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Fraktale Dimension. Boxcounting-Dimension[Bearbeiten] Bei der Boxcounting-Methode überdeckt man die Menge mit einem Gitter der Gitterbreite . Wenn die Zahl der von der Menge belegten Boxen ist, so ist die Box-Dimension Tatsächlich kann man andere Arten von Überdeckungen (Kreise bzw. Kugeln, sich überschneidende Quadrate, etc.) wählen und genauso berechnen, und das Ergebnis ist theoretisch dasselbe, in der numerischen Praxis (wenn man den Limes nicht ausrechnen kann) aber nicht unbedingt.

Yardstick-Methode[Bearbeiten] Diese Methode eignet sich nur für topologisch eindimensionale Mengen, also für Kurven. Und dem Radius dieser Kreise verfährt man weiter wie bei der Boxcounting-Methode. Minkowski-Dimension[Bearbeiten] Umgibt man eine Menge mit einer Minkowskiwurst der Dicke und misst deren -dimensionales Volumen , so lässt sich damit eine zu der Box-Dimension äquivalente Dimension definieren: Ähnlichkeits-Dimension[Bearbeiten] Mengen, die aus um den Faktor verkleinerten Versionen ihrer selbst bestehen, heißen selbstähnlich. .

Uncertainty exponent. Suppose we start with a random trajectory and perturb it by a small amount, , in a random direction. If the new trajectory ends up in a different basin from the old one, then it is called epsilon uncertain. If we take a large number of such trajectories, then the fraction of them that are epsilon uncertain is the uncertainty fraction, , and we expect it to scale exponentially with Thus the uncertainty exponent, , is defined as follows: The uncertainty exponent can be shown to approximate the box-counting dimension as follows: where N is the embedding dimension. C. Intrinsic dimension. In signal processing of multidimensional signals, for example in computer vision, the intrinsic dimension of the signal describes how many variables are needed to represent the signal.

For a signal of N variables, its intrinsic dimension M satisfies 0 ≤ M ≤ N. Usually the intrinsic dimension of a signal relates to variables defined in a Cartesian coordinate system. In general, however, it is also possible to describe the concept for non-Cartesian coordinates, for example, using polar coordinates. Example[edit] Let f(x1, x2) be a two-variable function (or signal) which is of the form f(x1,x2) = g(x1) for some one-variable function g which is not constant. A slightly more complicated example is f(x1,x2) = g(x1 + x2) f is still intrinsic one-dimensional, which can be seen by making a variable transformation x1 + x2 = y1 x1 - x2 = y2 which gives f(y1,y2) = g(y1) Since the variation in f can be described by the single variable y1 its intrinsic dimension is one. Formal definition[edit] g′(y)=g(By)

List of fractals by Hausdorff dimension. Deterministic fractals[edit] Random and natural fractals[edit] See also[edit] Notes and references[edit] Further reading[edit] Benoît Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman & Co; ISBN 0-7167-1186-9 (September 1982).Heinz-Otto Peitgen, The Science of Fractal Images, Dietmar Saupe (editor), Springer Verlag, ISBN 0-387-96608-0 (August 1988)Michael F.

External links[edit] Hausdorff dimension. In mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This means, for example, the Hausdorff dimension of a point is zero, the Hausdorff dimension of a line is one, and the Hausdorff dimension of the plane is two.

There are, however, many irregular sets that have noninteger Hausdorff dimension. The concept was introduced in 1918 by the mathematician Felix Hausdorff. Sierpinski triangle. Intuition[edit] The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. Every space filling curve hits some points multiple times, and does not have a continuous inverse. Formal definitions[edit] Hausdorff content[edit] In other words, . If.