Multifractals in Real World

Goran Zajic

ICT COLLEGEICT COLLEGE OF VOCATIONAL STUDIES

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

Data je linija. Podeli sa na 3.Ukloni se srednji deo.

Ponavlja se procedura za svaki deo.

• Cantor Set

Line is divided into 3 parts. The central part is removed.

The same rule is repeated for new created parts of original line.

Von Koch krivaFractals – Example 2

Line is divided into 3 parts. The central part is removed. Van Koch Curve

New four segments.

Von Koch pahuljicaFractals – Example 3

Line is divided into 3 parts. The casasas

Van Koch Snowflake

asddadsdasdasdadasdasdasdsadsadasdas

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.

)ln(

)ln(

r

NDS N – number of copies

r < 1 – scaling ratio

Connection between dimensions : Ds = d + 1

Fractal dimension

• Box-counting dimension (Db)

DDBB(()=1.278)=1.278

L=1=1/22

N=52

DDBB(()=1.25)=1.25

DDBB() == lnN/lnlnN/ln

N – number of colored boxes - dimension of box

Fractals – Example 1

Data je linija. Podeli sa na 3.Ukloni se srednji deo.

Ponavlja se procedura za svaki deo.

• Cantor Set

Line is divided into 3 parts. The central part is removed.

The same rule is repeated for new created parts of original line.

631,0)3/1ln(

)2ln(SD

D=1 (line), D<1 (fractal line)

N – number of copies(2)r < 1 – scaling ratio (1/3)

Von Koch krivaFractals – Example 2

Line is divided into 3 parts. The central part is removed. Van Koch Curve

New four segments.

262,1)3/1ln(

)4ln(SDN = 4, r =1/3

Fractal line(1D signal):1<DS<2

Fractal surface(2D signal, slika):

2<DS<3

Fractal volume:3<DS<4

Fractals – Example 4

Sierpinski Carpet

New nine quadratic fields. Central one is removed

893,1)3/1ln(

)8ln(SD

N =8 fieldsr =1/3 scaling ratio

D=2 (surface)D<2 (fractal surface)

Introduction to fractals““Fractal is a structure, composed of parts, which in Fractal is a structure, composed of parts, which in

somesomesense similar to the whole structure”sense similar to the whole structure”

B. MandelbrotB. Mandelbrot

Introduction to fractals““The basis of fractal geometry is the idea of self-The basis of fractal geometry is the idea of self-

similarity”similarity”S. BozhokinS. Bozhokin

Introduction to fractals““Nature shows us […] another level of complexity. Amount ofNature shows us […] another level of complexity. Amount of

different scales of lengths in [natural] structures is almost different scales of lengths in [natural] structures is almost infinite”infinite”

B. MandelbrotB. 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 forcalculation 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.

Real room impulse response

Structure of room impulse response

MA - Acoustics

Detection of early reflections in room impulse response

MA - Acoustics

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

Co-occurrence feature Wavelet feature

Shot boundary detection in MA alpha domain