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Leveraging Deterministic Chaos to Mitigate Combinatorial Explosions
Published in Larry B. Rainey, Mo Jamshidi, Engineering Emergence, 2018
Many real-world phenomena are self-similar (i.e., a smaller section of the whole can be scaled iteratively to represent the whole). Examples include trees, broccoli, cauliflower, and their respective branches; planets with multiple moons that can scale to a solar system with multiple planets; coastlines that can scale both upward and downward; and many other phenomena. Many phenomena can be represented by recursive generating functions, or Iterated Function Systems (IFS), which typically generate fractals [5,19,20]. The IFS generates a fractal shape by constructing many copies of itself and overlaying these to produce the fractal pattern. The copies are affine transformations of the original shape, and are typically contractive-affine, or a reduced size copy of the original. This IFS is the result of a set of contractive affine elements, by use of what is called the Hutchinson operator, which converges to a unique attractor [22]. The overlays of these affine transformations are mathematically the union of these functionally transformed shapes, which create a fractal topology.
Non-standardized Still Image Coding
Published in Yun-Qing Shi, Huifang Sun, Image and Video Compression for Multimedia Engineering, 2019
A fractal is a geometric form whose irregular details can be represented by some objects with different scale and angle, which can be described by a set of transformations such as affine transformations. Additionally, the objects used to represent the image irregular details have some form of self-similarity and these objects can be used to represent an image in a simple recursive way. An example of fractals is the Von Koch curve as shown in Figure 10.6. The fractals can be used to generate an image. The fractal image coding that is based on IFS is the inverse process of image generation with fractals; therefore, the key technology of fractal image coding is the generation of fractals with an IFS.
Nonstandard Still Image Coding
Published in Yun Q. Shi, Huifang Sun, for Multimedia Engineering, 2017
A fractal is a geometric form whose irregular details can be represented by some objects with different scale and angle, which can be described by a set of transformations such as affine transformations. Additionally, the objects used to represent the image irregular details have some form of self-similarity and these objects can be used to represent an image with simple recursive way. An example of fractals is Von Koch curve as shown in Figure 9.6. The fractals can be used to generate an image. The fractal image coding that is based on IFS is the inverse process of image generation with fractals; therefore, the key technology of fractal image coding is the generation of fractals with an IFS.
A Hybrid Fractal Metamaterial Antenna for Wireless Applications with Gain Enhancement
Published in IETE Journal of Research, 2023
Ritesh Kumar Saraswat, Mithilesh Kumar
The proposed hybrid fractal design is implemented by combining the Moore and Koch fractal curves. Fractal formation exhibiting the self-repetition phenomenon regarding iteration stages is identified with the Iterative Function System (IFS). IFS is a mathematical approach to provide the transformation, scaling, rotation formation for fractal geometries [18]. The transformation is defined by the following equation: where