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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 selfsimilarity properties in limited range of scales.
Fractals – Example 1
• Cantor Set
Line Data
is divided
into 3Podeli
parts. sa
Thenacentral
je linija.
3.
part is removed.
Ukloni se srednji deo.
The
same rule
is repeatedza
forsvaki
new deo.
created
Ponavlja
se procedura
parts of original line.
Fractals – Example 2
Von Koch kriva
Line is divided into 3 parts. The central
part is removed.
Van Koch Curve
New four segments.
Fractals
–
Example
3
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):
DS  
ln( N )
ln( r )
N – number of copies
r < 1 – scaling ratio
• Measured dimension (d)
– Set of strate line segments which cover the curve
of fractal structure.
– Smaller segments, better approximation of
structure curve.
Connection between dimensions : Ds = d + 1
Fractal dimension
• Box-counting dimension (Db)
e=1/22
L=1
N=52
N – number of colored boxes
e - dimension of box
DB(e) = lnN/lne
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 Data
is divided
into 3Podeli
parts. sa
Thenacentral
je linija.
3.
part is removed.
Ukloni se srednji deo.
DS  
The
same rule
is repeatedza
forsvaki
new deo.
created
Ponavlja
se procedura
parts of original line.
D=1 (line), D<1 (fractal line)
ln( 2)
 0,631
ln(1 / 3)
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
N = 4, r =1/3
DS  
ln( 4)
ln(1 / 3)
 1,262
Fractal volume:
3<DS<4
Fractals – Example 4
Sierpinski Carpet
New nine quadratic fields.
Central one is removed
N =8 fields
r =1/3 scaling ratio
DS  
ln( 8)
 1,893
ln( 1 / 3)
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

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