Enhanced BC Algorithm Incorporating a Novel Sampling Step and a Fractional Box Count

The Box-Counting (BC) method is one of the most commonly used algorithms for fractal dimension calculation of binary images in the fields of Engineering, Science, Medical Science, Geology and so on due to its simplicity and reliability. One of the issues related to fractal dimension is data sampling that involves a process where a certain size of box is taken from a given image and it has a direct effect on the precision of the fractal dimension estimation. The Geometric Step (GS) method, arithmetic step method, and divisor step method are the representative methods. The GS method is mainly used because of its efficiency. However, the GS method has some drawbacks in nature. If the image size is large, it provides insufficient data for regression analysis. It can be applied to the image of pixel size for 100 [%] pixel utilization. Application of the GS method to an image of may waste pixels in the calculation and degrade the estimation accuracy. In this thesis, a novel sampling method is proposed in order to resolve the shortcomings of the GS method on the basis of the intuitive observation that an estimate may have a higher degree of precision if more pixels are utilized in each step and a sufficiently large number of fitting data are guaranteed. The proposed sampling method is an improved version of the conventional GS method, called the modified GS (MGS) method. The MGS method selects some additional step sizes with higher pixel utilization rate among the middle values between the integer powers of 2 to constitute the overall step set with the GS method. Not all sampling methods including the MGS method can guarantee 100 [%] pixel utilization when the BC method is applied to images of an arbitrary size. This study suggests a novel fractional counting method to resolve the problem of pixel waste. The proposed counting method counts pixels of fractal within a discarded box (not of size) and adds its fractional count normalized by both the average pixel number of all boxes with size and step size to integer count. The performance of the enhanced BC method incorporating the MGS method and fractional counting method is verified on a set of deterministic fractal images whose theoretical dimensions are well known and compared it with those of the existing BC methods. The experimental results show that the proposed method outperforms the conventional BC method and triangle BC method.Contents List of Tables ⅲ List of Figures ⅳ Abstract ⅵ Chapter 1. Introduction 1.1 Motivation 1 1.2 Research objectives 3 1.3 Organization of the thesis 3 Chapter 2. Overview of Fractal Theory 2.1 Definition of fractal 5 2.2 Fractal dimension 7 2.3 Fractal geometry 9 2.3.1 Mandelbrot set and Julia set 10 2.3.2 Koch snowflake (Opened) 11 2.3.3 Apollonian gasket 12 2.3.4 Vicsek fractal 13 2.3.5 Sierpinski triangle 14 2.3.6 Rand cantor 15 2.3.7 Koch curve 85° 16 2.3.8 Sierpinski carpet 17 2.3.9 Hilbert curve 18 Chapter 3. Existing Box-Counting Methods 3.1 Conventional BC method 20 3.2 Triangle BC method 25 Chapter 4. Enhanced BC method 4.1 Existing sampling methods and their drawbacks 27 4.1.1 Sampling methods 27 4.1.2 Pixel utilization 30 4.1.3 Drawbacks of existing sampling methods 30 4.2 New sampling method 32 4.3 Fractional box count 35 4.4 Procedure of the enhanced BC method 38 Chapter 5. Experiments and Review 5.1 Experiments on deterministic fractal image 41 5.1.1 test image 41 5.1.2 Determination of 43 5.1.3 Experiment with images of pixels 44 5.1.4 Experiments on rotated image 45 5.1.5 Experiment with images of pixels 46 5.2 Experiments on non-deterministic fractal images 51 5.2.1 Converting color images to binary images 51 5.2.2 Coastline images 52 Chapter 6. Conclusion 56 References 58 Appendix 61Maste

A secure data authentication in wireless body area network for health monitoring using electrocardiogram-based key agreement

Wireless Body Area Network (WBAN) comprises of a set of biomedical sensors, which are implanted into or placed around a human body to serve a variety of network applications constantly. One of the applications, the ubiquitous health monitoring, has improved the ability of healthcare providers to deliver appropriate treatments to the patients either in hospitals or at homes. As the need of this application increases, several security issues also arise due to the nature of open wireless medium. Moreover, implementing an effective security mechanism uses a significant part of the available energy in a WBAN, whereby the sensors have limited resource constraints in terms of power consumption and memory space. Thus, this paper presents a new authentication protocol model that utilizes Electrocardiogram (ECG) signal as biometric as well as cryptographic key to ensure that the transmitted data are originated from the required WBAN. Due to the uniqueness and the permanence property of ECG signal, the proposed model is developed to achieve optimal security performance and required lightweight manners of to the resource-limited biomedical sensors. The simulation system is implemented based on the process of an improved fuzzy vault scheme with a new error correction algorithm, which results in reducing computational complexity, communication load and storage overhead when compared to several previous work

The fractal dimension of Islamic and Persian four-folding gardens

Since Benoit Mandelbrot (1924–2010) coined the term “fractal” in 1975, mathematical theories of fractal geometry have deeply influenced the fields of landscape perception, architecture, and technology. Indeed, their ability to describe complex forms nested within each other, and repeated towards infinity, has allowed the modeling of chaotic phenomena such as weather patterns or plant growth. Some human-designed patterns such as the ones developedby Islamic cultures have been found to follow similar principles of hierarchy, symmetry, and repetition. However, the application of these principles in the design of gardens is an underexplored field. This paper presents a comparative exploration of the four-fold garden design model—the chahár-bágh—typical of Persian and Islamic garden design by analyzing two case studies: Taj Mahal and Isfahan’s city plan. This four-fold pattern is known to not only have a religious reading but to be also linked with ideals of fair distribution. Using aninnovative compositional fractal analysis inspired by architecture, our results demonstrate that these gardens contain a high level of self-replication and scale invariance and that they exhibit a high fractal dimension. The novel application of this method of analysis to historical landscape plans allows us to assess to what extent fractal concepts were already in use before the European Renaissance and Mandelbrot’s explorations, and to speculate on their symbolism in the context of Islamic and Persian garden design. Specifically, we conclude that the fractal characteristics of these gardens might be intended as a representation of theinfinite divine but also of principles of fairness and equality. Moving forward, this approach could be applied to design spaces, namely in the infrastructural design of the urban fabric, which are both meaningful and environmentally just

On Multifractal Structure in Non-Representational Art

Multifractal analysis techniques are applied to patterns in several abstract expressionist artworks, paintined by various artists. The analysis is carried out on two distinct types of structures: the physical patterns formed by a specific color (``blobs''), as well as patterns formed by the luminance gradient between adjacent colors (``edges''). It is found that the analysis method applied to ``blobs'' cannot distinguish between artists of the same movement, yielding a multifractal spectrum of dimensions between about 1.5-1.8. The method can distinguish between different types of images, however, as demonstrated by studying a radically different type of art. The data suggests that the ``edge'' method can distinguish between artists in the same movement, and is proposed to represent a toy model of visual discrimination. A ``fractal reconstruction'' analysis technique is also applied to the images, in order to determine whether or not a specific signature can be extracted which might serve as a type of fingerprint for the movement. However, these results are vague and no direct conclusions may be drawn.Comment: 53 pp LaTeX, 10 figures (ps/eps

Differentiating population spatial behaviour using a standard feature set

Moving through space, consuming services at locations, transitioning and dwelling are all aspects of spatial behavior that can be recorded with unprecedented ease and accuracy using the GPS and other sensor systems on commodity smartphones. Collection of GPS data is becoming a standard experimental method for studies ranging from public health interventions to studying the browsing behavior of large non-human mammals. However, the millions of records collected in these studies do not lend themselves to traditional geographic analysis. GPS records need to be reduced to a single feature or combination of features, which express the characteristic of interest. While features for spatial behavior characterization have been proposed in different disciplines, it is not always clear which feature should be appropriate for a specific dataset. The substantial effort on subjective selection or design of feature may or may not lead to an insight into GPS datasets. In this thesis we describe a feature set drawn from three different mathematical heritages: buffer area, convex hull and its variations from activity space, fractal dimension of the recorded GPS traces, and entropy rate of individual paths. We analyze these features against six human mobility datasets. We show that the standard feature set could be used to distinguish disparate human mobility patterns while single feature could not distinguish them alone. The feature set can be efficiently applied to most datasets, subject to the assumptions about data quality inherent in the features

Chaos in music: historical developments and applications to music theory and composition

The Doctoral Dissertation submitted by Jonathan R. Salter, in partial fulfillment of the requirements for the degree Doctor of Musical Arts at the University of North Carolina at Greensboro comprises the following: 1. Doctoral Recital I, March 24, 2007: Chausson, Andante et Allegro; Tomasi, Concerto for Clarinet; Bartok, Contrasts; Fitkin, Gate. 2. Doctoral Recital II, December 2, 2007: Benjamin, Le Tombeau de Ravel ; Mandat, Folk Songs; Bolcom, Concerto for Clarinet; Kovacs, Sholem-alekhem, rov Fiedman! 3. Doctoral Recital III, May 3, 2009: Kalliwoda, Morceau du Salon; Shostakovich, Sonata, op. 94 (transcription by Kennan); Tailleferre, Arabesque; Schoen eld, Trio for Clarinet, Violin, and Piano. 4. Dissertation Document: Chaos in Music: Historical Developments and Applications to Music Theory and Composition. Chaos theory, the study of nonlinear dynamical systems, has proven useful in a wide-range of applications to scienti c study. Here, I analyze the application of these systems in the analysis and creation of music, and take a historical view of the musical developments of the 20th century and how they relate to similar developments in science. I analyze several 20th century works through the lens of chaos theory, and discuss how acoustical issues and our interpretation of music relate to the theory. The application of nonlinear functions to aspects of music including organization, acoustics and harmonics, and the role of chance procedures is also examined toward suggesting future possibilities in incorporating chaos theory in the act of composition. Original compositions are included, in both sheet music and recorded form

Genetic evolution and equivalence of some complex systems: fractals, cellular automata and lindenmayer systems

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid. Escuela Politécnica Superior, Departamento de Ingeniería informática.26-04-200

Sliding on a Massive Spinning Asteroid: Order and Chaos

In 2017 an interstellar visitor, \u27Oumuamua, was discovered by Robert Weryk at Haleakala Observatory, Hawai\u27i. Analysis of \u27Oumuamua\u27s light curve suggests it has a highly elongated shape. The dynamics of such irregularly shaped objects cannot be modeled with simple approximations; while Newton\u27s sphere theorem allows us to model the gravitational dynamics of spherically symmetric objects as point masses, a spherical approximation is insufficient to describe the interactions of highly elliptical objects like \u27Oumuamua -- and even Earth, which is not a true sphere but rather an oblate spheroid. Fortunately, there exist closed-form expressions for the gravitational potential of a spheroid. For this project, I use Mathematica to solve systems of nonlinear differential equations to model the motion of an object sliding on such a spheroid. Under certain conditions, the path of the slider becomes highly sensitive to initial conditions. This sensitivity manifests as chaos, which is apparent both qualitatively and quantitatively