### Sciences

## Subject: NUMERICAL ALGORITHMS FOR SIGNALS AND IMAGES PROCESSING (A.A. 2020/2021)

### master degree course in MATHEMATICS

Course year | 1 |
---|---|

CFU | 6 |

Teaching units |
Unit Elaborazione numerica di segnali e immagini
Models and Application (lesson)
- TAF: Compulsory subjects, characteristic of the class SSD: MAT/08 CFU: 6
Silvia BONETTINI |

Exam type | oral |

Evaluation | final vote |

Teaching language | English |

### Teachers

### Overview

The aim of this course is to provide the basic concepts of Digital Signal Processing (DSP), consisting in methods and algorithms for signal representation, analysis, compression and enhancement on a digital device.

### Admission requirements

- Differential calculus for real functions of real variables.

- Integral calculus for real functions of one real variable.

- Basics of linear algebra.

- Basics of numerical analysis.

- Basics of computer programming.

### Course contents

An audio track, an ECG, a seismic vibration, but also a picture or a radiography, all are examples of signals, in one or two dimensions (images).

From a mathematical point of view, signals are nothing but real functions of one or two real variables, which have to be represented on a digital device, such as a PC.

A first question to be answered is just how the information is modified when passing from a continuous domain, where function live, to a discrete and finite one, which is typically associated to a digital device.

Moreover, signals have often to be processed for improving their quality or extracting some interesting feature: for example, one could want to remove the surrounding noise from an audio track or extract the edges of objects represented in a picture.

Finally, compression techniques are needed for storing a signal optimizing the memory requirement.

Representation, processing, compression are three important task in signal analysis which will be presented in this course, including their mathematical foundation and the related numerical algorithms.

Some specific real world applications will also be addressed with the related numerical experience.

The main topics are listed below:

- Definition of digital signal

- Acquisition of a digital signal

- Fourier transform

- The sampling theorem

- Convolution operators

- Discrete Fourier Transform

- Circulant matrices

- Filtering techniques

- Image restoration

- Wavelet functions

- Signal compression

### Teaching methods

- Lectures in the classroom, with illustration of the content chapters by means of slides and blackboard. - Laboratory exercises on the numerical solution of the problems described during the lectures, for the practical verification of structured concepts that are the backbone of the course program.

### Assessment methods

Oral examination.

### Learning outcomes

This course enables the following skills:

- analysis of the frequency content of a signal

- application of filtering techniques in Matlab environment

- multiresolution analysis with the Matlab Wavelet Toolbox.

### Readings

[1] M.Bertero, P.Boccacci, 1998, Introduction to Inverse Problems in Imaging, IOP Publishing, Bristol.

[2] S. Mallat, 2009, A wavelet tour of signal processing, 3rd edition, Academic Press, Burlington (MA).