Overview and main functionalities of
Fraclab
Jacques Lévy Véhel 22 June 1998
This section presents an overview of the features available in
Fraclab
. A general presentation is made, followed
byabrief explanation of the functionalities asso ciated with each menu.
Contents
1 Overview 2
2 Description of Fraclab Functionalities 3
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 The main windowof
to ol
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Pop-up Menus Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3.1 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3.2 Fractal and Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.3 Signal Pro cessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.4 Miscellaneous tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 The View menu 5
3.1 The
Figure
sub-window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 The
Image mo de
sub-window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 The
Tools
sub-window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 General conventions and remarks 8
5 Known bugs 9
6 Homework 9
6.1 Analysis of a sto ck market log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.2 Synthetic Aperture Radar image denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.3 Optical image segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 Conclusion 15
8 References 15
1 Overview
Fraclab
is general purp ose signal pro cessing to olbox based on fractal and multifractal methods. It allows
to p erform many basic tasks in signal pro cessing, including estimation, detection, regularization, denoising,
modeling, segmentation and synthesis. Let us stress that
Fraclab
is not intended to process "fractal"
signals (whatever meaning is given to this word), but rather to apply fractal tools to the study of irregular
but otherwise arbitrary signals : just as e.g. gradient-based algorithms are often successfully applied for
image segmentation even when there are no mathematical or physical reasons for the original signal to
possess an ordinary derivative, a fractal analysis may yield useful insights for non fractal data. Of course,
it does not in general give relevant indications when the signal is mainly regular or smo oth, and reveals its
interest only if there is enough singularity in the data.
A comparison with classical signal processing may b e in order to make things clearer. In many cases, one
assumes that the meaningful information is regular in essence, and that the irregular aspect of the observed
data is due to noise coming from various sources: captor, thermal, coding, etc. A most useful tool is then
ltering, using for instance Fourier analysis, in order to get ridofthenoise. This approach has of course
proven extremely valuable in many applications.
However, there are cases where the irregular part of the observed data contains useful information that
cannot be recovered if only the smo oth part is kept. It can even be the case that most or all of the relevant
information is carried in the singular structure of the observation. Let us give some examples. It is well
known that some useful information about a heart condition is contained in the fractal dimension (more
precisely the correlation dimension, a feature related with the irregularityof the signal) of the ECG. The
lower this dimension, the worse the condition of the heart. Although it is p ossible to assess the heart
condition using classical metho ds, a regularity analysis seems to be a go od alternative in this case. A second
example is the case of radar images. These are dicult to pro cess b ecause of the presence of a sp ecic
noise, the sp eckle ("chatoiement" in French). However, speckle is not pure noise, but rather a genuine part
of the signal, caused bytheinterferometric nature of radar images. In this resp ect, it contains information
whichisessential about the imaged region. Although removing the speckle can be useful for purp oses of e.g.
segmentation, analyzing it is a necessary task for other applications, as for instance classication, simply
because the smo othed signal do es not contain the necessary information. From a broader p oint of view, one
may even argue that, though many image pro cessing techniques aim at getting rid of irregularities in the
data, the segmentation of simple, non noisy optical images should more logically be based on singularity
analysis: one is indeed mostly interested in singularities, since edges are basically discontinuities in the grey
levels. In that resp ect, the classical approaches, based on smo othing, do not appear as natural as is usually
assumed.
Manytoolsin
Fraclab
are thus designed to measure dierent kinds of irregularity, and use these measures
to perform signal pro cessing. The regularity is analyzed either from a global point of view, or from a lo cal
one. In the rst case,
Fraclab
allows to compute various fractional dimensions. In the second case, the
Holder exp onentisused. The exploitation of this local singularity information for signal pro cessing can b e
performed in two dierentways in
Fraclab
:
p
By keeping all the information, which basically means that, starting from the signal s(t), one builds a
new function a(t) which gives the Holder exponentofsatt. This is useful either when the singularity
function a(t) is simpler than the original signal, or for purposes of e.g. detection or denoising.
By using a global description of the singularity: while the use of fractional dimensions, suchasthe box
or regularization dimension will b e sucient if the signal is fractally homogeneous, in more general
situations, a ner analysis is needed: Multifractal analysis aims at extracting higher level information
from the singularity function asso ciated with the signal, in cases where a(t) is as complex as, or more
complex than s(t) (this happ ens for instance for self-ane functions), or if keeping trace of the singu-
larity information at eachpoint is not relevant (this is the case for instance in issues of classication):
In these situations, one computes a multifractal sp ectrum, which yields a global characterization of
the singularity structure. Usually, statistical or geometrical descriptions are used, leading to various
multifractal spectra. These multifractal sp ectra are for instance useful in classication problems or in
image segmentation.
2 Description of Fraclab Functionalities
2.1 Intro duction
Fraclab
can b e approached from three dierent p erspectives :
synthesis of fractal signals
,
fractal analysis
,
and
signal processing
. This separation is articial in a sense, since the tools associated with these three
streams overlap greatly, but it is conceptually helpful. Most functionalities can b e accessed either from a
fractal analysis or from a signal pro cessing point of view, and this help le will reect this situation.
In order to make
Fraclab
user-friendly, a graphic interface, called
to ol
,isprovided with this version. We
describe briey in the next sections the general organization of
to ol
,aswell as the main features of the
synthesis, analysis and signal pro cessing tools, as they app ear in the menus of
to ol
.
2.2 The main window of to ol
Once you launch
to ol
, the main window appears. It is divided into four zones :
UPPER PART : the p op-up menus
The
pop-up menus
allows to perform the various pro cessings available in
Fraclab
. These are briey
described in sections 2.3 below and detailed in the corresp onding parts of this help.
UPPER MIDDLE PART : the Variables and Details windows
The basic elements one manipulates in
Fraclab
are
structures
: The synthesis and the analysis tools all
produce and pro cess structures. A structure is a composite piece of information whichmay comprise
matrices of dierent size and other more complex elements. The upp er middle part of the
to ol
graphic interface is comp osed of two windows : the
Variables
window, which will display the name
of the structures generated in the course of using
Fraclab
. The highlighted name corresp onds to the
current active structure (i.e. theoneonwhich the pro cessings are made). The
Details
window shows
the building blo cks of the structure currently highlighted in the Variables window. To select a blo ck,
p
i.e. a sub-element of the structure (which may for instance be a matrix), just highlight it. This is
useful for instance for purp oses of visualization or if one wants to extract a particular building block
from a comp osite structure.
LOWER MIDDLE PART : les manipulation
Scan Workspace
allows you to transfer les from your matlab environment(workspace) into the
to ol
environment, while
Load
allows you to transfer les from your le system into the
to ol
environment.
Files formats that are currently recognized include mat-les, 1D ASCII signals and 2D tif images.
Save
allows you to save structures from
to ol
into your le system. Finally,
Clear
removes structures from
both the
to ol
and the matlab environment.
LOWER PART : Miscellaneous to ols
View
: This imp ortantmenu allows to perform various display-related op erations which are detailed
below.
Help
lists most of the routines available in
Fraclab
. To get a detailed help on a particular function,
double clickonit.
Quit
is what you guess.
Finally,you maywanttocheck the
Message
zone often, as error messages are usually displayed here.
In many occasions, a b eep is heard when a new message is sent. The
Erase
button lets you clear the
message zone.
2.3 Pop-up Menus Description
There are four kinds of p op-up menus : the rst one allows to p erform
synthesis
of "fractal" signals.
The second one deals with the
fractal and multifractal analysis
of signals and includes :
Fractional
Dimensions
,
1D Exponents Estimation
,
1D signals Multifractal Spectra
, and
Stable Motion
. The third group
is related to
Signal Pro cessing
and is comp osed of
Segmentation
and
Denoising
. Finally,
miscellaneous
to ols
are available, namely
Time Frequency and Time Scale analysis
and
Misc
, which allows to perform
various editing tasks. Let us describ e briey what can be done with these various menus. More detailed
explanations are given in the appropriate sections of this help.
2.3.1 Synthesis
Twotypes of signals can be generated : measures (i.e. an array of non negativedatathat add to one) or
functions. Measures are interesting in particular when one needs to takeinto account the resolution in an
explicit way. For b oth measures and signals, either deterministic or sto chastic data may b e generated. By
and large, this menu allows to synthesize a substantial subset of all classical fractal models described in the
literature : 1D and 2D fBm-s, mBm-s, (generalized) Weierstrass functions, stable motions, Wavelet based
1/f pro cess, multifractal measures, ...
2.3.2 Fractal and Multifractal Analysis
The most basic parameters that can b e computed are of course fractional dimensions. In the currentim-
plementation of
Fraclab
, only the box and regularization dimension are available. When one needs a lo cal
characterization of the signal, Hölder exponents are more pertinent, and the menu
1D Exponents Estimation
allows to estimate both p ointwise and lo cal exponents. In addition, a long range exponentmay b e computed,
as well as 2-microlo cal exponents.
1D signals Multifractal Spectra
oers various estimations pro cedures. Note
that the computation of Multifractal Spectra for 2D data is possible using the
Segmentation
menu. Finally,
Stable Motion
allows to test the stabilityofagiven pro cess and to estimate the relevant parameters.
2.3.3 Signal Pro cessing
Segmentation
allows to segment both 1D signals and images. In the former case, a mo deling based on a
generalization of IFS, called weakly self ane functions, is used. Images are segmented into edges or regions
of given regularity through multifractal analysis.
Denoising
allows to regularize and denoise 1D or 2D data
using various metho ds.
2.3.4 Miscellaneous to ols
TF-TS
allows to compute various time frequency representations of a signal, while
misc
oers basic structure
manipulations such as sums, extractions, ...
3 The View menu
Clicking on the
View
button op ens a new window, which serves as a control center for all the displays
(graphs + images) you mightwantto have. This window is composed of four sub-windows :
Figure
,
Image
mode
,
Tools
, and nally a originally blank region whichwillcontain the list of all op ened gures along with
the data displayed in each gure. This list allows you to select which gure or signal is currently active
by clicking on it. Only the active element will b e aected bythevarious commands available in the other
sub-windows of the
View
menu.
It is important to understand the dierence between a
View
and the data displayed in it. A
View
corresponds
to a Matlab gure that will display one or several graphs or images. The case of several graphs corresponds
to the use of the
sub-plot
command in Matlab. Since, in the case of multiple plots, you mightwant to apply a
dierent pro cessing to each sub-plot, you need to tell
Fraclab
which sub-plot is the current one. This is why
the list at the b ottom of the
View
windowwillshow the name of all the op ened views (i.e. Matlab gures)
numbered in the chronological order of their appearance, and, for each of these views, the name of the signals
that are shown in this gure as plots or sub-plots. This will allowyou to select either a whole gure or one
of its sub-plot. For instance, some viewing options suchas
hold
,
rotate
or
zoom
are available only when a
particular signal is highlighted, and are grayed out when the whole view is highlighted. Note nally that
clicking in the list either on a view or on one of the signals that it contains will bring the corresponding
window to the foreground, a useful feature if you havemany opened windows or/and if you do not remember
which signal is which.
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