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The box-counting (BC) algorithm is applied to calculate fractal dimensions of four fractal sets. The sets are contaminated with an additive noise with amplitude $\gamma =10^{-5}-10^{-1}$. The accuracy of calculated numerical values of the fractal dimensions is analyzed as a function of $\gamma$ for different sizes of the data sample. In particular, it has been found that even in case of pure fractals ($\gamma =0$) as well as for tiny noise ($\gamma \approx 10^{-5}$) one has considerable error for the calculated exponents of order 0.01. For larger noise the error is growing up to 0.1 and more, with natural saturation limited by the embedding dimension. This prohibits the power-like scaling of the error. Moreover, the noise effect cannot be cured by taking larger data samples.

An optimal box-counting algorithm for estimating the fractal dimension of a nonempty set which changes over time is given. This nonstationary environment is characterized by the insertion of new points into the set and in many cases the deletion of some existing points from the set. In this setting, the issue at hand is to update the box-counting result at appropriate time intervals with low computational cost. The proposed algorithm tackles the dynamic box-counting problem by using computational geometry methods. In particular, we use a sequence of compressed Box Quadtrees to store the data points. This storage permits the fast and efficient application of our box-counting approach to compute what we call the "dynamic fractal dimension". For a nonempty set of points in the d-dimensional space ℝ^d (for constant d ≥ 1), the time complexity of the proposed algorithm is shown to be O(n log n) while the space complexity is O(n), where n is the number of considered points. In addition, we show that the time complexity of an insertion, or a deletion is O(log n), and that the above time and space complexity is optimal. Experimental results of the proposed approach illustrated on the well-known and widely studied Hénon map are presented.

The moment-based box counting method of multifractal analysis is widely used for estimating generalized dimensions, D_q, from two-dimensional grayscale images. An evaluation of the accuracy of this method is needed to establish confidence in the resulting estimates of D_q. We estimated D_q from q = -10 to +10 for 23 random geometrical multifractal fields with different grid sizes, and known analytical D_q versus q functions. The fields were transformed to give normalized grayscale values between zero and one. Comparison of the estimated and analytical functions indicated the moment-based box counting method overestimates D_q by as much as 6.9% when q ≪ 0. The root mean square error, RMSE, for the entire range of q values examined ranged from 7.81 × 10^-6 to 1.35 × 10^-1, with a geometric mean of 6.50 × 10^-3. The RMSE decreased with decreasing grid size and increasing heterogeneity. These trends appear to be largely due to the presence of zeros in the normalized grayscale fields. Variations in the slope of the log-transformed partition function, ln[χ(q,δ)], with box size resulted in the overestimation of D_q when q ≪ 0. An alternative procedure for estimating D_q was developed based on the numerical first derivatives of ln[χ(q,δ)]. Using this approach the maximum deviation in D_q values was only 1.2%, while the RMSE varied from 3.11 × 10^-6 to 2.72 × 10^-2, with a geometric mean of 2.57 × 10^-4. When analyzing normalized grayscale fields, moment-based estimates of D_q should be interpreted with care. An order of magnitude increase in the accuracy of D_q can be achieved for such fields if the numerical first derivatives of ln[χ(q,δ)] are used in the analysis instead of standard linear regression.

In this study, we illustrate the fractal nature of the wake shed by a periodically flapping filament. Such wake structure is a combination of primary vortex shedding resulting in the von Kármán vortex street, a series of concentrated vortex dipoles formed when the trailing edges of filaments reach their maximum amplitudes and small eddies form along the shear layer connected with the concentrated vortices due to the shear layer instability. The vortex dynamics of the flapping filament are visualized and imaged experimentally using a soap-film flow tunnel with a high-speed camera and a low pressure sodium lamp as a light source. The wake fractal geometry is measured using the standard box-counting method and it is shown that the fractal dimension of the soap pattern boundaries in the wake is D = 1.38 ± 0.05, which agrees well with those measured for fully developed turbulences and other shear flow phenomena. The invariant of the fractality in the wake induced by the flapping filament thus provides another illustration of the geometrical self-similarity and nonlinear dynamics of chaotic fluid flows.

Fractal geometry can adequately represent many complex and irregular objects in nature. The fractal dimension is typically computed by the box-counting procedure. Here I compute the box-counting and the Kaplan-Yorke dimensions of the 14-dimensional models of the Drosophila circadian clock. Clockwork Orange (CWO) is transcriptional repressor of direct target genes that appears to play a key role in controlling the dynamics of the clock. The findings identify these models as strange attractors and highlight the complexity of the time-keeping actions of CWO in light-day cycles. These fractals are high-dimensional counterexamples of the Kaplan-Yorke conjecture that uses the spectrum of the Lyapunov exponents.

The analysis of fractal patterns has grown during the past years, mainly due to the wide range of applications to diverse scientific areas where fractals have been explored. It turns out that the key tool to study the complexity of a given system is the Fractal Dimension (FD), since this is its main invariant which throws quite useful information about the complexity that it presents when being examined with enough level of detail. In the proposed method, we adopt a hyper-surface partition method which considers the hyper-surface as continuous and divides the image into non-overlapped blocks. We present a novel counting method in the RGB color domain. The experimental results demonstrate that the proposed method shows its robustness and it can be performed as a reliable FD estimation approach for the color images. The running time of the proposed method is much shorter than that of other algorithms.

Fractal Dimension (FD) estimation in digital image analysis has received much attention due to its dimensional significance and therefore has become an active area of research over the year. The earlier FD-based techniques often followed traditional box-counting and its different variation of differential box-counting (DBC) paradigms, in which the proper choice of box count has remained a major concern. However, most of the state-of-the-art DBC variants suffer from considerable limitations like over-counting (OC), under-counting (UC), and limited their application only to square-shaped images, and it is still a major research problem! In this backdrop, the current investigation proposes a generalized box-counting (graylevel invariant DBC); and compares it with other state-of-the-art techniques. The proposed model is evaluated on five benchmark texture datasets (which include real and generated synthetic images) and obtained better results than the existing methods and achieved all desired outcomes by eliminating both OC and UC problems. This algorithm works for any arbitrarily sized (both squared and rectangular) images. It gives a higher rate of accuracy in terms of less fitting error in detecting exact surface roughness from given datasets.

The bocage landscape has evolved very radically during the last fifty years with the agricultural developement. The shapes of the hedgerow lattice have many consequences for example, in hydrological or ecological functions of this landscape. So we attempt to characterize the network structure by means of fractal geometry. The box-counting method permits to get significantly different results to analyse this structure with fractal parameters (dimension, upper and lower cuts off). Combining them with other parameters, we plan to model the functionning of this landscape.