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# Transformation Theory for Computer Science

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## Introduction

This course provides an overview of the use of integral transformations for computer science applications.  In particular, the first part of the course is dedicated to the study of Fourier/Hilbert series and transforms, paying particular attention to numerical calculations.  Other transforms, such as Gabor filters are described, not only to illustrate research methods in pattern recognition, but as a motivation for multiresolution theory and wavelets. The second part of the course is dedicated to the study and application of the wavelet transform. In particular, present research using wavelets for image analysis and compression is described.

Upon completion of this course, a student should be able to write programs in Matlab or GNU/Octave to analyze signals and/or images, useful for contemporary pattern recognition in computer science.

### Learning outcomes

1. Knowing the theory and techniques of Fourier transforms and Wavelet solutions to integrate artificial intelligence problems.
2. Deepening the mathematical theory of complex vector spaces, both analytically and numerically, to understand how and when to use processed to solve real problems.
3. Show the connection between the Fourier theory and Wavelets.
4. Present techniques for the Fourier transformation of short time to solve problems in the analysis of voice, audio and images.
5. Deepening the use of wavelets to problems that are of short duration in time.
6. Develop wavelet algorithms for multi-resolution problems.

### Teachers:

• David Olivieri  (dnolivieri at gmail.com)
• Mª José Lado Touriño (mrpepa at uvigo.es)

• None in 2010

## Course Contents

1. (David). Vector spaces
- Basic 3-dim vector algebra
- n-dim linear algebra
- Hilbert Space
- Eigenvalue problem and numerical methods
2. (David). Fourier Transforms
- Fourier series expansions
- Fourier Integrals: convolutions, correlations
- FFT algorithm and numerical algorithms
- power spectrum estimation, maximum entropy
- STFT  and phase Space
- motivation for multiresolution
- optimal filtering
- Applications to computer science problems
3. (David). Other Transforms
- Gabor transform
- Laplace transforms
- Chirp transform
4. (Pepa). Wavelets
- Multiresolution Theory

## Course Activities

1. Classroom Lessons (2 ECTS).- It will consist mainly on magistral sessions and readings to introduce the work to be done in the other activities.
2. Experiments and Practices (1 ECTS).- It will consist on the develop of little examples and exercises supervised by the teachers.
3. Seminars (2 ECTS).- It will consist mainly in the presentation of a concrete item by students in small groups.
4. Conferences (0,5 ECTS).- It will consist on the elaboration of an original scientific article where a practical application of the information theory is explained, and its oral exposition.
5. Tutorials (0,5 ECTS).- It will consist in the solution of some problems proposed by the teachers, and the pursuit of the seminars and conferences.

## Course Assessment

#### Evaluation Procedure for any student

Problem sets will be assigned each week and students will be required to submit solutions within an allotted time period. The homework problem sets will cover the material discussed in class and are designed to gain practical experience with concepts.

The evaluation of the course is based upon the level of successful execution of the homework solutions.  Along with solving the assigned problems, a percentage of the evaluation of each homework set will be allotted to both presentation of the results as well as a proposed solution to a related problem that each student may propose.   There will be a penalty for late submissions. The grading system will be the following:

Successful completion:              6 points max.
presentation of solutions:          1 point
Proposed original problem:        3 points max.
Late submission:                     -1 /each week

At the end of each subcourse, students will be selected to defend their proposed problems or discussions.  The successful defense of this problem will count as a complete homework set.

89f4e081ec52155ddf8f9c4597c2a42d