Multifractals in Real World ICT COLLEGE ICT COLLEGE OF VOCATIONAL STUDIES Multifractals in Real World Goran Zajic

Agenda Introductions to fractals Fractals in architecture Introduction to multifractals Multifractals in real world Application in biomedical engeenering Application in acoustics Application in video processing

Fractals 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.

Self-similarity 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.

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.

Fractals – Example 1 Cantor Set Data je linija. Podeli sa na 3. Line is divided into 3 parts. The central part is removed. Data je linija. Podeli sa na 3. Ukloni se srednji deo. The same rule is repeated for new created parts of original line. Ponavlja se procedura za svaki deo.

Fractals – Example 2 Von Koch kriva Van Koch Curve Line is divided into 3 parts. The central part is removed. Van Koch Curve New four segments.

Fractals – Example 3 Van Koch Snowflake Von Koch pahuljica asddadsdasdasdadasdasdasdsadsadasdas Von Koch pahuljica Line is divided into 3 parts. The casasas Van Koch Snowflake New four segments.

Fractals – Example 4 Sierpinski Carpet New nine quadratic fields. Central one is removed

Fractal dimension 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)

Fractal dimension 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. N – number of copies r < 1 – scaling ratio Connection between dimensions : Ds = d + 1

Fractal dimension Box-counting dimension (Db) DB(e) = lnN/ln N – number of colored boxes  - dimension of box DB(e) = lnN/ln DB(e)=1.278 DB(e0)=1.25

Fractals – Example 1 Cantor Set N – number of copies(2) r < 1 – scaling ratio (1/3) Line is divided into 3 parts. The central part is removed. Data je linija. Podeli sa na 3. Ukloni se srednji deo. The same rule is repeated for new created parts of original line. Ponavlja se procedura za svaki deo. D=1 (line), D<1 (fractal line)

Fractal line(1D signal): Fractals – Example 2 Von Koch kriva Line is divided into 3 parts. The central part is removed. Van Koch Curve New four segments. Fractal line(1D signal): 1<DS<2 Fractal surface (2D signal, slika): 2<DS<3 Fractal volume: 3<DS<4 N = 4, r =1/3

Fractals – Example 4 Sierpinski Carpet N =8 fields New nine quadratic fields. Central one is removed N =8 fields r =1/3 scaling ratio D=2 (surface) D<2 (fractal surface)

Introduction to fractals “Fractal is a structure, composed of parts, which in some sense similar to the whole structure” B. Mandelbrot

Introduction to fractals “The basis of fractal geometry is the idea of self-similarity” S. Bozhokin

Introduction to fractals “Nature shows us […] another level of complexity. Amount of different scales of lengths in [natural] structures is almost infinite” B. Mandelbrot

Fractals in Architecture Visualization of object in different planes and scale. Fractal dimension is used for object description and comparison.

Multifractals Fractal dimension is not the same in all scales

Multifractal Analysis 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.

Multifractal Analysis (MA) 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)

Multifractal Analysis (MA) There are several ways to determine the multifractal parameters and one of the most common is called box-counting method. Histogram based algorithm for calculation of MA singularity spectrum.

Multifractal Analysis (MA) Legendre multifractal singularity spectrum

MA - Biomedical engineering Random signals (self-similarity). PMV versus Healthy classification PMV (Prolaps Mitral Valve) heart beat anomaly. PMV signal has weak statistical properties.

Heart beat signal with PMV anomaly.

MA - Biomedical engineering Analysis of Multifractal singularity spectrum

Transformation of MA spectrum to angle domain and classification

MA - Acoustics 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.

MA - Acoustics Real room impulse response Structure of room impulse response

MA - Acoustics Detection of early reflections in room impulse response

MA - Video processing 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.

MA - Video processing Co-occurrence feature Wavelet feature

MA - Video processing Shot boundary detection in MA alpha domain Co-occurrence feature Wavelet feature