The approximated fractal dimension estimation methods can be split into two approaches. In the digital representation, the fractals are shown in a discrete space (image grid) which derails the Hausdorff dimension calculus once the fractal is handled as a disconnected set of points, compromising seriously the resultant dimension. These methods became still more important with the representation of fractal objects in digital images. As a consequence, along the years, a lot of numerical methods were developed in order to estimate the fractal dimension in these cases. Such a need is not satisfied for example in some Lindenmayer fractals as the three-dimensional weeds.
Indeed, as the Hausdorff dimension is calculated by a covering process using infinitesimal sets, it is necessary for this calculus to know a well defined mathematical rule used in the formation of the fractal. This concept of dimension is an analytical measure whose calculus uses to be complex and even impossible in many cases. The fractal dimension concept was originally defined in as being the Hausdorff dimension of the fractal object. The self-similarity property is also widely observable in natural objects such as the leaves of a tree or the stream of a river and this fact strongly encourages researchers in using fractal dimension to characterize objects from the real world, as in the applications previously cited. As a consequence of this multiscale characteristic, this measure can still be used to detect the level of self-similarity in the analyzed fractal object, that is, it shows how much the object is composed of similar copies of itself at different scales, an intrinsic characteristic of fractals. Thus, the fractal dimension characterizes the fractal object with a number which is not dependent on the observation scale. The importance of the fractal dimension and consequently its large number of applications can be explained by the fact that this measure gauges, at different scales, how the fractal fills the space in which it is immersed. The examples of fractal dimension applications are in a wide range and can be found in areas as distinct as medicine, texture analysis, geology, botany, , materials engineering, electronic, physics, histology, soil analysis, plant diseases, polymer analysis, among many others. In the last years, fractal dimension has become an important tool for characterizing objects, forms and surfaces in some areas of science.
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