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

Numerical Studies on Collective Motion and Polymer Statistics

Ph.DDOCTOR OF PHILOSOPH

Coordination et robustesse des systèmes dynamiques multi-agents

This thesis presents a study on multi-agent systems. Such systems find numerous applications such as multi-vehicle control in robotics, the design of smart distributed energy networks and the modeling of opinion dynamics. In a first part, we present new results regarding consensus theory which extend the recent work from Hendrickx and Tsitsiklis on cut-balanced consensus. Then, we apply the consensus system to the control of a fleet of vehicles. We present several results regarding velocity alignment (flocking). This study is based upon a graph robustness analysis in order to preserve the connectivity of the interaction network. This concept is of main importance in this study. In the last part, we state results froma collaborative work with a sociologist regarding the social network linked to the controversy concerning o-road motorized leisure in France. We study the link between the national and local scenes. To do so, we use large graph visualization tools and actor centrality measures.Nous nous intéressons à l'étude de la dynamique des réseaux composés d'une multitude d'agents. Les motivations de ce travail trouvent leurs sources dans de nombreux domaines et notamment la biologie avec l'étude de l'émergence de comportements collectifs cohérents chez les animaux (vol en formation d'oiseaux migrateurs). Considérons un certain nombre d’agents (animaux) dont le comportement dynamique individuel peut être modélisé par une équation différentielle. Les agents communiquent : les liens sont représentés sous la forme d’un graphe dont les sommets sont les agents du système. Chaque agent a la connaissance de l’état des agents auxquels il est connecté et ajuste sa dynamique à l’aide de cette information. Des comportements collectifs peuvent alors émerger comme par exemple le phénomène de flocking (tous les agents se déplacent dans la même direction). Plusieurs modèles d'interaction ont été proposés, les plus connus étant le modèle de Viscek (1995) ou le modèle de Cucker-Smale (2007). L'étude de ces modèles repose généralement sur des méthodes d'analyse de stabilité des systèmes dynamiques ou des systèmes hybrides, lorsque le graphe de communication évolue dans le temps. Nous souhaitons dans cette thèse évaluer la robustesse de l'émergence de ces comportements collectifs en étudiant l'influence de divers facteurs: paramètres du modèle, topologie du graphe, nombre d'agents, présences de perturbations. Nous nous intéresserons notamment au phénomène de scission du groupe d'agents en plusieurs groupes d'agents coordonnés