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HW10
A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.
A fractal often has the following features: • It has a fine structure at arbitrarily small scales.v • It is too irregular to be easily described in traditional Euclidean geometric language. • It is self-similar (at least approximately or stochastically). • It has a Hausdorff dimension which is greater than its topological dimension. • It has a simple and recursive definition.
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms. Images of fractals can be created using fractal generating software. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, i.e., it is quite possible to zoom into an empty region and produce a blank image.
2) Fractal dimension
Image: The first four iterationsof the Koch snowflake
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal dimensions are the Rényi dimension, the Hausdorff dimension and packing dimension. Practically, the box-counting dimension and correlation dimension are widely used, partly due to their ease of implementation.
Although for some classical fractals all these dimensions do coincide, in general they are not equivalent. For example, the dimension of the Koch snowflakehas a topological dimension, but it is by no means a curve: the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but neither is it like a piece of the plane or any other. It could be said that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the speculation that its dimension might best be described in a sense by a number between one and two. This is just one simple way of imagining the idea of fractal dimension.
Fig.(1) Defining dimension from a unit object
There are two main approaches to generate a fractal structure. One is growing from a unit object (Fig. 1), and the other is to construct the subsequent divisions of an original structure, like the Sierpinski triangle (Fig.2). Here we follow the second approach to define the dimension of fractal structures.
If we take an object with linear size equal to 1 residing in Euclidean dimension D, and reduce its linear size by the factor 1 / l in each spatial direction, it takes N = lD number of self similar objects to cover the original object(Fig.(1)). However, the dimension defined by:
(where the logarithm can be of any base) is still equal to its topological or Euclidean dimension.[1] By applying the above equation to fractal structure, we can get the dimension of fractal structure (which is more or less the Hausdorff dimension) as a non-whole number as expected.
where N(ε) is the number of self-similar structures of linear size ε needed to cover the whole structure.
For instance, the fractal dimension of the Sierpinski triangle (Fig.(2)) is given by,
Fig.(2) Sierpinski triangle derived by recursively dividing original structure
3) Fractals and Chaos in Geology and Geophysics, Turcotte, Donald L., Cornell University, New York)
Abstract: Now in a greatly expanded second edition, this book relates fractals and chaos to a variety of geological and geophysical applications and introduces the fundamental concepts of fractal geometry and chaotic dynamics. In this new edition, Turcotte expands coverage of self-organized criticality and includes statistics and time series to provide a broad background for the reader. Topics include drainage networks and erosion, floods, earthquakes, mineral and petroleum resources, fragmentation, mantle convection, and magnetic field generation. The author introduces all concepts at the lowest possible level of mathematics consistent with their understanding, so that the reader requires only a background in basic physics and mathematics. He includes problems for the reader to solve. This book will appeal to a broad range of readers interested in complex natural phenomena.
4) The defocus error is essential, especially for a microscopic imaging system with a small depth of field. Jun Wang
Abstract: In this paper, we study correlation between fractal dimensions and contrast of the images. Our study reveals the peak of fractal dimensions from a sequence of images, having different translations along the optical axis, gives the location of the best focal plane. This is in agreement with the fact that a highest contrast of the image gives the best focal plane of an imaging system [7]. This proposed technique allows measurement of the defocus error with a sensitivity of a few microns, well within the system’s depth of field. This feature makes the technique potentially useful in an optical and electronic microscopic imaging system by reducing the defocus aberration.
5) Nonparametric trend estimation in the presence of fractal noise: Application to fMRI time-series analysis
Babak Afshinpour, Gholam-Ali Hossein-Zadeha, and Hamid Soltanian-Zadeh,
Abstract: Unknown low frequency fluctuations called “trend” are observed in noisy time-series measured for different applications. In some disciplines, they carry primary information while in other fields such as functional magnetic resonance imaging (fMRI) they carry nuisance effects. In all cases, however, it is necessary to estimate them accurately. In this paper, a method for estimating trend in the presence of fractal noise is proposed and applied to fMRI time-series. To this end, a partly linear model (PLM) is fitted to each time-series. The parametric and nonparametric parts of PLM are considered as contributions of hemodynamic response and trend, respectively. Using the whitening property of wavelet transform, the unknown components of the model are estimated in the wavelet domain. The results of the proposed method are compared to those of other parametric trend-removal approaches such as spline and polynomial models. It is shown that the proposed method improves activation detection and decreases variance of the estimated parameters relative to the other methods.
Keywords: Trend; Nonparametric estimation; fMRI time-series; Fractal noise
6) From Digital Circuit Analysis and Design with Simulink®Modeling
Steven T. Karris
Error detecting and correcting codes are codes that can detect and also correct errors occurring during the transmission of binary data.
Cyclic codes are very practical because of the simplicity with which can be synthesized and can be encoded and decoded using simple feedback shift registers. A class of cyclic codes is the well−known Bose−Chauduri codes and these codes are very effective in detecting and correcting randomly occurring multiple errors.
• The addition or subtraction operation when calculating polynomials in CRCs is the modulo−2 addition and it is performed by Exclusive−OR gates.
A positive pulse is a waveform in which the normal state is logic 0 and changes to logic 1 momentarily to produce a clock pulse. A negative pulse is a waveform in which the normal state is logic 1 and changes to logic 0 momentarily to produce a clock pulse. In either case, the positive edge is the transition from 0 to 1 and the negative edge is the transition from 1 to 0.
• Edge triggering uses only the positive or negative edge of the clock pulse. Triggering occurs during the appropriate clock transition. Edge triggered flip flops are employed in applications where incoming data may be random. The SN74LS74 IC device is a positive edge triggered D−type flip flop. The SN74LS70 device is a JK flip flop which is triggered at the positive edge of the clock pulse. The SN74LS76 IC device is a JK flip flop which is triggered at the negative edge of the clock pulse. The SN74LS70 JK flip flop is preferable in applications where the incoming data is not synchronized with the clock whereas the SN74LS76 JK flip flop is more suitable for synchronous operations such as synchronous counters.
• The analysis of synchronous sequential logic circuits is facilitated by the use of state tables which consist of three vertical sections labeled present state, flip flop inputs, and next state. The present state represents the state of each flip flop before the occurrence of a clock pulse. The flip flop inputs section lists the logic levels (zeros or ones) at the flip flop inputs which are determined from the given sequential circuit. The next state section lists the states of the flip flop outputs after the clock pulse occurrence.
• The design of synchronous counters is facilitated by first constructing a state diagram and then the state table consisting of the present state, next state, and flip flop inputs sections. The present state and next state sections of the state table are easily constructed from the state diagram. The flip flops input section is derived from the present state and next state sections with the aid of the transition tables.
• A register is a group of binary storage devices such as flip flops used to store words of binary information. thus, a group of n flip flops appropriately connected forms an n−bit register capable of storing n bits of information. A register may also contain some combinational circuitry to perform certain tasks. Transfer of information from one register to another can be either synchronous transfer or asynchronous transfer.
• Binary words can be transferred from one register to another either in parallel or serial mode. In a parallel transfer all bits of the binary word are transferred simultaneously with a single transfer command. Some registers are provided with an additional command line referred to as the read command in addition to the transfer command which is often referred to as the load command.
