Multiresolution Area-based Fractal Dimension Estimation of Signals Applied to EEG Data

In this paper, we present an approach to estimate fractal complexity of discrete time signal waveforms based on computation of area bounded by sample points of the signal at different time resolutions. The slope of best straight line fit to the graph of log(A(rk)A / rk(2)) versus log(l/rk) is estimated, where A(rk) is the area computed at different time resolutions and rk time resolutions at which the area have been computed. The slope quantifies complexity of the signal and it is taken as an estimate of the fractal dimension (FD). The proposed approach is used to estimate the fractal dimension of parametric fractal signals with known fractal dimensions and the method has given accurate results. The estimation accuracy of the method is compared with that of Higuchi's and Sevcik's methods. The proposed method has given more accurate results when compared with that of Sevcik's method and the results are comparable to that of the Higuchi's method. The practical application of the complexity measure in detecting change in complexity of signals is discussed using real sleep electroencephalogram recordings from eight different subjects. The FD-based approach has shown good performance in discriminating different stages of sleep.

Signal Characterization using Fractal Dimension

Fractal Dimensions (FD) are popular metrics for characterizing signals. They are used as complexity measuresin signal analysis applications in various fields. However, proper interpretation of such analyses has not been thoroughly addressed. In this paper, we study the effect of various signal properties on FD and interpret results in terms of classical signal processing concepts such as amplitude, frequency,number of harmonics, noise power and signal bandwidth. We have used Higuchi’s method for estimating FDs. This study helps in gaining a better understanding of the FD complexity measure for various signal parameters. Our results indicate that FD is a useful metric in estimating various signal properties. As an application of the FD measure in real world scenario, the FD is used as a feature in discriminating seizures from seizure free intervals in intracranial EEG data recordings and the FD feature has given good discrimination performance.

Complexity analysis of EEG in patients with schizophrenia using fractal dimension

We computed Higuchi's fractal dimension (FD) of resting, eyes closed EEG recorded from 30 scalp locations in 18 male neuroleptic-naive, recent-onset schizophrenia (NRS) subjects and 15 male healthy control (HC) subjects, who were group-matched for age. Schizophrenia patients showed a diffuse reduction of FD except in the bilateral temporal and occipital regions, with the reduction being most prominent bifrontally. The positive symptom (PS) schizophrenia subjects showed FD values similar to or even higher than HC in the bilateral temporo-occipital regions, along with a co-existent bifrontal FD reduction as noted in the overall sample of NRS. In contrast, this increase in FD values in the bilateral temporo-occipital region was absent in the negative symptom (NS) subgroup. The regional differences in complexity suggested by these findings may reflect the aberrant brain dynamics underlying the pathophysiology of schizophrenia and its symptom dimensions. Higuchi's method of measuring FD directly in the time domain provides an alternative for the more computationally intensive nonlinear methods of estimating EEG complexity.

Signal characterization using fractal dimension

Fractal Dimensions (FD) are one of the popular measures used for characterizing signals. They have been used as complexity measures of signals in various fields including speech and biomedical applications. However, proper interpretation of such analyses has not been thoroughly addressed. In this paper, we study the effect of various signal properties on FD and interpret results in terms of classical signal processing concepts such as amplitude, frequency, number of harmonics, noise power and signal bandwidth. We have used Higuchi's method for estimating FDs. This study may help in gaining a better understanding of the FD complexity measure itself, and for interpreting changing structural complexity of signals in terms of FD. Our results indicate that FD is a useful measure in quantifying structural changes in signal properties.

Use of running fractal dimension for the analysis of changing patterns in electroencephalograms

Running fractal dimensions were measured on four channels of an electroencephalogram (EEG) recorded from a normal volunteer. The changes in the background activity due to eye closure were clearly differentiated by the fractal method. The compressed spectral array (CSA) and the running fractal dimensions of the EEG showed corresponding changes with respect to change in the background activity. The fractal method was also successful in detecting low amplitude spikes and the changes in the patterns in the EEG. The effects of different window lengths and shifts on the running fractal dimension have also been studied. The utility of fractal method for EEG data compression is highlighted.

Computing Fractal Dimension of Signals using Multiresolution Box-counting Method

In this paper, we have developed a method to compute fractal dimension (FD) of discrete time signals, in the time domain, by modifying the box-counting method. The size of the box is dependent on the sampling frequency of the signal. The number of boxes required to completely cover the signal are obtained at multiple time resolutions. The time resolutions are made coarse by decimating the signal. The loglog plot of total number of boxes required to cover the curve versus size of the box used appears to be a straight line, whose slope is taken as an estimate of FD of the signal. The results are provided to demonstrate the performance of the proposed method using parametric fractal signals. The estimation accuracy of the method is compared with that of Katz, Sevcik, and Higuchi methods. In ddition, some properties of the FD are discussed.

Post-seizure EEG fractal dimension of first ECT predicts antidepressant response at two weeks

Background: Duration of seizure by itself is an insufficient criterion for a therapeutically adequate seizure in ECT. Therefore, measures of seizure EEG other than its duration need to be explored as indices of seizure adequacy and predictors of treatment response. We measured the EEG seizure using a geometrical method-fractal dimension (FD) and examined if this measure predicted remission. Methods: Data from an efficacy study on melancholic depressives (n = 40) is used for the present exploration. They received thrice or once weekly ECTs, each schedule at two energy levels - high or low energy level. FD was computed for early-, mid- and post-seizure phases of the ictal EEG. Average of the two channels was used for analysis. Results: Two-thirds of the patients (n = 25) were remitted at the end of 2 weeks. As expected, a significantly higher proportion of patients receiving thrice weekly ECT remitted than in patients receiving once weekly ECT. Smaller post-seizure FD at first ECT is the only variable which predicted remission status after six ECTs. within the once weekly ECT group too, smaller post-seizure FD was associated with remission status. Conclusions: Post-seizure FD is proposed as a novel measure of seizure adequacy and predictor of treatment response. Clinical implications: Seizure measures at first ECT may guide selection of ECT schedule to optimize ECT. Limitations: The study examined short term antidepressant effects only. The results may not be generalized to medication-resistant depressives. (C) 1999 Elsevier Science B.V. All rights reserved.

Fractal dimension method for delamination detection in composite beams

A new delaminated composite beam element is formulated for Timoshenko as well as Euler-Bernoulli beam models. Shape functions are derived from Timoshenko functions; this provides a unified formulation for slender to moderately deep beam analyses. The element is simple and easy to implement, results are on par with those from free mode delamination models. Katz fractal dimension method is applied on the mode shapes obtained from finite element models, to detect the delamination in the beam. The effect of finite element size on fractal dimension method of delamination detection is quantified.

Matrix Crack Detection in Composite Plate with Spatially Random Material Properties using Fractal Dimension

Fractal dimension based damage detection method is investigated for a composite plate with random material properties. Composite material shows spatially varying random material properties because of complex manufacturing processes. Matrix cracks are considered as damage in the composite plate. Such cracks are often seen as the initial damage mechanism in composites under fatigue loading and also occur due to low velocity impact. Static deflection of the cantilevered composite plate with uniform loading is calculated using the finite element method. Damage detection is carried out based on sliding window fractal dimension operator using the static deflection. Two dimensional homogeneous Gaussian random field is generated using Karhunen-Loeve (KL) expansion to represent the spatial variation of composite material property. The robustness of fractal dimension based damage detection method is demonstrated considering the composite material properties as a two dimensional random field.

Matrix crack detection in spatially random composite structures using fractal dimension

Fractal dimension based damage detection method is studied for a composite structure with random material properties. A composite plate with localized matrix crack is considered. Matrix cracks are often seen as the initial damage mechanism in composites. Fractal dimension based method is applied to the static deformation curve of the structure to detect localized damage. Static deflection of a cantilevered composite plate under uniform loading is calculated using the finite element method. Composite material shows spatially varying random material properties because of complex manufacturing processes. Spatial variation of material property is represented as a two dimensional homogeneous Gaussian random field. Karhunen-Loeve (KL) expansion is used to generate a random field. The robustness of fractal dimension based damage detection methods is studied considering the composite plate with spatial variation in material properties.

A note on fractal dimensions of biomedical waveforms

In this paper, we study performance of Katz method of computing fractal dimension of waveforms, and its estimation accuracy is compared with Higuchi's method. The study is performed on four synthetic parametric fractal waveforms for which true fractal dimensions can be calculated, and real sleep electroencephalogram. The dependence of Katz's fractal dimension on amplitude, frequency and sampling frequency of waveforms is noted. Even though the Higuchi's method has given more accurate estimation of fractal dimensions, the study suggests that the results of Katz's based fractal dimension analysis of biomedical waveforms have to be carefully interpreted.

Estimating the Hausdorff-Besicovitch dimension of boundary of basin of attraction in helicopter trim

Helicopter trim involves solution of nonlinear force equilibrium equations. As in many nonlinear dynamic systems, helicopter trim problem can show chaotic behavior. This chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot. This study focuses on the boundary of the basin of attraction obtained for a set of control inputs. We analyze the boundary by considering it at different magnification levels. The magnified views reveal intricate geometries. It is also found that the basin boundary exhibits the characteristic of statistical self-similarity, which is an essential property of fractal geometries. These results led the authors to investigate the fractal dimension of the basin boundary. It is found that this dimension is indeed greater than the topological dimension. From all the observations, it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature. (C) 2012 Elsevier Inc. All rights reserved.

Search on a fractal lattice using a quantum random walk

The spatial search problem on regular lattice structures in integer number of dimensions d >= 2 has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here, we investigate the spatial search problem on fractals of noninteger dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.

Small-angle x-ray scattering study of the aggregation of gold nanoparticles during formation at the toluene-water interface

We report the results of an in situ small-angle x-ray scattering (SAXS) study of the aggregation of gold nanoparticles formed by an interfacial reaction at the toluene-water interface. The SAXS data provide a direct evidence for aggregate formation of nanoparticles having 1.3 nm gold core and an organic shell that gives a core-core separation of about 2.5 nm. Furthermore, the nanoparticles do not occupy all the cites of 13-member cluster. This occupancy decreases with reaction time and indicate reorganization of the clusters that generates planner disklike structures. A gradual increase in fractal dimension from 1.82 to 2.05 also indicate compactification of cluster aggregation with reaction time, the final exponent being close to 2 expected for disklike aggregates.