Care, 2020;28(4):381-390.
    PMID: 31796717 DOI: 10.3233/THC-191965
    BACKGROUND: The human brain controls all actions of the body. Walking is one of the most important actions that deals with the movement of the body. In fact, the brain controls and regulates human walking based on different conditions. One of the conditions that affects human walking is the complexity of path of movement. Therefore, the brain activity should change when a person walks on a path with different complexities.

    OBJECTIVE: In this research we benefit from fractal analysis to study the effect of complexity of path of movement on the complexity of human brain reaction.

    METHODS: For this purpose we calculate the fractal dimension of the electroencephalography (EEG) signal when subjects walk on different paths with different fractal dimensions (complexity).

    RESULTS: The results of the analysis show that the complexity of brain activity increases with the increment of complexity of path of movement.

    CONCLUSION: The method of analysis employed in this research can also be employed to analyse the reaction of the human heart and respiration when subjects move on paths with different complexities.

    Matched MeSH terms: Fractals
  3. Detection of abnormality in wireless capsule endoscopy images using fractal features
    Jain S, Seal A, Ojha A, Krejcar O, Bureš J, Tachecí I, et al.
    Comput Biol Med, 2020 12;127:104094.
    PMID: 33152668 DOI: 10.1016/j.compbiomed.2020.104094
    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.
    Matched MeSH terms: Fractals
  4. Fulltext Fractal dimension algorithm for automatic detection of oil spills in RADARSAT-1 SAR
    Marghany, M., Cracknell, A.P., Hashim, M.
    ASM Science Journal, 2009;3(1):7-16.
    MyJurnal
    This paper introduces a method for modification of the formula of the fractal box counting dimension. The method is based on the utilization of the probability distribution formula in the fractal box count. The purpose of this method is to use it for the discrimination of oil spill areas from the surrounding features e.g. sea surface and look-alikes in RADARSAT-1 SAR data. The result showed that the new formula of the fractal box counting dimension was able to discriminate between oil spills and look-alike areas. The low wind area had the highest fractal dimension peak of 2.9, as compared to the oil slick and the surrounding rough sea. The maximum error standard deviation of the low wind area was 0.68 which performed with a 2.9 fractal dimension value.
    Matched MeSH terms: Fractals
  5. A novel automated diagnostic system for classification of myocardial infarction ECG signals using an optimal biorthogonal filter bank
    Sharma M, Tan RS, Acharya UR
    Comput Biol Med, 2018 11 01;102:341-356.
    PMID: 30049414 DOI: 10.1016/j.compbiomed.2018.07.005
    Myocardial infarction (MI), also referred to as heart attack, occurs when there is an interruption of blood flow to parts of the heart, due to the acute rupture of atherosclerotic plaque, which leads to damage of heart muscle. The heart muscle damage produces changes in the recorded surface electrocardiogram (ECG). The identification of MI by visual inspection of the ECG requires expert interpretation, and is difficult as the ECG signal changes associated with MI can be short in duration and low in magnitude. Hence, errors in diagnosis can lead to delay the initiation of appropriate medical treatment. To lessen the burden on doctors, an automated ECG based system can be installed in hospitals to help identify MI changes on ECG. In the proposed study, we develop a single-channel single lead ECG based MI diagnostic system validated using noisy and clean datasets. The raw ECG signals are taken from the Physikalisch-Technische Bundesanstalt database. We design a novel two-band optimal biorthogonal filter bank (FB) for analysis of the ECG signals. We present a method to design a novel class of two-band optimal biorthogonal FB in which not only the product filter but the analysis lowpass filter is also a halfband filter. The filter design problem has been composed as a constrained convex optimization problem in which the objective function is a convex combination of multiple quadratic functions and the regularity and perfect reconstruction conditions are imposed in the form linear equalities. ECG signals are decomposed into six subbands (SBs) using the newly designed wavelet FB. Following to this, discriminating features namely, fuzzy entropy (FE), signal-fractal-dimensions (SFD), and renyi entropy (RE) are computed from all the six SBs. The features are fed to the k-nearest neighbor (KNN). The proposed system yields an accuracy of 99.62% for the noisy dataset and an accuracy of 99.74% for the clean dataset, using 10-fold cross validation (CV) technique. Our MI identification system is robust and highly accurate. It can thus be installed in clinics for detecting MI.
    Matched MeSH terms: Fractals
  6. Ion conductive polymer electrolyte membranes and simulation of their fractal growth patterns
    Amir S, Mohamed N, Hashim Ali S
    Sains Malaysiana, 2011;40:1123-1127.
    Due to their high ionic conductivity, solid polymer electrolyte (SPE) systems have attracted wide spread attention as the most appropriate choice to fabricate all-solid-state electrochemical devices, namely batteries, sensors and fuel cells. In this work, ion conductive polymer electrolyte membranes have been prepared for battery fabrication. However, fractals were found to grow in these polymer electrolyte membranes weeks after they were prepared. It was believed that the formation of fractal aggregates in these membranes were due to ionic movement. The discovery of fractal growth pattern can be used to understand the effects of such phenomenon in the polymer electrolyte membranes. Digital images of the fractal growth patterns were taken and a simulation model was developed based on the Brownian motion theory and a fractal dialect known as L-system. A computer coding has been designed to simulate and visualize the fractal growth.
    Matched MeSH terms: Fractals
  7. Malaysian household income distribution: a fractal point of view
    Fatimah Abdul Razak, Faridatulazna Ahmad Shahabuddin
    Sains Malaysiana, 2018;47:2187-2194.
    Malaysian Household Income Survey data provided by the Malaysian Department of Statistics is used to provide evidence
    that the upper tails of the household income distribution follows a fractal based distribution known as power-law.
    Inequality measures are then applied to ascertain the levels of inequality based on this distribution. In addition to that,
    we analyzed the data in terms of different classes of occupation, obtained power-law exponents for each class and then
    highlighted the inequality between these classes.
    Matched MeSH terms: Fractals
  8. Detection of Voice Pathology using Fractal Dimension in a Multiresolution Analysis of Normal and Disordered Speech Signals
    Ali Z, Elamvazuthi I, Alsulaiman M, Muhammad G
    J Med Syst, 2016 Jan;40(1):20.
    PMID: 26531753 DOI: 10.1007/s10916-015-0392-2
    Voice disorders are associated with irregular vibrations of vocal folds. Based on the source filter theory of speech production, these irregular vibrations can be detected in a non-invasive way by analyzing the speech signal. In this paper we present a multiband approach for the detection of voice disorders given that the voice source generally interacts with the vocal tract in a non-linear way. In normal phonation, and assuming sustained phonation of a vowel, the lower frequencies of speech are heavily source dependent due to the low frequency glottal formant, while the higher frequencies are less dependent on the source signal. During abnormal phonation, this is still a valid, but turbulent noise of source, because of the irregular vibration, affects also higher frequencies. Motivated by such a model, we suggest a multiband approach based on a three-level discrete wavelet transformation (DWT) and in each band the fractal dimension (FD) of the estimated power spectrum is estimated. The experiments suggest that frequency band 1-1562 Hz, lower frequencies after level 3, exhibits a significant difference in the spectrum of a normal and pathological subject. With this band, a detection rate of 91.28 % is obtained with one feature, and the obtained result is higher than all other frequency bands. Moreover, an accuracy of 92.45 % and an area under receiver operating characteristic curve (AUC) of 95.06 % is acquired when the FD of all levels is fused. Likewise, when the FD of all levels is combined with 22 Multi-Dimensional Voice Program (MDVP) parameters, an improvement of 2.26 % in accuracy and 1.45 % in AUC is observed.
    Matched MeSH terms: Fractals*
  9. Fractal-based classification of electroencephalography (EEG) signals in healthy adolescents and adolescents with symptoms of schizophrenia
    Namazi H, Aghasian E, Ala TS
    Technol Health Care, 2019;27(3):233-241.
    PMID: 30829625 DOI: 10.3233/THC-181497
    Brain activity analysis is an important research area in the field of human neuroscience. Moreover, a subcategory in this field is the classification of brain activity in terms of different brain disorders. Since the Electroencephalography (EEG) signal is, in fact, a non-linear time series, employing techniques to investigate its non-linear structure is rather crucial. In this study, we evaluate the non-linear structure of the EEG signal between healthy and schizophrenic adolescents using fractal theory. The results of our analysis revealed that in terms of all recording channels, the EEG signal of healthy subjects is more complex compared to the ones suffering from schizophrenia. The statistical analysis also indicated that there is a significant difference in the complex structure of the EEG signal between these two groups of subjects. We also utilized approximate entropy in our analysis in order to verify the obtained results of the fractal analysis. The result of the entropy analysis suggested that EEG signal for healthy subjects is less random compared to the EEG signal in schizophrenic individuals. In addition, the employed methodology in this research can be further investigated in order to classify the brain activity in terms of other brain disorders, where one can explore how the complex structure of the EEG signal alters between them.
    Matched MeSH terms: Fractals*
  10. Fulltext Mathematical Modelling and Prediction of the Effect of Chemotherapy on Cancer Cells
    Namazi H, Kulish VV, Wong A
    Sci Rep, 2015;5:13583.
    PMID: 26316014 DOI: 10.1038/srep13583
    Cancer is a class of diseases characterized by out-of-control cells' growth which affect DNAs and make them damaged. Many treatment options for cancer exist, with the primary ones including surgery, chemotherapy, radiation therapy, hormonal therapy, targeted therapy and palliative care. Which treatments are used depends on the type, location, and grade of the cancer as well as the person's health and wishes. Chemotherapy is the use of medication (chemicals) to treat disease. More specifically, chemotherapy typically refers to the destruction of cancer cells. Considering the diffusion of drugs in cancer cells and fractality of DNA walks, in this research we worked on modelling and prediction of the effect of chemotherapy on cancer cells using Fractional Diffusion Equation (FDE). The employed methodology is useful not only for analysis of the effect of special drug and cancer considered in this research but can be expanded in case of different drugs and cancers.
    Matched MeSH terms: Fractals
  11. Fulltext Fractal-Fractional Mathematical Model Addressing the Situation of Corona Virus in Pakistan
    Shah K, Arfan M, Mahariq I, Ahmadian A, Salahshour S, Ferrara M
    Results Phys, 2020 Dec;19:103560.
    PMID: 33200064 DOI: 10.1016/j.rinp.2020.103560
    This work is the consideration of a fractal fractional mathematical model on the transmission and control of corona virus (COVID-19), in which the total population of an infected area is divided into susceptible, infected and recovered classes. We consider a fractal-fractional order

    SIR

    type model for investigation of Covid-19. To realize the transmission and control of corona virus in a much better way, first we study the stability of the corresponding deterministic model using next generation matrix along with basic reproduction number. After this, we study the qualitative analysis using "fixed point theory" approach. Next, we use fractional Adams-Bashforth approach for investigation of approximate solution to the considered model. At the end numerical simulation are been given by matlab to provide the validity of mathematical system having the arbitrary order and fractal dimension.
    Matched MeSH terms: Fractals
  12. Complexity-Based Decoding of the Coupling Among Heart Rate Variability (HRV) and Walking Path
    Mujib Kamal S, Babini MH, Krejcar O, Namazi H
    Front Physiol, 2020;11:602027.
    PMID: 33324242 DOI: 10.3389/fphys.2020.602027
    Walking is an everyday activity in our daily life. Because walking affects heart rate variability, in this research, for the first time, we analyzed the coupling among the alterations of the complexity of walking paths and heart rate. We benefited from the fractal theory and sample entropy to evaluate the influence of the complexity of paths on the complexity of heart rate variability (HRV) during walking. We calculated the fractal exponent and sample entropy of the R-R time series for nine participants who walked on four paths with various complexities. The findings showed a strong coupling among the alterations of fractal dimension (an indicator of complexity) of HRV and the walking paths. Besides, the result of the analysis of sample entropy also verified the obtained results from the fractal analysis. In further studies, we can analyze the coupling among the alterations of the complexities of other physiological signals and walking paths.
    Matched MeSH terms: Fractals
  13. Fulltext Characterization of alumina using small angle neutron scattering (sans)
    Megat Harun Al Rashid Megat Ahmad, Abdul Aziz Mohamed, Azmi Ibrahim, Che Seman Mahmood, Putra, Edy Giri Rachman, Muhammad Rawi Muhammad Zin, et al.
    Journal of Nuclear and Related Technologies, 2007;2005(1):19-23.
    MyJurnal
    Alumina powder was synthesized from an aluminum precursor and studied using small angle neutron scattering (SANS) technique and complemented with transmission electron microscope (TEM). XRD measurement confirmed that the alumina produced was of high purity and highly crystalline D-phase. SANS examination indicates the formation of mass fractals microstructures with fractal dimension of about 2.8 on the alumina powder.
    Matched MeSH terms: Fractals
  14. Fulltext Characterization of skeletons of simulated drought and flood of water bodies
    Dinesh, S.
    ASM Science Journal, 2010;4(1):62-73.
    MyJurnal
    Studies conducted on the various geometric properties of skeletons of water bodies have shown highly promising results. However, these studies were made under the assumption that water bodies were static objects and that they remained constant over time. Water bodies are actually dynamic objects; they go through significant spatio-temporal changes due to drought and flood. In this study, the characterization of skeletons of simulated drought and flood of water bodies was performed. It was observed that as the drought level increased from 1 to 9, the average length of the skeletons decreased due to reduction in the size of the water bodies and increase in the number of water bodies. As the drought level increased from 9 to 15, the average length of the skeletons increased further due to vanishing of small water bodies. Flood caused an increase in the average length of the skeletons due to merging of adjacent water bodies. Power law relationships were observed between the average length of the skeletons of the simulated drought/flood and the level of drought/flood. The scaling exponent of these power laws which was named as a fractal dimension, indicated the rate of change of the average length of the skeletons of simulated drought/flood of water bodies over varying levels of drought/flood. However, errors observed in the goodness of fit of the plots indicated that monofractals were not sufficient to characterise the skeletons of simulated drought and flood of water bodies. Multifractals and lacunarity analysis were required for more accurate characterisation.
    Matched MeSH terms: Fractals
  15. Fulltext A Review on the Extraction of Quantitative Retinal Microvascular Image Feature
    Kipli K, Hoque ME, Lim LT, Mahmood MH, Sahari SK, Sapawi R, et al.
    Comput Math Methods Med, 2018;2018:4019538.
    PMID: 30065780 DOI: 10.1155/2018/4019538
    Digital image processing is one of the most widely used computer vision technologies in biomedical engineering. In the present modern ophthalmological practice, biomarkers analysis through digital fundus image processing analysis greatly contributes to vision science. This further facilitates developments in medical imaging, enabling this robust technology to attain extensive scopes in biomedical engineering platform. Various diagnostic techniques are used to analyze retinal microvasculature image to enable geometric features measurements such as vessel tortuosity, branching angles, branching coefficient, vessel diameter, and fractal dimension. These extracted markers or characterized fundus digital image features provide insights and relates quantitative retinal vascular topography abnormalities to various pathologies such as diabetic retinopathy, macular degeneration, hypertensive retinopathy, transient ischemic attack, neovascular glaucoma, and cardiovascular diseases. Apart from that, this noninvasive research tool is automated, allowing it to be used in large-scale screening programs, and all are described in this present review paper. This paper will also review recent research on the image processing-based extraction techniques of the quantitative retinal microvascular feature. It mainly focuses on features associated with the early symptom of transient ischemic attack or sharp stroke.
    Matched MeSH terms: Fractals
  16. Age-related rarefaction in retinal vasculature is not linear
    Che Azemin MZ, Ab Hamid F, Aminuddin A, Wang JJ, Kawasaki R, Kumar DK
    Exp Eye Res, 2013 Nov;116:355-358.
    PMID: 24512773 DOI: 10.1016/j.exer.2013.10.010
    The fractal dimension is a global measure of complexity and is useful for quantifying anatomical structures, including the retinal vascular network. A previous study found a linear declining trend with aging on the retinal vascular fractal dimension (DF); however, it was limited to the older population (49 years and older). This study aimed to investigate the possible models of the fractal dimension changes from young to old subjects (10-73 years). A total of 215 right-eye retinal samples, including those of 119 (55%) women and 96 (45%) men, were selected. The retinal vessels were segmented using computer-assisted software, and non-vessel fragments were deleted. The fractal dimension was measured based on the log-log plot of the number of grids versus the size. The retinal vascular DF was analyzed to determine changes with increasing age. Finally, the data were fitted to three polynomial models. All three models are statistically significant (Linear: R2 = 0.1270, 213 d.f., p 
    Matched MeSH terms: Fractals
  17. Automated classification of patients with coronary artery disease using grayscale features from left ventricle echocardiographic images
    Acharya UR, Sree SV, Muthu Rama Krishnan M, Krishnananda N, Ranjan S, Umesh P, et al.
    Comput Methods Programs Biomed, 2013 Dec;112(3):624-32.
    PMID: 23958645 DOI: 10.1016/j.cmpb.2013.07.012
    Coronary Artery Disease (CAD), caused by the buildup of plaque on the inside of the coronary arteries, has a high mortality rate. To efficiently detect this condition from echocardiography images, with lesser inter-observer variability and visual interpretation errors, computer based data mining techniques may be exploited. We have developed and presented one such technique in this paper for the classification of normal and CAD affected cases. A multitude of grayscale features (fractal dimension, entropies based on the higher order spectra, features based on image texture and local binary patterns, and wavelet based features) were extracted from echocardiography images belonging to a huge database of 400 normal cases and 400 CAD patients. Only the features that had good discriminating capability were selected using t-test. Several combinations of the resultant significant features were used to evaluate many supervised classifiers to find the combination that presents a good accuracy. We observed that the Gaussian Mixture Model (GMM) classifier trained with a feature subset made up of nine significant features presented the highest accuracy, sensitivity, specificity, and positive predictive value of 100%. We have also developed a novel, highly discriminative HeartIndex, which is a single number that is calculated from the combination of the features, in order to objectively classify the images from either of the two classes. Such an index allows for an easier implementation of the technique for automated CAD detection in the computers in hospitals and clinics.
    Matched MeSH terms: Fractals
  18. An integrated index for automated detection of infarcted myocardium from cross-sectional echocardiograms using texton-based features (Part 1)
    Sudarshan VK, Acharya UR, Ng EY, Tan RS, Chou SM, Ghista DN
    Comput Biol Med, 2016 Apr 1;71:231-40.
    PMID: 26898671 DOI: 10.1016/j.compbiomed.2016.01.028
    Cross-sectional view echocardiography is an efficient non-invasive diagnostic tool for characterizing Myocardial Infarction (MI) and stages of expansion leading to heart failure. An automated computer-aided technique of cross-sectional echocardiography feature assessment can aid clinicians in early and more reliable detection of MI patients before subsequent catastrophic post-MI medical conditions. Therefore, this paper proposes a novel Myocardial Infarction Index (MII) to discriminate infarcted and normal myocardium using features extracted from apical cross-sectional views of echocardiograms. The cross-sectional view of normal and MI echocardiography images are represented as textons using Maximum Responses (MR8) filter banks. Fractal Dimension (FD), Higher-Order Statistics (HOS), Hu's moments, Gabor Transform features, Fuzzy Entropy (FEnt), Energy, Local binary Pattern (LBP), Renyi's Entropy (REnt), Shannon's Entropy (ShEnt), and Kapur's Entropy (KEnt) features are extracted from textons. These features are ranked using t-test and fuzzy Max-Relevancy and Min-Redundancy (mRMR) ranking methods. Then, combinations of highly ranked features are used in the formulation and development of an integrated MII. This calculated novel MII is used to accurately and quickly detect infarcted myocardium by using one numerical value. Also, the highly ranked features are subjected to classification using different classifiers for the characterization of normal and MI LV ultrasound images using a minimum number of features. Our current technique is able to characterize MI with an average accuracy of 94.37%, sensitivity of 91.25% and specificity of 97.50% with 8 apical four chambers view features extracted from only single frame per patient making this a more reliable and accurate classification.
    Matched MeSH terms: Fractals
  19. Radioactivity and heat production characterization by natural gamma ray well logging and multi-fractal techniques in banting district, Malaysia
    Asfahani J, Samuding K, Mostapa R, Othman O
    Appl Radiat Isot, 2021 Jan;167:109296.
    PMID: 33022484 DOI: 10.1016/j.apradiso.2020.109296
    Natural gamma ray well logging technique is used to characterize the radioactivity (GR) laterally and vertically in Banting district, SW of Malaysia. Seven drilled boreholes, along N-S profile with their natural gamma ray records are utilized to compute the heat production (HP) parameter, based on the Bucker and Rybach relationship.The analysis of 3467 measured points in those boreholes indicates that GR varies between 6.24 API and 358.4 API with an average of 79.95 API, while HP varies between 0.086 and 5.65 μw/m3 with an average of 1.25 μw/m3.The multi-fractal Concentration-Number (C-N) is used to characterize the radioactivity and heat production variations and to isolate different GR and HP populations in the study region. The high radioactivity and heat production ranges are mainly related to the silty clay layers, accompanied by uranium and thorium.
    Matched MeSH terms: Fractals
  20. Complexity-based classification of EEG signal in normal subjects and patients with epilepsy
    Namazi H, Aghasian E, Ala TS
    Technol Health Care, 2020;28(1):57-66.
    PMID: 31104032 DOI: 10.3233/THC-181579
    Analysis of human brain activity is an important topic in human neuroscience. Human brain activity can be studied by analyzing the electroencephalography (EEG) signal. In this way, scientists have employed several techniques that investigate nonlinear dynamics of EEG signals. Fractal theory as a promising technique has shown its capabilities to analyze the nonlinear dynamics of time series. Since EEG signals have fractal patterns, in this research we analyze the variations of fractal dynamics of EEG signals between four datasets that were collected from healthy subjects with open-eyes and close-eyes conditions, patients with epilepsy who did and patients who did not face seizures. The obtained results showed that EEG signal during seizure has greatest complexity and the EEG signal during the seizure-free interval has lowest complexity. In order to verify the obtained results in case of fractal analysis, we employ approximate entropy, which indicates the randomness of time series. The obtained results in case of approximate entropy certified the fractal analysis results. The obtained results in this research show the effectiveness of fractal theory to investigate the nonlinear structure of EEG signal between different conditions.
    Matched MeSH terms: Fractals
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