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Introduction | p. xi |
A Fractal World | p. 1 |
Fractals pervade into geography | p. 2 |
From geosciences to physical geography | p. 3 |
Urban geography: a big beneficiary | p. 6 |
Forms of fractal processes | p. 10 |
Some fractal forms that make use of the principle of allometry | p. 11 |
Time series and processes are also fractal | p. 12 |
Rank-size rules are generally fractal structures | p. 14 |
First reflections on the link between power laws and fractals | p. 14 |
Brief introduction into power laws | p. 15 |
Some power laws recognized before the fractal era | p. 17 |
Conclusion | p. 19 |
Auto-similar and Self-affine Fractals | p. 21 |
The rarity of auto-similar terrestrial forms | p. 22 |
Yet more classes of self-affine fractal forms and processes | p. 24 |
Brownian, fractional Brownian and multi-fractional Brownian motion | p. 25 |
Levy models | p. 32 |
Four examples of generalizations for simulating realistic forms | p. 35 |
Conclusion | p. 37 |
From the Fractal Dimension to Multifractal Spectrums | p. 39 |
Two extensions of the fractal dimension: lacunarity and codimension | p. 40 |
Some territorial textures differentiated by their lacunarity | p. 40 |
Codimension as a relative fractal dimension | p. 41 |
Some corrections to the power laws: semifractals, parabolic fractals and log-periodic distributions | p. 43 |
Semifractals and double or truncated Pareto distributions | p. 43 |
The parabolic fractal model | p. 45 |
Log-periodic distributions | p. 46 |
A routine technique in medical imaging: fractal scanning | p. 48 |
Multifractals used to describe all the irregularities of a set defined by measurement | p. 50 |
Definition and characteristics of a multifractal | p. 50 |
Two functions to interpret: generalized dimension spectrum and singularity spectrum | p. 52 |
An approach that is classical in geosciences but exceptional in social sciences | p. 54 |
Three potential generalizations | p. 56 |
Conclusion | p. 57 |
Calculation and Interpretation of Fractal Dimensions | p. 59 |
Test data representing three categories of fractals: black and white maps, grayscale Landsat images and pluviometric chronicle series | p. 60 |
A first incontrovertible stage: determination of the fractal class of the geographical phenomenon studied | p. 62 |
Successive tests using Fourier or .wavelet decompositions | p. 63 |
p. 73 | |
Some algorithms for the calculation of the fractal dimensions of auto-similar objects | p. 75 |
Box counting, information and area measurement dimensions for auto-similar objects | p. 75 |
A geographically inconclusive application from perception | p. 78 |
The fractal dimensions of objects and self-affine processes | p. 80 |
A multitude of algorithms | p. 80 |
High irregularity of decadal rainfall for Barcelona and Beirut | p. 84 |
Conclusion | p. 85 |
The Fractal Dimensions of Rank-size Distributions | p. 87 |
Three test series: rainfall heights, urban hierarchies and attendance figures for major French museums | p. 88 |
The equivalence of the Zipf, Pareto and Power laws | p. 89 |
Three strategies for adjusting the rank-size distribution curve | p. 92 |
A visual approach using graphs | p. 92 |
Adjusting the only linear part of the curve | p. 95 |
Choosing the best adjustment, and therefore the most pertinent law | p. 96 |
Which rank-size distribution should be used for Italian towns, the main French agglomerations and all French communes? | p. 98 |
Conclusion | p. 101 |
Calculation and Interpretation of Multifractal Spectrums | p. 103 |
Three data sets for testing multifractality: a chronicle series, a rank-size distribution and satellite images | p. 104 |
Distinguishing multifractal and monofractal phenomena | p. 104 |
An initial imperfect visual test | p. 105 |
A second statistical test: generalized correlation dimensions | p. 107 |
Various algorithms for calculation of the singularity spectrum | p. 111 |
Generalized box-counting and variogram methods | p. 111 |
Methods derived from wavelet treatment | p. 112 |
Interpretation of singularity spectrums | p. 113 |
Possible generalizations of the multifractal approach | p. 116 |
Conclusion | p. 118 |
Geographical Explanation of Fractal Forms and Dynamics | p. 121 |
Turbulence generates fractal perturbations and multifractal pluviometric fields | p. 122 |
The fractality of natural hazards and catastrophic impacts | p. 126 |
Other explanations from fields of physical geography | p. 128 |
A new geography of populations | p. 129 |
Harmonization of town growth distributions | p. 131 |
Development and urban hierarchies | p. 132 |
Understanding the formation of communication and social networks | p. 136 |
Conclusion | p. 137 |
Using Complexity Theory to Explain a Fractal World | p. 139 |
A bottomless pit debate | p. 140 |
General mechanisms for explaining power laws | p. 143 |
Four theories on fractal universality | p. 144 |
Critical self-organization theory | p. 144 |
Bejan's constructal theory | p. 151 |
Nottale's scale relativity theory | p. 153 |
A general theory of morphogenesis | p. 154 |
Chaos and fractal analysis theory | p. 163 |
Conclusion | p. 164 |
Land-use Planning and Managing a Fractal Environment | p. 167 |
Fractals, extreme values and risk | p. 168 |
Under estimated hazards in preliminary risk assessments | p. 168 |
Fractal networks, fighting epidemics and Internet breakdowns | p. 171 |
Fractals, segmentation and identification of objects in image processing | p. 173 |
New image processing tools | p. 173 |
p. 177 | |
Fractals, optimization and land management | p. 177 |
Fractal beauty and landscapinng | p. 179 |
Conclusion | p. 180 |
Conclusion | p. 183 |
Some tools and methods for quantifying and qualifying multiscale coarseness and irregularity | p. 184 |
A recap on geographical irregularities and disparities | p. 186 |
A paradigm that gives rise to new land-use management practices | p. 189 |
Appendices | p. 191 |
Preliminary thoughts on fractal analysis software | p. 191 |
Instructions for the following programs | p. 192 |
Software programs for the visual approach of a satellite or cartographic series or image | p. 193 |
Software programs for calculating fractal dimensions for a chronicle or frequency series | p. 198 |
Software programs for calculating the fractal dimensions of a satellite image or map | p. 208 |
Software programs for calculating multifractal spectrums of a series and an image | p. 213 |
Bibliography | p. 221 |
Index | p. 239 |
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