Introductions to fractals



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Introductions to fractals

  • Introductions to fractals

  • Fractals in architecture

  • Introduction to multifractals

  • Multifractals in real world

    • Application in biomedical engeenering
    • Application in acoustics
    • Application in video processing


The fractal concept has been introduced by Benoit Mandelbrot in the middle of last century.

  • The fractal concept has been introduced by Benoit Mandelbrot in the middle of last century.

  • Fractals can be defined as structures with scalable property or as set of objects, entities that are similar to the whole unit.



Fractals have self-similarity property.

  • Fractals have self-similarity property.

  • A structure is self-similar if it has undergone a transformation whereby the dimensions of the structure were all modified by the same scaling factor.

  • Relative proportions of the shapes sides and internal angles remain the same.



Two types of fractals:

  • Two types of fractals:

  • Deterministic fractals : artifitial fractals generated using specific rule for transformation (self-similarity exist in all scales).

  • Random fractals: Nature fractals with self-similarity properties in limited range of scales.



Cantor Set

  • Cantor Set



Von Koch kriva

  • Von Koch kriva



Von Koch pahuljica

  • Von Koch pahuljica





Fractal dimension is describing how a set of items are filing the 'space'

  • Fractal dimension is describing how a set of items are filing the 'space'

  • Three types of Fractal dimension:

  • Self-similarity dimension (Ds)

  • Measured dimension (d)

  • Box-counting dimension (Db)



Self-similarity dimension (Ds):

  • Self-similarity dimension (Ds):

  • Measured dimension (d)

    • Set of strate line segments which cover the curve of fractal structure.
    • Smaller segments, better approximation of structure curve.


Box-counting dimension (Db)

  • Box-counting dimension (Db)



Cantor Set

  • Cantor Set



Von Koch kriva

  • Von Koch kriva













Fractal dimension is not the same in all scales

  • Fractal dimension is not the same in all scales



Presents the way of describing irregular objects and phenomena.

  • Presents the way of describing irregular objects and phenomena.

  • Multifractal formalism is based on the fact that the highly nonuniform distributions, arising from the nonuniformity of the system, often have many scalable features including self-similarity describing irregular objects and phenomena.



Studying the so-called long-term dependence (long range dependency), dynamics of some physical phenomena and the structure and nonuniform distribution of probability,

  • Studying the so-called long-term dependence (long range dependency), dynamics of some physical phenomena and the structure and nonuniform distribution of probability,

  • MA can be used for characterization of fractal characteristics of the results of measurements.

  • Multifractal analysis studies the local and global irregularities of variables or functions in a geometrical or statistical way.

  • Multifractal formalism describes the statistical properties of these singular results of measurements in the form of their generalized dimensions (local property) and their singularity spectrum (global)



There are several ways to determine the multifractal parameters and one of the most common is called box-counting method.

  • There are several ways to determine the multifractal parameters and one of the most common is called box-counting method.





Random signals (self-similarity).

  • Random signals (self-similarity).

  • PMV versus Healthy classification

  • PMV (Prolaps Mitral Valve) heart beat anomaly.

  • PMV signal has weak statistical properties.









Random signals (self-similarity).

  • Random signals (self-similarity).

  • Detection of early reflections in room impulse response

  • Aplication of Inverse MA.

  • Signal is tranform into MA alpha domain.

  • Detection of reflections is performed on alpha values.







Random signals (self-similarity)

  • Random signals (self-similarity)

  • Shot boundary detection

  • Color and texture features are extracted from video frames.

  • Inverse MA is implemented on time series of specific feature elements.








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