New Mathematical Wavelet Technologies in Undergraduate and Graduate Engineering Education

 

CASTOVA, Nina 1 & KUCERA, Radek2

1Department of Applied Mathematics, FEI, Technical University Ostrava, tr. 17. listopadu, nina.castova@vsb.cz
2Department of Mathematics, HGF, Technical University Ostrava, tr. 17. listopadu, radek.kucera@vsb.cz

 

Abstract: In last decade the wavelet-based techniques are widely used in science and engineering. Although mathematical background of the wavelet theory is rather complicated its practical application is not so difficult. Effective application of the wavelet technologies naturaly assumes some capability to understand the theoretical background of the comertial software. Moreover, the capability to work out own computer programs gives another opportunity to prepare numerical experiments with aim to get more experience and deeper insight into both theory and applications. Our experience shows that students are able to contribute to the research if the subject is explained them in an adequate way. In our talk, we shall present materials prepared with assistence of the students. It turned out that they often brought a new view on the problem and motivation for further research.

Keywords: undergraduate, graduate education, integral transform, wavelet

 

1  Introduction

The wavelet transform is one of the latest methods in mathematical theory of series, in signal processing and in their applications. This transform represents considerable qualitative progress in comparison with the Fourier transform and its modifications. Above all the wavelet transform is effective in data processing when data files are large (signals, images). Its typical applications are compression, decompression, filtration and statistical processing. Application of the wavelet methods is also effective for solution of the inverse problems that are connected with analysis of complicated deconvolution equations. In addition the wavelet transform eliminates or suppresses undesirable effects, which are produced sometimes by classical transforms as a by-product.

For example the mathematical background of the Fourier transform conducts sometimes to practically inapplicable results. Namely, the Fourier basis is formed by the sinus waves and therefore every basis function influences uniformly the analysed signal on the whole time scale. This global effect is very disturbing in case of a signal with dynamic behaviour. The high frequency or amplitude at a short time interval induces occurrence of high-frequency components in the Fourier spectrum. Their influence has to be eliminated then in the regions with the low dynamic by further spectral components. Hence we obtain high data volume for acceptable signal description and consequently, complications with the computer implementation could appear. At the same time the error propagation can debase the computed results.

The usual technique, which suppresses the undesirable effects, is the windowed Fourier transform. We put a suitable weight function - window - in the Fourier integral. This function " glides" along the analysed signal and reduces the high-frequency influences outside this window. The classical windows are independent on the analysed signal and their widths are constant.

The wavelet transform adapts the width of the window in the time as well as in the frequency according to behaviour of the examined signal and therefore belongs to the time-frequency analysis. An expansion of the window in the time domain leads to a dilatation of the window in the frequency domain and vice versa. The wavelet transform works similarly as the windowed Fourier transform. In addition it automatically adapts this process with regard to the local character of the examined signal.

We emphasise the interdisciplinary character of the wavelet methods that we can support by present-day or by prepared applications on all technological faculties of VSB - Technical University. Particularly, we can mention the analysis and interpretation of the seismic and other geophysical data (on the Faculty of Mining and the Faculty of Civil Engineering), the analysis of signals in heavy-current electrical engineering, medical electrical engineering and in the field of measurement and control, signal coding (Faculty of Electrical Engineering and Information Science, Faculty of Mechanical Engineering) or study of transition phenomena in the optical fibre theory (Faculty of Electrical Engineering and Information Science), at turbulent flow (Faculty of Mechanical Engineering) or material diagnostics (Faculty of Metallurgy and Material Engineering, Faculty of Mining).

To follow the above mentioned trends, teaching of the basics of the wavelet methods was introduced already in the school year of 1995 - 96. These methods are contained in the subject Integral Transforms and Spectral Analysis in the PhD study. Experience gained during the extended classes resulted in more complex implementation of the wavelet methods to classes as well as to professional practice. The development project No. 0637 Application of Wavelet Transformation prepared by the Czech Republic University Development Fund and the grant project GACR 205/1233 (on solution and preparation of which the Geonics Institute of the Czech Republic Science Academy also participated) have been a part of implementation of the wavelet methods into the professional practice.

Our major target was to develop a teaching methodology of the wavelet theories and their applications by implementation of specialised subjects into both the engineering degree and the PhD studies as well as by suitable implementation of selected parts into the existing system of the classes. To reach this target it was also necessary to gain appropriate software for data processing and to carry out the wavelet transforms. This target was followed by other partial aims:

  1. Selection of basic application directions for implementation of wavelet methods in co-operation with guarantors of both the engineering degree and the PhD studies and the related conception preparation of students' professional activities.

  2. Innovation of selected subjects in the existing teaching plan (syllabus) and preparation for implementation of new specialised subjects.

  3. Preparation of a syllabus for teaching the basics of the wavelet theories completed with a selection of typical applications.

An integral part of the preparation was also development and adjustment of the appropriate software that was developing into two individual directions:

It should be also mentioned here that the new software packet MATLAB, version 5, together with its accessory Wavelet-Toolbox was bought from the financial means reserved for solution of the subject development program.

Our effort to reach the designed target as soon as possible resulted also to searching for contacts with other teachers and research workers from other universities and institutes. We sent queries to universities in Prague (Charles' University, CVUT), Plzen, Brno (VUT), Bratislava and Kosice. We found out that the Charles' University in Prague is the only university teaching by the wavelet transform. We established contacts with Doc. K. Najzar, CSc. from the Numeric Mathematics Department of the Faculty of Mathematics and Physics (Charles' University in Prague) who provided us with its own brief wavelet syllabus. Similarly we established contacts with Dr. Piotr Wojdyllo from the Institute of Mathematics of the Warsaw University, who sent us a pre-print "Why a Wavelet is a Wavelet?".

In the Russian Federation we succeeded to gain some information from Mr. professor G. V. Korovin, DrSc. who is a corresponding member of the Russian Science Academy and Environmental Centre in Moscow and also from Mrs. doc. T. N. Saburova from the Technologic University - Moscow Institute of Steel and Alloys. The gained information and written materials enable us to orient in the field of wavelet transform and their development direction.

2  Selection of application directions

Selection of application directions was partially given by structure and interest of students of the PhD studies, however it was taken into the consideration that these students are not concentrated on applied mathematics but they are students of technical subjects. Selection of topics was mirrored in titles of semester works in the subject of applied mathematics. This selection was always made after consultation of a PhD student with his/her supervisor. These works are required at the end of a mathematics course as a part of the final test and their selection is recommended so that they can be used later on when preparing a thesis. We think that in every respect these works contribute to better knowledge of mathematics and its application in practice. The following theses were processed and defended before examiners within the 1997 - 1999 period (the subject within the framework of which the thesis was processed is shown in brackets). More difficult tasks were completed by a group of students and every student had its own part of the task.

We mention some titles of the of the semester works:

Some of the theses of the 4th year students (engineering degree studies, the subject Discrete Transforms) can also be listed here as they show suitable selection of the application directions for implementation of wavelet methods into the classes.

3  Innovation of the Selected Subjects

As stated in the Introduction, to follow the latest trends in mathematical theory of series, teaching of the basics of the wavelet methods was introduced already in the school year of 1995 - 96. These methods were contained in the subject " Integral Transforms and Spectral Analysis" in the PhD study.

In the following period we extended the wavelet methods also to other subjects of the applied mathematics. New modern solution procedures of signal filtration, compression and signal analysis have been implemented into the curriculum of subjects, which follows the basic mathematics course in the years 1 and 2. The subject called "Integral Transforms" is in the year 3 and the subject called "Discrete Transforms" is in the year 4. These subjects have been modified and the Senate approved their curriculum. The subject "Integral Transforms" is compulsory for all electrical engineering branches. "Discrete Transforms" is a voluntary subject, there were 10 students last year passing this subject.

A lecture "Introduction to the wavelet theory", where basic ideas and practical examples of problem solutions in the field of signal filtration are shown, completed the curriculum of the "Integral Transforms" - subject for the students of engineering degree studies at the Faculty of Electrical Engineering and Information Science.

The curriculum of the "Discrete Transforms" subject have been partially modified and in the present time it contains the following chapters:

Topicality of the problem has been proven by an increased interest in dissertations in this field, here are three of them as an example:

  1. Application of the wavelet transformation in the time and frequency data analysis;

  2. Parallel implementation of wavelet packet decompositions and their use;

  3. Using of the wavelet transformation to smoothing stochastic signals.

There was a subject called "Integral and discrete transforms" established within the frame of the PhD studies which contains of basis of continuous and discrete wavelet transforms (same as in the subject "Discrete transforms"). This subject is destined for such PhD students who did not have the subject "Discrete transforms" during their regular studies. And besides that this subject is variable in accordance with specification and interest of the PhD students. The subject "Discrete transforms and spectral analysis has been divided into two subjects for the time reasons. All chapters containing information about continuous and discrete transforms have been included in the subject of "Integral and discrete transforms", chapters containing information about wavelet transforms in stochastic signals processing have been transferred to the subject "Selected essays in mathematics II". These chapters are particularly specialised on processing of the non-Gauss random processes and on forming coherent functions by means of wavelets.

The interest of PhD students from specialised departments in wavelet transform ended in co-operation when trying to solve particular problems. Nowadays four topics are being processed in this way. For other interested students and employees of specialised departments a series of lectures about the theory and application of wavelet transform has been prepared. These lectures were presented at the seminar "Applied Mathematics" by the development project researchers and by K. Najzar, CSc. from the Charles' University in Prague. We would like to say thank you to Mr. K. Najzar for his help and participation.

4  Processing of the Syllabus and the Wavelet-Toolbox Manual

The syllabus consists of four separate sections.

The first syllabus section "Convolution and its filtration properties" is devoted to the basic terminology of the integral and discrete transform theory and filtration theory. This especially means the terms of convolution as an integral transform and distribution as a functional (generalised function). These terms are sometimes understood a little bit differently in technical sciences than in mathematics. In this section a general procedure of forming the windowed (scale) functions on the distribution basis is given. This terminology is very important to understand the following sections of the syllabus.

The second section with the title"Windowed transforms and some orthogonal discrete systems" continues the previous topic. It contains motivation for construction and definition of the windowed transforms, overview of the windowed transforms, their Fourier transforms and their use. Implementation of the mentioned transforms can be carried out by two ways, either by discrete formulas of numerical integration or directly by means of the discrete definition, i.e. by orthogonal (or mostly orthogonal) system functions. We found out that the second alternative has better results in registration of bigger anomalies (singularities). This procedure could be also very suitable for the definition of the discrete wavelet transform. The most important orthogonal discrete systems are listed in the syllabus. These systems are used not only for implementation of the transform but also for the reason of discrete transform definition. Computer numeric implementation and graphic presentation have been carried out by D. Horak, who is a student of the year 4 in the Faculty of Electrical Engineering and Information Science, study branch of Engineering Information Science and Applied Mathematics.

The last section of the syllabus describes the continuous and the discrete wavelet transform, their implementation and applications. Its title shows that the last section is devoted to the computer implementation of the discrete wavelet transform and is completed with practical teaching examples. This section also contains wavelet definition, way of wavelet construction, wavelet transform definition, examples of use and discussion of wavelet transform advantages in comparison with Fourier or windowed transforms respectively. There is given an overview of selected mother wavelet properties as well.

This syllabus will be used as a basis of university textbooks for the subject " Integral and Discrete Transforms" for both the engineering degree and the PhD studies. The syllabus is accessible to all students on a school computer net page of the Applied Mathematics Department.

J. Kubica, a student of the year 5 (school year 1997 - 1998), prepared under our management the " Wavelet-Toolbox User's Manual" for the use of the MATLAB software product. The first version of this manual is destined for students using this professional program. The Wavelet-Toolbox User's Manual is provided at the Applied Mathematics Department for the disposal of students passing appropriate courses as well as for all other interested persons from other departments.

Our efforts resulted into improving and increasing the quality of teaching new modern topics. Some of our partial results, reached in co-operation with our students, were presented on conferences and also published within the frame of the workshop " Modern Mathematical Methods in the Engineering" (June 1998). On the 6th conference of students VSTEZ in Vyskov on May 18 - 20, 1998, Mr. P. Praks, who was the year 4 student of the study branch of Engineering Information Science and Applied Mathematics successfully presented the results of his work (prepared under our management) on construction of wavelet precondition for the linear algebraic equations system.

5  History of Wavelets

Wavelets have generated a tremendous interest in both theoretical and applied areas, especially over the past ten years. However, the basic idea – representation of general functions in terms of simpler, fixed building blocks at different scales and positions – is already old about seventy years.

In 30's, the Littlewood-Paley techniques were developed to effective substitute the techniques based on Fourier series and Fourier transforms. In 50's and 60's, these developed into powerful tools for studying other things, such as a solution of partial equations and integral equations. It was realised by means of Calderon-Zygmund theory, an area of harmonic analysis. In 70's, the so-called atomic decomposition was widely used, especially in Hardy space theory. In the early 80's, Stromberg discovered the first orthogonal wavelets. This was done in the context of trying to further understand Hardy spaces, as well as other spaces used to measure the size and smoothness of functions. At the same time A. Grossmann and J.Morlet studied independently the wavelet transform in its continuous form and they are authors of the name "wavelet" for the building blocks. In the early to mid 80's, several groups realised that tools from Calderon-Zygmund theory and Littelwood-Paley representations had discrete analogies and could give unified view of many of the results in harmonic analysis. P.G. Lemarie and Y. Meyer, independent of Stromberg, constructed new orthogonal wavelet expansions. Soon, I. Daubechies gave a construction of wavelets, non-zero only on a finite interval and with arbitrary high, but fixes, regularity. During 90's many authors published many works about this topics and enriched the wavelet theory by new ideas. We mention the main types of wavelets below. Here we repeat the instructive scheme from [SS98] that shows the capability of the wavelets.

Figure 1: Many areas of science, engineering, and mathematics have contributed to the development of wavelets. Some of these are indicated surrounding the centre bubble.

6  Example

As example we show the result obtained by the student of 5 year, J. Kubica. He tests the time-frequency analyses of modelled signal using the wavelet packet decomposition based on standard and stacionary wavelet transform and compare it with the Fourier analysis

Modelled signal was constructed of up to 4 sinusoids acting on different intervals:

We processed Fourier analysis, standard and. stationary wavelet packet analysis of modelled signal using discrete wavelet decomposition filters based on the orthogonal Daubechies wavelet, a biorthogonal wavelet and the Meyer wavelet. Results can be viewed in following figures. We give also composed Fourier spectrum of standard wavelet packet decomposition coefficients for comparison

There are drawn wavelet functions on the following figures.

Fig.2: Daubechies wavelet

 

Fig.3: Biorthogonal wavelet functions

 

Fig. 4: Discrete Meyer wavelet

 

Fig.5: Modelled signal

 

Fig. 6: Fourier transform of modelled signal

 

Fig.7: Wavelet transform based on Daubechies wavelet (grey scale)

 

Fig.8: Wavelet transform based on Daubechies wavelet (frequencies)

 

Fig.9: Wavelet transform based on Daubechies wavelet (frequencies and time)

 

Fig.10: Stacionary wavelet transform based on Daubechies wavelet (grey scale)

 

Fig.11: Stacionary wavelet transform based on Daubechies wavelet (frequencies)

 

Fig.12: Stacionary wavelet transform based on Daubechies wavelet (frequencies and time)

 

Fig.13: Biorthogonal wavelet transform (grey-scale)

 

Fig.14: Biorthogonal wavelet transform (frequencies)

Fig.15: Biorthogonal wavelet transform (frequencies and time)

 

Fig.16: Meyer wavelet transform (grey-scale)

 

Fig.17: Meyer wavelet transform (frequencies)

 

Fig.18: Meyer wavelet transform (frequencies and time)

 

References

[K94]  KAISER G.: A Friendly Guide to Wavelets. Boston-Basel-Berlin: Birkhaser, 1994. 300 pp. ISBN 0-8176-3711-7.

[K93]  KOORNWINDER T.: Wavelets: An Elementary Treadment of Theory and Applications. Singapore-New Jersey-London-HonkKong: World Science. 1993. 225 pp. ISBN 981-02-1388-3.

[SN97]  STRANG G., NGUYEN T.: Wavelets and Filter Banks. Wellesley: Cambridge Press, 1997. 520 pp. ISBN 0-9614088-7-1.

[SS98]  SWELDENS W., SCHRODER P.: Building Your Own Wavelets at Home. Available from www: <URL: http://cm.bell-labs.com/who/wim/papers/athome/index.html>.

 

This paper is produced as a part of research supported by grant No. 201/96/0665 of the Grant Agency of the Czech Republic and research project No. CEZ: J17/98:272400019.