Wavelets and fractals: overview of their similarities based on different application areas (original) (raw)
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On the wavelet transformation of fractal objects
Journal of Statistical Physics, 1988
The wavelet transformation is briefly presented. It is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half-plane near the boundary of its domain of analyticity. As an example, a “Weierstrass-like” fractal function is considered, for which the wavelet transform is related to a Jacobi theta function. Some of the scalings of this theta function are analyzed, and give some information about the scaling behavior of this fractal.
Fractals Study and Its Application
Proceeding of the Electrical Engineering Computer Science and Informatics
The overall of this paper is a review of fractal in many areas of application. The review exposes fractal definition, analysis, and its application. Most applications discussed are based on analysis from geometric and image processing studies. Patterns of some fractals will be discussed. Some simulation results are supplied to illustrate the discussion. Simulation resulted are from various software and tools. Some principles of fractals with informative patterns have been simulated. Whereas the simulations could support some recommendations for prospective purposes and applications. The prospective application may help in predictive pattern of many fields. The predictive pattern will lead to pattern control and pattern disruptions.
Fractal analysis of aftershock sequence of the Bhuj earthquake: A wavelet-based approach
2005
Wavelet-based fractal analysis of the aftershock data of the disastrous Bhuj earthquake (26 January 2001) has been carried out to understand the behaviour of the aftershocks. The Omori's curve suggests that the decay of aftershocks follows a power law relation with time. The p-value, known as 'decay constant' is found to be 0.88. The b-value of the region as obtained by wavelet variance analysis is close to 1, which is in agreement with the normal value for tectonically active regions. The results obtained from the multi-scale analysis of the aftershock sequence using the wavelets indicate that the fractal behaviour of aftershock data sustains up to certain scales only. Using the wavelet variance, the fractal dimension of the source is obtained as 2.06, which indicates that it is a 2D plane that is being filled up by the source fractures. The slip ratio that determines the fraction of total slip occurring on the primary fault is computed to be 0.48, which reveals that 48% of the total slip has occurred on the primary fault.
Facies Recognition Using Wavelet Based Fractal Analysis On Compressed Seismic Data
6th International Congress of the Brazilian Geophysical Society
We have used a Wavelet Based Fractal Analysis (WBFA) and a Waveform Classifier (WC) to recognize lithofacies at the Oritupano A field (Oritupano-Leona Block, Venezuela). The WBFA was applied first to Sonic, Density, Gamma Ray and Porosity well logs in the area. The logs that give the best response to the WBFA are the Gamma Ray and NPHI (porosity) logs. In the case of the logs, the lithological content could be associated to the fractal parameters: slope, intercept and fractal dimension. The map obtained using the fractal dimension shows tendencies that generally agree with the depositional patterns previously observed in conventional geological maps. According to the results obtained in this study, zones with fractal dimension values lower than 0.9 correspond to sandstone channels. Values between 0.9 and 1.2 coincide with the interdistributary deltaic shelf and values greater than 1.2 might be associated with zones of greater shale content. The WBFA and WC results obtained for the seismic data show no relation with the lithofacies. The lost of low and high frequencies in these seismic data, as well as phase problems, could be the reasons for this behavior.
Fractal representation of images via the discrete wavelet transform
Eighteenth Convention of Electrical and Electronics Engineers in Israel, 1995
Fractal representation of images is based on mappings between similar regions within an image (also known as IFS). Such a representation can be applied to image coding and to increase image resolution. One o f the main drawbacks of conventional fractal representation is the fact that the mappings are between blocks. As a result, the reconstructed image may suffer from disturbing blockiness. In this work we present a method for mapping similar regions within an image in the wavelet domain. we first show how to use the Haar wavelet transform coefficients to find mappings which are identical to conventional blockwise mappings. The union o f these mappings, between sets of wavelet coefficients, can be interpreted as a prediction of higher bands of a signal from its lower band. Changing the mother-wavelet to other than Haar, creates mappings which are between regions which smoothly decay towards their borders, thus reducing the blockiness, as well as improving the PSNR of the reconstructed image.
The b value of earthquakes is very useful to forecast the occurrence of aftershocks in a given region. The b value characterizes the release of energy due to stress accumulation in the rocks through an earthquake and is a direct indicator for the prediction of aftershocks in the region. Wavelet based fractal analysis is used in this study to determine the b value by calculating the fractal dimension.This method guarantees high accuracy results through a limited dataset.The objective of this work was to demonstrate an elegant method for the determination of the b value after an earthquake and predict the occurrence of aftershocks with high accuracy. Repeated earthquakes were analyzed between 2003 and 2011 in Turkey and the b value was found for these earthquakes. The results gave an indication that the b value of the mainshock and its aftershocks are different and aftershocks occur in the region when the b value of the mainshock deviates significantly from0.5, and aftershocks keep occurring until the b value of the earthquake approaches close to 0.5 for this region.
Fractal/Wavelet representation of objects
2008 3rd International Conference on Information and Communication Technologies: From Theory to Applications, 2008
A Fractal model equipped with detail concept like the one used in wavelet transforms is introduced and used to represent objects in a more efficient way . This new representation can be used to deform object (locally and globally) and to manipulate the geometric texture of these objects. This fractal model based on Projected IFS attractors allows the definition of free form fractal shapes controlled by a set of points. The projected IFS is a type of IFS (Iterated Function System) which mixes free forms models with IFS models. The details concept idea taken from wavelet theory represents the geometric texture of the object. This concept is introduced by wavelet transform. The wavelet transform represents a signal in hierarchic manner. The signal is divided in two parts: one representing the signal in different scales, and the other representing the details of this signal. We proposed a model based on projected IFS and used the idea of details introduced by wavelet theory. An approximation step is first done to fit the model to the object, this step is formulated as a non-linear fitting problem and resolved using a modified Levenberg-Marquardt minimization method. Our goal is to change the representation of objects from an ordered set of data(points, pixels,..) to a set of control data and a vector of details such that this new representation facilitate the manipulation of objects. In this work, we focus on 2D curves.
FRACTALS AND FRACTAL DESIGN IN ARCHITECTURE
Fractal geometry defines a rough or fragmented geometric shapes that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. In short, irregular details or patterns are repeated themselves in even smaller scale. Fractal geometry deal with the concept of self-similarity and roughness in the nature. The most important properties of fractals are repeating formations, self-similarity, a non-integer dimension, and so called fractional size which can be defined by a parameter in irregular shapes. Fractals are formed by a repetition of patterns, shapes or a mathematical equation. Formation is dependent on the initial format. Not only in nature, fractals are also seen in the study of various disciplines such as physics, mathematics, economics, medicine and architecture. For a variety of reasons, in different cultures and geography, many times the fractal pattern had reflected on creating the architecture. In the computer-aided architectural design area, fractals are considered as a subset for the representation of knowledge for design aid and syntactic science of the grammatical form. If compared with the grammar of shapes, the number of rules used in the production process of fractals is defined as less, with number of repetitions as more and self-similarity feature, it can be a tool to help qualified geometric design. A simple form produced with fractal geometry with ultimate repetition is being transformed into an algorithmic complex. This algorithm with an initial state and a production standard that applies to this initial state produces self-similar formats. In this study, the development of the fractals from the past to the present, the use of fractals in different research areas and the investigation of examples of fractal properties in the field of architecture has been researched.
Analysis of Image Compression Using Fractal Wavelet Techniques
—In this paper we show the two implementations of fractal (Pure-fractal and Wavelet fractal image compression algorithms) which have been applied on the images in order to investigate the compression ratio and corresponding quality of the images using peak signal to noise ratio (PSNR). And in this paper we also set the threshold value for reducing the redundancy of domain blocks and range blocks, and then to search and match. By this, we can largely reduce the computing time. In this paper we also try to achieve the best threshold value at which we can achieve optimum encoding time. Keywords: Fractal image coding; Wavelet; Iterated Function System; Wavelet; Mean Square Error; Compression Ratio.