Time Series Analysis, Fractals and Wavelets

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Time Series Analysis, Fractals and Wavelets
Prof. Abul Hasan Siddiqi Department of
Mathematical Sciences
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Time Series Analysis
The analysis of experimental data (observed) data
that have been observed at different points in
time is known as time series analysis
In this topic one likes to observe / study the
following properties of the time series (signal)

• Trends

• Seasonalities (Periodicities)

• Business Cycles

• Abrupt changes

• Drift

• De-noise
• (unwanted components)

• Forecasting / Prediction

• Self Similarity

• Compression

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Situation where time series can occur
In the filed of economics/ banking system where
one is exposed stock market quotations or monthly
unemployment figures or foreign exchange rates
In social sciences we find population changes
time series such as birth rate time series or
university / school enrollments
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In medical sciences we need to study time series
of influenza cases during certain period of time,
blood pressure measurements traced over time for
evaluating drugs used in treating hyper-tention.
Electocardiogram (ECG) data and functional
magnetic resonance imaging of brain wave time
series patterns to study how the brain reacts to
certain stimuli under various experimental
conditions
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In physical, engineering and environmental
sciences we come across with a lot of time series
for example

• Time series measurements acquired in the
  atmospheric boundary layer

• Time series of rain fall effecting the
  agricultural products and quite useful for flood
  control

• Time series of surface evaluation corresponding
  to wind generated waves measured near the shore
  areas of sea/lake/river

• Time series temperature variation and wind
  pressure

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• Time series appearing in machining process
  control/shutter, wear and breakage such as
  chatter in metal cutting, the condition
  monitoring of rotating machinery attempts to
  detect and diagnose machinery faults from
  vibration signals picked up usually from the
  machine casing

• Time series of Nuclear Reactor

• Time series of ultra sound and vibrations

• Time series of blood flow sounds, heart sounds
  and rates and lung sounds

• Time series of global warming

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• Time series of speech data

• El Nino and fish population

• Time series of earth quakes

• Time series of explosions

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Methods Employed in the Past
Methods of Statistics and Fourier Analysis
Recently wavelet and Fractal Methods are used. A
few standard references are
R. H. Shumway D. S. Stoffer, Time Series
Analysis and its Applications, Springer 2000
A. Arneodo, Wavelet Fractal Methods, Oxford
University Press 1996
R.Gencay, F. Seluk and B. Wishter, Academic
Press 2001
Paul S. Addison The Illustrated Wavelet Transform
Handbook Introductory Theory and Application in
Science, Engineering, Medicine and Finance,
Institute of Physics Publication, Publishing
Bristol and Philadelphia 2002
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Wavelet Analysis of a time series is the study
of the above mentioned properties by the breaking
up of the signal (time series) into shifted and
scaled version of the original (mother) wavelet
As mention earlier scaling a wavelet simply means
stretching (or compressing) it.
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One can make a plot on which the x-axis represent
position along the signal (time) the y-axis
represents scale and the color at each x, y point
represent the magnitude of each wavelet
coefficients. See figure below
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The coefficients plots resembles a bumpy surface
view from above it will look like figure 1 if we
look at the same surface from the side.
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Figure 1
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The continuous wavelet transform plots are
precisely the time-scale view of the signal. Low
scale a implies compressed wavelet ? Rapidly
changing details ? High frequency w, High scale a
implies stretch wavelet ? slowly changing ,
coarse features ? low frequency ?
If we think of this surface in cross section as a
one-dimensional signal, then it is reasonable to
think of the signal as having components of
different scales large features carved by the
impacts of large meteorites, and finer features
abraded by small meteorites.
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Here is a case where thinking in terms of scale
makes much more sense than thinking in terms of
frequency. Inspection of the CWT coefficients
plot for this signal reveals patterns among
scales and shows the signals possibly fractal
nature.
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Figure 2
Even though this signal is artificial, many
natural phenomena from the intricate branching
of blood vessels and trees, to the jagged
surfaces of mountains and fractured metals lend
themselves to an analysis of scale.
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Inspection of the wavelet Fourier transform for
signal in figure 2 reveals pattern among scales
and shows the signals Fractal nature
One stage Filtering Approximations and Details
For many signals, the low frequency content is
the more important part. It gives signal its
identity. The high frequency content, on the
other hand, imparts flavor or nuance. For example
in human voice if the high frequency components
are removed the voice sounds different, but still
tell whats being said. How ever if we removed
enough of the low frequency components we hear
gibberish.
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Thus in wavelet analysis we can divide signal in
parts, one type called approximations and other
called details. The approximations are the high
scale low frequency components of the signal ,
the details are the low scale high frequency
components of the signal.
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Multiple-Level Decomposition The decomposition
process can be iterated, with successive
approximations being decomposed in turn, so that
one signal is broken down into many lower
resolution components. This is called the wavelet
decomposition tree.
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Looking at a signals wavelet decomposition tree
can yield valuable information.
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Number of Levels Since the analysis process is
iterative, in theory it can be continued
indefinitely. In reality, the decomposition can
proceed only until the individual details consist
of a single sample or pixel. In practice, youll
select a suitable number of levels based on the
nature of the signal, or on a suitable criterion
such as entropy
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Applications of Wavelet Methods for

• Detecting discontinuity and breakdown points

• Detecting long term evolution

• Detecting self similarity

• Identifying pure frequencies

• Suppressing signals

• De-Nosing signals

• Compressing signals

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Steps
Use graphical interface tools for the MATLAB
command line type wavemenu
Click on wavelets 1-D (or other tool as
appropriate)
Load the sample analysis by selecting the submenu
item by file ? Demo Analysis
Note One can also use the different options
provided in the graphical interface to look at
the different components of the signal to
compress or de-noise the signal. To examine
signal statistics or to zoom in and out on
different signal features.
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One can load the corresponding MAT-file from the
MATLAB command line and use the wavelet toolbox
functions to investigate further this sample
signals. The MAT-files are located in the
directory tool box/wavelet/wavedemo. There are
many signals in the wave demo directory that one
can analyze.
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Wavelet Families
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Haar
Any discussion of wavelets begins with Haar
wavelet, the first and simplest. Haar wavelet is
discontinuous, and resembles a step function.
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Daubechies
Ingrid Daubechies, one of the brightest stars in
the world of wavelet research, invented what are
called compactly supported orthonormal wavelets
thus making discrete wavelet analysis
practicable. The names of the Daubechies family
wavelets are written dbN, where N is the order,
and db the surname of the wavelet. The db1
wavelet, as mentioned above, is the same as Haar
wavelet. Here are the wavelet functions psi of
the next nine members of the family
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Biorthogonal This family of wavelets exhibits the
property of linear phase, which is needed for
signal and image reconstruction. By using two
wavelets, one for decomposition (on the left
side) and the other for reconstruction (on the
right side) instead of the same single one,
interesting properties are derived.
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Coiflets Built by I. Daubechies at the request of
R. Coifman. The wavelet function has 2N moments
equal to 0 and the scaling function has 2N-1
moments equal to 0. The two functions have a
support of length 6N-1. You can obtain a survey
of the main properties of this family by typing
waveinfo('coif') from the MATLAB command line.
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Symlets The symlets are nearly symmetrical
wavelets proposed by Daubechies as modifications
to the db family. The properties of the two
wavelet families are similar. Here are the
wavelet functions psi.
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Morlet This wavelet has no scaling function, but
is explicit.
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Mexican Hat This wavelet has no scaling function
and is derived from a function that is
proportional to the second derivative function of
the Gaussian probability density function.
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Meyer The Meyer wavelet and scaling function are
defined in the frequency domain.
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WAVELETS FRACTALS

• Sharp signal Transitions create large amplitude
  wavelets coefficients

• Singularity are detected by following across
  scales the local minimum of the wavelet transform

• In images, high amplitude wavelet coefficients
  indicate the position of edges, which are sharp
  variations of the image intensity

• Different scales provide the contours of image
  structures of varying sizes. Such multiscale edge
  detection is particularly effective for pattern
  recognition in computer vision

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Introduction to Fractals and their Applications
Fractals comes from Latin word Fractus meaning
broken, describe objects that are too irregular
to fit into traditional geometrical settings.

• Fractals occur as graphs of functions. Indeed
  various phenomena display fractal features when
  plotted as functions of time. Examples include
  atmospheric pressure, labels of reservoir and
  prices of the stock market, usually when recorded
  over fairly long time spans.

• The zooming capability of the wavelet transform
  not only locates isolated singular events, but
  can also characterize more complex multi-fractal
  signals having non-isolated singularities.

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Construction of Middle Third Cantor F, by
repeated removal of middle third of the
intervals. Note FL and FR are the left and right
copies of F scaled by a factor 1/3
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As we know dimension of R is 1
R2 is 2 R3 is 3 Rn is n
A natural question arise do we have set or space
whose dimension is rational number. The question
is what will be notion of such dimension?
Notion of Lebesgue measure and Hausdorff measure
are well known
Iterated function system, Fractals and Chaos are
closely related
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Hausdorff measure spectrum, a new concept
recently studied by Dr. Fahima Nekkah of Montreal
University is more useful concept then Fractal
dimension. Wavelet transform of multifractals or
Fractal function are quite useful for studying
very very irregular time series or structure.
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• Mandelbrot B.B. Mandelbrot. The Fractal Geometry
  Nature, W.H. Freeman and Co., San Francisco,
  1982
• Recognized the existence of multi-fractals
  (fractals as functions of measure) in most
  corners of nature
• Scaling one part of a multi-fractal produces a
  signal that is statistically similar to the whole
  This self-similarity appears in the wavelet
  transform, which modifies the analyzing scale

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From the global wavelet transform decay, one can
measure the singularity distribution of
multifractals. This is particularly important in
analyzing their properties and testing models
that explain the formation of multifractals
appearing in diverse fields such as
Thermodynamics, Statistical mechanics,
Environment, Financial engineering and
Mathematics, particularly Mechanical Sciences,
see for example

• Arneodo et al.
• The Thermodynamics of Fractals Revisited with
  Wavelets Physica A 213 (1995) 232-275,
• Wavelet-Based Multifractal Formation Image Anel
  Stereol 20 1-6, 2001
• Wavelet-based Multifractal Formalism On the
  Anisotropic Structure of Galactia, H., Preprint
  2003

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• G. Papanicolou and K. Solna
• Wavelet based Estimation of Local
  Kolmogorav Turbulences, Preprint 1999

• Erhan Bayraktar et al.
• Estimating the Fractal Dimension of the
  S4 P 500 Index using Wavelet Analysis, Preprint,
  June 2003

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• The Wavelet Transform takes advantage of
  multifractal self-similarities, in order to
  compute the distribution of their singularities.
  This singularity spectrum is used to analyze
  multifractal properties.

• Signals that are singular at almost every point
  called multi-fractals are also encountered in the
  maintenance of economic records, physiological
  data including heart records, electromagnetic
  fluctuations in galactic radiation noise,
  textures in images of natural terrain, variations
  of traffic flow, etc.

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• Fractal functions or Multifractals are those
  functions whose graph are fractal sets P.R.
  Massopust Fractal Functions, Fractal Surfaces and
  Wavelets, Academic Press, 1994, Hubhard Barbara,
  Wavelets., 1999 According to some authors, when
  the characteristics of a fractal evolve with time
  and become local, the signal is known as a
  multifractal.

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Hausdorff Measure
Diameter of a set U
Greatest distance between any two points
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Let F is a subset of Rn and s is a non-negative
number. For any we define
is a cover of F
is called the s-dimensional Hausdorff Measure of F
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is called Hausdorff dimension
Dimension of the middle third counter set s if
slog2/log3 0.6309
In general Hausdorff dimension of the middle
counter set lies between a half and one.
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Box Counting Dimensions (Kolmogrove Entropy/
Entropy dimension/ capacity dimension/ metric
dimension/ Minkowski dimension)
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Box dimension of F
Box dimension of middle third counter set F
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Singularity spectrum
Let S be the set of all points where
the point wise lipschitz regularity of f is equal
to . The spectrum of singularity of f
is the fractal dimension of S .
The singularity spectrum gives the proportion of
Lipschitz singularities that appear at any
scale a.
Dr. Fahima Nekkah Presentation
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BEYOND THE FRACTAL CHARACTRIZATION OF POROUS
MEDIA THE MODIFIED AUTO-CORRELATION METHOD
Fahima Nekka and Jun Li Faculté de
Pharmacie Université de Montréal
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INTRODUCTION

• Heterogeneous media, such as composites, porous
  materials and blend polymers have complex
  microstructures.
• These microstructure variations have important
  consequences on bulk properties.
• These physical structures can share the same
  fractal dimension in spite of their different
  appearance.

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Problematic

• Fractal dimension is a first order parameter of
  complexity which can degenerate
• Very different structures share the same fractal
  dimension.
• WHY?
• In the simple case of similarity dimension
  D log N/log(1/r), this equation does
  not uniquely define D.
• D and N define the parts length but not their
  positions!
• There is a need to develop more advanced
  quantifiers

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Two carpets having the same D
log(8)/log(3)1.892 but completely different
structures.
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Purpose Classification

• Texture is an important characteristic of a
  structure.
• One aspect of texture is expressed by the size
  distributions of pores and their locations within
  the structure.
• This aspect is manifest in many domains
• Material Sciences Porosity, microarchitecture
• Cosmology galaxy distributions.
• Purpose taking into account these holes
  properties in order to differentiate structures
  having the same fractal dimension but different
  texturesour method HMSF

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The idea is really simple!
Intuitively, the oscillations of this
intersection inform about the presence and the
characters of holes of the structures.
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The Hausdorff Measure Spectrum Functions (HMSF)

• We introduce HMSF as a new way to distinguish
  sets having the same fractal dimension.
• HMSF is based on Hausdorff measure of the
  translation of the set through itself in a
  continuous manner.
• Since translation is made continuously on each
  point (local) and the Hausdorff measure (global)
  is estimated ? HMSF extract the whole information
  on the set.
• The indicator function of the intersection of a
  set with its translate can be viewed as a
  two-point joint moment (autocovariance).
• This explains in a way why HMSF completes the
  information obtained from pointwise descriptors
  like D.

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A tractable model

• Let . When t varies
  between -1 and 1, the Hausdorff measure of I (t)
  at dimension s log 2/log 3 can only get a
  value of .
• Question Is it possible to determine the exact
  forms of the translation elements
• Response We proved that theres an infinite tree
  structure between Tn, where the number of
  branches to any knot of the tree is infinite,
  they are given by

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Numerical construction of HMSF

• A direct calculation of HMSF is laborious and
  analytically possible only for
• some cases. So we propose three
  algorithms to estimate HMSF.
• Similarity algorithm built on the similarity
  properties of a fractal set which are inherited
  by HMSF itself, Fig. 1
• Interpolation algorithm HMSF is symmetric and
  discontinuous everywhere but can be approximated
  by continuous functions, Fig. 2
• Recursive algorithm HMSF M(t) satisfies the
  recursive properties. For Cantor third set
  
  
• if
  
  
  

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Application of the classification algorithm

• HMSF can be used to distinguish different sets
  having the same Hausdorff dimension.
• The sets as examples are constructed from the
  initiator I 0, 1 for simplicity. Their
  generators are defined respectively by the IFS on
  the right
• HMSF see Fig. 3

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How to use HMSF to differentiate between sets of
the same fractal dimension?

• We take structures (a), (b), (c), (d) and (e),
  all having the same D log N/log b log2/log3.
• We suggest two successive steps to characterize
  the structure.
• Step1 Translation Invariance Based Method
  (TIBM) take values preserved by translation,
  each value representing a level. The graph of
  these levels, in terms of the shift elements are
  compared.
• If Step 1 is not conclusive, then go to Step2
• Step2 Fixed Level Based Method (FLBM) compare,
  for a given level (the first one is enough), the
  HMSF values for various sets. Then, quantify
  difference in these values by averaging their
  distances from the accumulation point.

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Results from step 1

• From TIBM The graph of translation invariant
  values in terms of shift numbers succeeds in
  differentiating between (a), (c) and (b) (or
  (d)), see Fig. 4.
• However, TIBM levels are the same for (a) and (e)
  as well as for (b) and (d).
• This last fact does not allow one to conclude
  that (a) and
• (e) or (b) and (d) are the same.
• We have yet to go a further step in our
  exploration and use the FLBM.

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Results from step 2

• From FLBM This method compares, for a given
  level, the HMSF values of the concerned sets.
• In fact, the first level, which contains the
  whole information of HMSF, is enough.
• We plotted the first four fixed levels (from 0
  to 3)
• of the HMSF of (a) and (e). See Fig. 5

• Graphically, the difference is already obvious on
  level one. This difference can be quantified by
  averaging weighted distances between shift values
  and the shift accumulation point at the first
  level, giving thus level indexes associated to
  each set. For example, using a dyadic sequence
  weights 1/2ii , we get the value 0.5962 for (a)
  and 0.6248 for (e).
• This index, from one part, is able to
  differentiate between sets and, from the other
  part, indicates the degree of homogeneity of the
  set the higher the index, the more homogeneous
  is the set.

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Fig.1 Algorithme de similarité
Fig.2 Algorithme dinterpolation
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HMSF
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TIBM
FLBM
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Porous media

• We can synthesize porous media using fat fractal
  models they have similar size-dependent power
  distribution of pores.
• Fractal dimension for these sets is equal to 1,
  and then cannot be used to distinguish between
  them.
• We combine our method, HMSF, to the
  regularization dimension, which is more sensitive
  to variations than the box-counting dimension.

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Results for the simulated porous media

• We apply the regularization dimension (RD)
  directly to porous structures corresponding to
  the hole sizes 1/3, 1/4, 1/5, and 1/15.
• We also apply the regularization dimension to the
  HMSF of these same structures.
• RESULT the combination of our method HMSF to RD
  gives better results
• difference in RD values in terms of hole size is
  amplified and the graph is more linear and
  monotonic.

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Comparison of RD applied directly to the
structures with RD combined to the HMSF
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The inverse problem in porous media

• Many characterization techniques provide
  morphological information in the form of
  correlation functions.
• From this, a real-space microstructural model is
  needed to understand and predict material
  properties and to assure that this way of
  characterization is adequate.
• The problem given a correlation function, find
  the corresponding microstructure.

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The modified auto-correlation function
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Conclusion and future directions

• For structures having the same fractal dimension,
  we succeeded in differentiating them using their
  HMSF.
• Fat models,as synthetic models of porous media
  have also been characterized by a combination of
  our method with the regularization dimension.
• The developed method offers a more precise
  description of fine texture generally
  indistinguishable by existing methods.
• Generalization to 2-D structures and the inverse
  problem are under study.
• Acknowledgements This work has been supported by
  NSERC Individual Grants.