Authors | |

Publisher | Productivity Press Inc |

Year | 01/12/2019 |

Edition | First |

Pages | 194 |

Version | hardback |

Readership level | General/trade |

Language | English |

ISBN | 9781498741354 |

Categories | Differential calculus & equations, Geometry, Analytic topology, Applied mathematics |

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Most books on fractals focus on deterministic fractals as the impact of incorporating randomness and time is almost absent. Further, most review fractals without explaining what scaling and self-similarity means. This book introduces the idea of scaling, self-similarity, scale-invariance and their role in the dimensional analysis. For the first time, fractals emphasizing mostly on stochastic fractal, and multifractals which evolves with time instead of scale-free self-similarity, are discussed. Moreover, it looks at power laws and dynamic scaling laws in some detail and provides an overview of modern statistical tools for calculating fractal dimension and multifractal spectrum.

Fractals

Preface

Scaling, Scale-invariance and Self-similarity

Dimensions of physical quantity

Buckingham Pi-theorem

Examples to illustrate significance of P-theorem

Similarity

Self-similarity

Dynamic scaling

Scale-invariance: Homogeneous function

Power-law distribution

**Fractals**

Introduction

Euclidean geometry

**Fractals**

Space of Fractal

Construction of deterministic fractals

Stochastic Fractal

Introduction

A brief description of stochastic process

Dyadic Cantor set (DCS): Random fractal

Kinetic Dyadic Cantor set

Stochastic dyadic Cantor set

Numerical Simulation

Stochastic fractal in aggregation with stochastic self-replication

Discussion and summary

Multifractality

Introduction

The Legendre transformation

Theory of multifractality

Multifractal formalism in fractal

Cut and paste model on Sierpinski carpet

Stochastic multifractal

Weighted planar stochastic lattice model

Algorithm of the weighted planar stochastic lattice (WPSL)

Geometric properties of WPSL

Geometric properties of WPSL

Multifractal formalism in kinetic square lattice

Fractal and Multifractal in Stochastic Time Series

Introduction

Concept of scaling law, monofractal and multifractal time series

Stationary and Non-stationary time series

Fluctuation analysis on monofractal stationary and non-stationary time series

Fluctuation analysis on stationary and non-stationary multifractal time series

Discussion

Application in Image Processing

Introduction

Generalized fractal dimensions

Image thresholding

Performance analysis

Medical image processing

Mid-sagittal plane detection

References

Scaling, Scale-invariance and Self-similarity

Dimensions of physical quantity

Buckingham Pi-theorem

Examples to illustrate significance of P-theorem

Similarity

Self-similarity

Dynamic scaling

Scale-invariance: Homogeneous function

Power-law distribution

Introduction

Euclidean geometry

Space of Fractal

Construction of deterministic fractals

Stochastic Fractal

Introduction

A brief description of stochastic process

Dyadic Cantor set (DCS): Random fractal

Kinetic Dyadic Cantor set

Stochastic dyadic Cantor set

Numerical Simulation

Stochastic fractal in aggregation with stochastic self-replication

Discussion and summary

Multifractality

Introduction

The Legendre transformation

Theory of multifractality

Multifractal formalism in fractal

Cut and paste model on Sierpinski carpet

Stochastic multifractal

Weighted planar stochastic lattice model

Algorithm of the weighted planar stochastic lattice (WPSL)

Geometric properties of WPSL

Geometric properties of WPSL

Multifractal formalism in kinetic square lattice

Fractal and Multifractal in Stochastic Time Series

Introduction

Concept of scaling law, monofractal and multifractal time series

Stationary and Non-stationary time series

Fluctuation analysis on monofractal stationary and non-stationary time series

Fluctuation analysis on stationary and non-stationary multifractal time series

Discussion

Application in Image Processing

Introduction

Generalized fractal dimensions

Image thresholding

Performance analysis

Medical image processing

Mid-sagittal plane detection

References