The Science of Fractal Images

1 Fractals in nature: From characterization to simulation.- 1.1 Visual introduction to fractals: Coastlines, mountains and clouds.- 1.1.1 Mathematical monsters: The fractal heritage.- 1.1.2 Fractals and self-similarity.- 1.1.3 An early monster: The von Koch snowflake curve.- 1.1.4 Self-similarity and dimension.- 1.1.5 Statistical self-similarity.- 1.1.6 Mandelbrot landscapes.- 1.1.7 Fractally distributed craters.- 1.1.8 Fractal planet: Brownian motion on a sphere.- 1.1.9 Fractal flakes and clouds.- 1.2 Fractals in nature: A brief survey from aggregation to music.- 1.2.1 Fractals at large scales.- 1.2.2 Fractals at small scales: Condensing matter.- 1.2.3 Scaling randomness in time: 1/f?-noises.- 1.2.4 Fractal music.- 1.3 Mathematical models: Fractional Brownian motion.- 1.3.1 Self-affinity.- 1.3.2 Zerosets.- 1.3.3 Self-affinity in higher dimensions : Mandelbrot landscapes and clouds.- 1.3.4 Spectral densities for fBm and the spectral exponent ?.- 1.4 Algorithms: Approximating fBm on a finite grid.- 1.4.1 Brownian motion as independent cuts.- 1.4.2 Fast Fourier Transform filtering.- 1.4.3 Random midpoint displacement.- 1.4.4 Successive random additions.- 1.4.5 Weierstrass-Mandelbrot random fractal function.- 1.5 Laputa: A concluding tale.- 1.6 Mathematical details and formalism.- 1.6.1 Fractional Brownian motion.- 1.6.2 Exact and statistical self-similarity.- 1.6.3 Measuring the fractal dimension D.- 1.6.4 Self-affinity.- 1.6.5 The relation of D to H for self-affine fractional Brownian motion.- 1.6.6 Trails of fBm.- 1.6.7 Self-affinity in E dimensions.- 1.6.8 Spectral densities for fBm and the spectral exponent ?.- 1.6.9 Measuring fractal dimensions: Mandelbrot measures.- 1.6.10 Lacunarity.- 1.6.11 Random cuts with H ? 1/2: Campbell's theorem.- 1.6.12 FFT filtering in 2 and 3 dimensions.- 2 Algorithms for random fractals.- 2.1 Introduction.- 2.2 First case study: One-dimensional Brownian motion.- 2.2.1 Definitions.- 2.2.2 Integrating white noise.- 2.2.3 Generating Gaussian random numbers.- 2.2.4 Random midpoint displacement method.- 2.2.5 Independent jumps.- 2.3 Fractional Brownian motion : Approximation by spatial methods.- 2.3.1 Definitions.- 2.3.2 Midpoint displacement methods.- 2.3.3 Displacing interpolated points.- 2.4 Fractional Brownian motion : Approximation by spectral synthesis.- 2.4.1 The spectral representation of random functions.- 2.4.2 The spectral exponent ? in fractional Brownian motion.- 2.4.3 The Fourier filtering method.- 2.5 Extensions to higher dimensions.- 2.5.1 Definitions.- 2.5.2 Displacement methods.- 2.5.3 The Fourier filtering method.- 2.6 Generalized stochastic subdivision and spectral synthesis of ocean waves.- 2.7 Computer graphics for smooth and fractal surfaces.- 2.7.1 Top view with color mapped elevations.- 2.7.2 Extended floating horizon method.- Color plates and captions.- 2.7.3 The data and the projection.- 2.7.4 A simple illumination model.- 2.7.5 The rendering.- 2.7.6 Data manipulation.- 2.7.7 Color, anti-aliasing and shadows.- 2.7.8 Data storage considerations.- 2.8 Random variables and random functions.- 3 Fractal patterns arising in chaotic dynamical systems.- 3.1 Introduction.- 3.1.1 Dynamical systems.- 3.1.2 An example from ecology.- 3.1.3 Iteration.- 3.1.4 Orbits.- 3.2 Chaotic dynamical systems.- 3.2.1 Instability: The chaotic set.- 3.2.2 A chaotic set in the plane.- 3.2.3 A chaotic gingerbreadman.- 3.3 Complex dynamical systems.- 3.3.1 Complex maps.- 3.3.2 The Julia set.- 3.3.3 Julia sets as basin boundaries.- 3.3.4 Other Julia sets.- 3.3.5 Exploding Julia sets.- 3.3.6 Intermittency.- 4 Fantastic deterministic fractals.- 4.1 Introduction.- 4.2 The quadratic family.- 4.2.1 The Mandelbrot set.- 4.2.2 Hunting for Kc in the plane - the role of critical points.- 4.2.3 Level sets.- 4.2.4 Equipotential curves.- 4.2.5 Distance estimators.- 4.2.6 External angles and binary decompositions.- 4.2.7 Mandelbrot set as one-page-dictionary of Julia sets.- 4.3 Generalizations and extensions.- 4.3.1 Newton's Method.- 4.3.2 Sullivan classification.- 4.3.3 The quadratic family revisited.- 4.3.4 Polynomials.- 4.3.5 A special map of degree four.- 4.3.6 Newton's method for real equations.- 4.3.7 Special effects.- 5 Fractal modelling of real world images.- 5.1 Introduction.- 5.2 Background references and introductory comments.- 5.3 Intuitive introduction to IFS: Chaos and measures.- 5.3.1 The Chaos Game : 'Heads', 'Tails' and 'Side'.- 5.3.2 How two ivy leaves lying on a sheet of paper can specify an affine transformation.- 5.4 The computation of images from IFS codes.- 5.4.1 What an IFS code is.- 5.4.2 The underlying model associated with an IFS code.- 5.4.3 How images are defined from the underlying model.- 5.4.4 The algorithm for computing rendered images.- 5.5 Determination of IFS codes: The Collage Theorem.- 5.6 Demonstrations.- 5.6.1 Clouds.- 5.6.2 Landscape with chimneys and smoke.- 5.6.3 Vegetation.- A Fractal landscapes without creases and with rivers.- A.1 Non-Gaussian and non-random variants of midpoint displacement.- A.1.1 Midpoint displacement constructions for the paraboloids.- A.1.2 Midpoint displacement and systematic fractals: The Takagi fractal curve, its kin, and the related surfaces.- A.1.3 Random midpoint displacements with a sharply non-Gaussian displacements' distribution.- A.2 Random landscapes without creases.- A.2.1 A classification of subdivision schemes: One may displace the midpoints of either frame wires or of tiles.- A.2.2 Context independence and the "creased" texture.- A.2.3 A new algorithm using triangular tile midpoint displacement.- A.2.4 A new algorithm using hexagonal tile midpoint displacement.- A.3 Random landscape built on prescribed river networks.- A.3.1 Building on a non-random map made of straight rivers and watersheds, with square drainage basins.- A.3.2 Building on the non-random map shown on the top of Plate 73 of "The Fractal Geometry of Nature".- B An eye for fractals.- Dietmar Saupe.- C A unified approach to fractal curves and plants.- C.1 String rewriting systems.- C.2 The von Koch snowflake curve revisited.- C.3 Formal definitions and implementation.- D Exploring the Mandelbrot set.- B An eye for fractals.- Yuval Fisher.- D.1 Bounding the distance to M.- D.2 Finding disks in the interior of M.- D.3 Connected Julia sets.