Fractals

In this paper, the fractal features of natural images are used to construct an image authentication scheme, which can detect whether an image is maliciously tampered (cutting, wiping, modification, etc.) or not and can even locate the tampered regions. For the original image, the fractal transformation is applied to each of the image blocks, and some of the transformation parameters are quantized and used as the authentication code. The authentication code can be stored or transmitted secretly. To authenticate an image, the new authentication code is computed from the image with the similar method, and then compared with the stored or received code. A metric is proposed to decide whether an image block is tampered or not. Comparative experiments show that the authentication scheme can detect malicious tampering, is robust against such common signal processing as JPEG compression, fractal coding, adding noise or filtering, and thus, obtains competent performances compared with existing image authentication schemes.

We present a general result on absolute continuity of biased non-uniform self-similar measure on the real line, given by n different contraction and translation rates. The result holds generically in the sense of Lebesgue measure on a certain part of the parameter domain.

The decision problem of satisfiability of Boolean expression in k-conjunctive normal form (kSAT) is a typical NP-complete problem. In this paper, by mapping the whole Boolean expressions in k-conjunctive normal form onto a unit hypercube, we visualize the problem space of kSAT. The pattern of kSAT is shown to have self-similarity which can be deciphered in terms of graph directed iterated function system. We provide that the Hausdorff dimension of the pattern of kSAT is equal to the box-counting dimension and increases with k. This suggests that the time complexity of kSAT increases with k.

The core of this research is related to the human crowd problem. Some major problems are congestion, emergency evacuation, and fatal catastrophe. In fact, it has been realized that many crowd related problems can be resolved by influencing (controlling) human flow with providing various control measures. Thus, the problem itself becomes the motivation for this research and the solution is approached through model and simulation within the virtual environment.

The relationship between fractal pattern and crowd behavior is produced with respect to the crowd behavior model. It exposes a comparison between the crowd paths resulting from the model developed and the fractal formation created to imitate the crowd paths. A new approach for crowd behavior modeling is proposed based on fractal features and, thus, gives new understanding about a relationship between fractal pattern and crowd behavior. Several innovations of fractal patterns that can be used in crowd behavior model applications adds to the novelty of the research contribution. Ultimately, the new method of modeling the crowd can be obtained and at the same time the new direction of crowd modeling research can be directed.

Ultrasonic transducers composed of a periodic piezoelectric composite are generally accepted as the design of choice in many applications. Their architecture is normally very regular and this is due to manufacturing constraints rather than performance optimization. Many of these manufacturing restrictions no longer hold due to new production methods such as computer controlled, laser cutting, and so there is now freedom to investigate new types of geometry. In this paper, the plane wave expansion model is utilized to investigate the behavior of a transducer with a self-similar architecture. The Cantor set is utilized to design a 2-2 configuration, and a 1-3 configuration is investigated with a Sierpinski carpet geometry. Ideally a single longitudinal mode in the thickness direction will drive the transducer in a piston-like fashion. In this paper it was found that by increasing the fractal generation level, the bandwidth surrounding the main thickness mode will increase, but there will be a corresponding reduction in the amplitude of the electrical conductance. It is also shown that a shift in the frequency of operation of the device can be achieved by altering the spatial periodicity of the electrical excitation.

This paper modifies the escape-time method, analyzes the process that the points inside a period bulb of the generalized Mandelbrot set (the generalized M set) after iterating a certain pre-period process finally enter into the periodic orbit. All the points with same pre-period form a circularity structure which shapes similar as the period bulb; as pre-period grows the circularity structure diffuses towards the period bulb boundary until crams the whole period bulb. Besides, we present a particular experimental observation of the rule of the pre-period bulb graphic of the generalized M set and its growing process and offer the theory proof.

We compared two models of aggregation which represent the sol gel transition of base-catalyzed colloidal gels. The two models are a Langevin dynamics approach of this transition and the DLCA model (diffusion limited cluster-cluster aggregation). For large numbers of particles, we compared the fractal dimension of the gelling network via the structure factor of the bulk. We observed that the Langevin dynamics approach did give more realistic fractal dimension than the DLCA model. This is due to the fact that Langevin dynamics evolves as a function of a physical time.

The construction of a host of interesting patterns over one and two dimensions, as transformations of multifractal measures via fractal interpolating functions related to simple affine mappings, is reviewed. It is illustrated that, while space-filling fractal functions most commonly yield limiting Gaussian distribution measures (bells), there are also situations (depending on the affine mappings' parameters) in which there is no limit. Specifically, the one-dimensional case may result in oscillations between two bells, whereas the two-dimensional case may give rise to unexpected circle map dynamics of an arbitrary number of two-dimensional circular bells. It is also shown that, despite the multitude of bells over two dimensions, whose means dance making regular polygons or stars inscribed on a circle, the iteration of affine maps yields exotic kaleidoscopes that decompose such an oscillatory pattern in a way that is similar to the many cases that converge to a single bell.

Recently, a lot of work has been devoted to the study of the Lipschitz equivalence between self-similar sets. Many related results on self-similar sets of the real line have been well-known under the condition that the similarity ratios equal each other. In this paper, we generalize some results to a more general setting. We mainly study the Lipschitz equivalence between two general Sierpinski carpets in ℝ².

In this work, we propose a new method to describe fractal shapes using parametric L-systems. This method consists of introducing scaling factors in the production rules of the parametric L-system grammars. We present a turtle monoid on which we base our calculations to show the exact mathematical relation between L-systems and iterated function systems (IFS); we then establish the conditions for the scaling factors to produce plants' and curves' fractal shapes from parametric L-systems. We demonstrate that with specific values of the scaling factors, we find the exact relationship established by Prusinkiewicz and Hammel between L-systems and IFS. Finally, we present some examples of fractal plant forms and curves created using parametric L-systems with scaling factors.