FracLab-Mat.zip_Fraclab MATLAB_MATLAB Fraclab_fracl_fraclab

% Overview and main functionalities of Fraclab % Jacques Lévy Véhel % 22 June 1998 % % This section presents an overview of the features available in Fra­ % clab. A general presentation is made, followed by a brief explanation % of the functionalities associated with each menu. % ______________________________________________________________________ % % Table of Contents: % % 1. Overview % % 2. Description of Fraclab Functionalities % % 2.1. Introduction % % 2.2. The main window of fltool % % 2.3. Pop-up Menus Description % % 2.3.1. Synthesis % % 2.3.2. Fractal and Multifractal Analysis % % 2.3.3. Signal Processing % % 2.3.4. Miscellaneous tools % % 3. The View menu % % 3.1. The Figure sub-window % % 3.2. The Image mode sub-window % % 3.3. The Tools sub-window % % 4. General conventions and remarks % % 5. Known bugs % % 6. Homework % % 6.1. Analysis of a stock market log % % 6.2. Synthetic Aperture Radar image denoising % % 6.3. Optical image segmentation % % 7. Conclusion % % 8. References % ______________________________________________________________________ % % 1. Overview % % Fraclab is general purpose signal processing toolbox based on fractal % and multifractal methods. It allows to perform many basic tasks in % signal processing, 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 smooth, and reveals its interest only if there is enough % singularity in the data. % % A comparison with classical signal processing may be 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 filtering, using for % instance Fourier analysis, in order to get rid of the noise. 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 % smooth 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 irregularity of the signal) of the ECG. The % lower this dimension, the worse the condition of the heart. Although % it is possible to assess the heart condition using classical methods, % a regularity analysis seems to be a good alternative in this case. A % second example is the case of radar images. These are difficult to % process because of the presence of a specific noise, the speckle % ("chatoiement" in French). However, speckle is not pure noise, but % rather a genuine part of the signal, caused by the interferometric % nature of radar images. In this respect, it contains information which % is essential about the imaged region. Although removing the speckle % can be useful for purposes of e.g. segmentation, analyzing it is a % necessary task for other applications, as for instance classification, % simply because the smoothed signal does not contain the necessary % information. From a broader point of view, one may even argue that, % though many image processing 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 respect, the % classical approaches, based on smoothing, do not appear as natural as % is usually assumed. % % Many tools in Fraclab are thus designed to measure different kinds of % irregularity, and use these measures to perform signal processing. % The regularity is analyzed either from a global point of view, or from % a local one. In the first case, Fraclab allows to compute various % fractional dimensions. In the second case, the Holder exponent is % used. The exploitation of this local singularity information for % signal processing can be performed in two different ways in Fraclab : % % · 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 exponent of s at t. 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, such as the box or regularization dimension % will be sufficient if the signal is ``fractally homogeneous'', in % more general situations, a finer analysis is needed: Multifractal % analysis aims at extracting higher level information from the % singularity function associated with the signal, in cases where % a(t) is as complex as, or more complex than s(t) (this happens for % instance for self-affine functions), or if keeping trace of the % singularity information at each point is not relevant (this is the % case for instance in issues of classification): In these % situations, one computes a multifractal spectrum, which yields a % global characterization of the singularity structure. Usually, % statistical or geometrical descriptions are used, leading to % various multifractal spectra. These multifractal spectra are for % instance useful in classification problems or in image % segmentation. % % 2. Description of Fraclab Functionalities % % 2.1. Introduction % % Fraclab can be approached from three different perspectives : % synthesis of fractal signals, fractal analysis, and signal processing. % This separation is artificial in a sense, since the tools associated % with these three streams overlap greatly, but it is conceptually % helpful. Most functionalities can be accessed either from a fractal % analysis or from a signal processing point of view, and this help file % will reflect this situation. % % In order to make Fraclab user-friendly, a graphic interface, called % fltool, is provided with this version. We describe briefly in the next % sections the general organization of fltool, as well as the main % features of the synthesis, analysis and signal processing tools, as % they appear in the menus of fltool. % % 2.2. The main window of fltool % % Once you launch fltool, the main window appears. It is divided into % four zones : % % · UPPER PART : the pop-up menus % % The pop-up menus allows to perform the various processings % available in Fraclab. These are briefly described in sections 2.3 % below and detailed in the corresponding 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 process % structures. A structure is a composite piece of information which %