Measuring Fractal Dimension: Morphological Estimates And Iterative Optimization (original) (raw)
Related papers
Estimation of fractal dimension through morphological decomposition
Chaos, Solitons & Fractals, 2004
Set theory based morphological transformations have been employed to decompose a binary fractal by means of discrete structuring elements such as square, rhombus and octagon. This decomposition provides an alternative approach to estimate fractal dimensions. The fractal dimensions estimated through this morphological decomposition procedure by employing different structuring elements are considerably similar. A color-coding scheme is adapted to identify the several sizes of decomposed non-overlapping disks (DNDs) that could be fit into a fractal. This exercise facilitates to test the number-radius relationship from which the fractal dimension has been estimated for a Koch Quadric, which yield the significantly similar values of 1.67 ± 0.05 by three structuring elements. In addition to this dimension, by considering the number of DNDs of various orders (radii) and the mean diameter of disks (MDDs) of corresponding order, two topological quantities namely number ratio (R B) and mean diameter ratio (R L) are computed, employing which another type of fractal dimension is estimated as log R B log R L. These results are in accord with the fractal dimensions computed through number-radius relationship, and connectivity network of the Koch Quadric that is reported elsewhere.
Determination of fractal dimensions for geometrical multifractals
Physica A: Statistical Mechanics and its Applications, 1989
Two independent approaches, the box counting and the sand box methods are used for the determinatiov of the generalized dimensions (Dq) associated with, the geometrical structure of growing deterministic fractals. We find that the muitifractal nature of the geometry results in an unusually slow convergence of the numerically calculated Dq's to their true values. Our study demonstrates that the above-mentioned two methods are equivalent only if the sand box method is applied with an averaging over randomly selected centres. In this case the latter approach provides better estimates of the generalized dimensions.
A study of the different methods usually employed to compute the fractal dimension
2002
The fractal dimension (FD) is a widely used magnitude having a basic formulation in terms of the Hausdor measure. However, there are a lot of practical deÿnitions mostly used to compute the FD of a given system not yet having a mathematically rigorous approach. In this paper, we analyze these alternative FDs and present a mathematical formalism for all of them obtaining practical expressions that can be used to compute each FD so far described. The conditions for applying these expressions are also analyzed. Finally, we have applied all the deÿnitions to compute the FD of fractal aggregates formed at the air-liquid interface.
Measurements of fractal dimension by box-counting: a critical analysis of data scatter
Physica A: Statistical Mechanics and its Applications, 1998
The multifractal concept was introduced in the 1980s by Mandelbrot. This theory arose from the analysis of complex and=or discontinuous objects. In this study, we analyzed the data scatter obtained by a modiÿed box-counting method. Considering the curved shape of the data scatter, it is noticeable that there is more than one slope corresponding to di erent fractal behavior of an object. In this work, to discriminate di erent fractal dimensions from data scatter obtained by box counting, we suggest a rigorous selection of data points. The results show that large "'s usually characterize the embedding surface of the whole object and that small "'s approximate the dimension of the substructure for discontinuous objects. They also show that a dimension can be associated with a density distribution of singularities.
Fractal Dimension Computational Methods
In this method the fractal object is covered with a grid of n-dimensional boxes with side lengths and then counting the number of boxes that contain a part of the fractal object, say N( ). The fractal surface is covered with boxes of recursively different sizes. The slope m is obtained from the logarithmic plot of the number of boxes used to cover the fractal against the box size and the fractal dimension D B is given by. D B=−m (1.1) This dimension is also known as the box or Minkowski dimension. For a smooth one-dimensional curve, it is expected that
Physica A: Statistical Mechanics and its Applications, 2001
In this paper we present an algorithm to estimate the Hausdor fractal dimension. The algorithm uses a recursive formula with a fast enough convergence. The accuracy of results is independent on the size, i.e., degree of deÿnition of the fractal set. This fact is particularly useful when studying real physical fractals with a low deÿnition, such as colloidal aggregates of small size. The di erent tests reveal no dependence of the results on the irregularities of the fractal. Thus, self-similarity or statistical similarity of the fractal set does not a ect results. The proposed algorithm gives correct values for all the fractal dimension of the tested sets. Finally, the algorithm was used to evaluate the HÃ enon attractor fractal dimension and was applied to an experimental system obtained from a two-dimensional aggregation of latex colloidal particles.
A Unified Approach To Fractal Dimensions
International Journal of Cognitive Informatics and Natural Intelligence, 2007
Many scientific papers treat the diversity of fractal dimensions as mere variations on either the same theme or a single definition. There is a need for a unified approach to fractal dimensions for there are fundamental differences between their definitions. This paper presents a new description of three essential classes of fractal dimensions based on: (1) morphology, (2) entropy, and (3) transforms, all unified through the generalized-entropy-based Rényi fractal dimension spectrum. It discusses practical algorithms for computing 15 different fractal dimensions representing the classes. Although the individual dimensions have already been described in the literature, the unified approach presented in this paper is unique in terms of (1) its progressive development of the fractal dimension concept, (2) similarity in the definitions and expressions, (3) analysis of the relation between the dimensions, and (4) their taxonomy. As a result, a number of new observations have been made, and new applications discovered. Of particular interest are behavioral processes (such as dishabituation), irreversible and birth-death growth phenomena (e.g., diffusion-limited aggregates (DLAs), dielectric discharges, and cellular automata), as well as dynamical non-stationary transient processes (such as speech and transients in radio transmitters), multi-fractal optimization of image compression using learned vector quantization with Kohonen's self-organizing feature maps (SOFMs), and multi-fractal-based signal denoising.
Review of the Software Packages for Estimation of the Fractal Dimension
2015
The fractal dimension is used in variety of engineering and especially medical fields due to its capability of quantitative characterization of images. There are different types of fractal dimensions, among which the most used is the box counting dimension, whose estimation is based on different methods and can be done through a variety of software packages available on internet. In this paper, ten open source software packages for estimation of the box-counting (and other dimensions) are analyzed, tested, compared and their advantages and disadvantages are highlighted. Few of them proved to be professional enough, reliable and consistent software tools to be used for research purposes, in medical image analysis or other scientific fields.
A comparison of fractal dimension algorithms using synthetic and experimental data
ISCAS'99. Proceedings of the 1999 IEEE International Symposium on Circuits and Systems VLSI (Cat. No.99CH36349), 1999
The fractal dimension (FD) of a waveform represents a powerful tool for transient detection. In particular, in analysis of electroencephalograms (EEG) and electrocardiograms (ECG), this feature has been used to identify and distinguish specific states of physiologic function. A variety of algorithms are available for the computation of FD. In this study, the most common methods of estimating the FD of biomedical signals are analyzed and compared. The analysis is performed over both synthetic data and intracranial EEG (IEEG) data recorded during pre-surgical evaluation of individuals with epileptic seizures. The advantages and drawbacks of each technique are highlighted. The effects of window size, number of overlapping points, and signal to noise ratio (SNR) are evaluated for each method. This study demonstrates that a careful selection of FD algorithm is required for specific applications.
Practical application of fractal analysis: problems and solutions
Geophys. J. Int., 132, 275–282, 1998
Fractal analysis is now common in many disciplines, but its actual application is often a¡ected by methodological errors which can bias the results. These problems are commonly associated with the evaluation of the fractal dimension D and the range of scale invariance R.