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2 triangles, example to show how fractal compression works. Fractal compression is a lossy compression method for digital images, based on fractals.The method is best suited for textures and natural images, relying on the fact that parts of an image often resemble other parts of the same image. [1]
Fractal branching of trees. Fractal analysis is assessing fractal characteristics of data.It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from phenomena including topography, [1] natural geometric objects, ecology and aquatic sciences, [2] sound, market fluctuations ...
One well-known method of determining lacunarity for patterns extracted from digital images uses box counting, the same essential algorithm typically used for some types of fractal analysis. [1] [4] Similar to looking at a slide through a microscope with changing levels of magnification, box counting algorithms look at a digital image from many ...
Sierpiński Carpet - Infinite perimeter and zero area Mandelbrot set at islands The Mandelbrot set: its boundary is a fractal curve with Hausdorff dimension 2. (Note that the colored sections of the image are not actually part of the Mandelbrot Set, but rather they are based on how quickly the function that produces it diverges.)
Hausdorff dimension (exact value) Hausdorff dimension (approx.) Name Illustration Remarks Calculated: 0.538: Feigenbaum attractor: The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic map for the critical parameter value =, where the period doubling is infinite.
f 3 x n + 1 = 0.2 x n − 0.26 y n y n + 1 = 0.23 x n + 0.22 y n + 1.6. This transformation encodes the self-similarity of the entire fern with the bottom left leaf. In the matrix representation, it is seen to be a near-90° counterclockwise rotation, scaled down to approximately 30% size with a translation in the positive y direction. In the ...
In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical traps are points, lines, circles, flower shapes and even raster images. Orbit traps are typically used to colour two dimensional fractals ...
[1] Fractal generating software creates mathematical beauty through visualization. Modern computers may take seconds or minutes to complete a single high resolution fractal image. Images are generated for both simulation (modeling) and random fractals for art. Fractal generation used for modeling is part of realism in computer graphics. [2]