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Titre: Traitement du Signal et Ondelettes Master Ingénierie Mathématique

Auteurs: Alain Yger

Ecole/Université: Néant

Résumé: Voir le document

Extrait du sommaire:

1 Introduction : signaux et images 1
1.1 De l’analogique au digital 1
1.2 Du déterministe à la stochastique 4
1.3 L’analyse de Fourier 5
1.3.1 Spectre d’un signal ou d’une image d’énergie finie 5
1.3.2 Un premier souci : le principe d’incertitude 9
1.3.3 Le spectre des signaux périodiques 10
1.3.4 Un souci numérique : le problème de l’échantillonnage 12
2 L’outil Fourier 15
2.1 La transformée de Fourier discrète d’une signal digital ou d’une image 15
2.1.1 Définition de la transformation ; notion de dft 15
2.1.2 Aspects algorithmiques de la dft ; algorithmes de fft 19
2.2 Sommes d’exponentielles et transformation de Fourier 25
2.3 Corrélation et algorithme autorégressif 30
2.3.1 Matrice de covariance, corrélation, densité spectrale de puissance 30
2.3.2 La recherche des paramètres d’autorégressivité 34
2.4 Les transformations du type Wigner-Ville 37
3 L’analyse temps-échelles et ses applications 45
3.1 Pourquoi analyser en temps et enéchelles ? 45
3.2 Le principe de l’analyse temps-échelles continue (Continuous Wavelet Transform) 46
3.3 Deux applications de la transformation temps-échelles continue 54
3.3.1 Séparation signal/bruit 54
3.3.2 Restauration d’un signal depuis les extréma locaux de jCWT[s]j 57
3.4 La décomposition de Franklin d’un signal digital 59
3.5 La perception visuelle » d’une information et l’algorithme pyramidal 62
3.6 Le concept d’analyse multi-résolution 68
3.6.1 L’instrument d’analyse 68
3.6.2 De l’information à son « résumé ». 70
3.6.3 La mère de l’analyse multi-résolution 73
3.6.4 De l’information à ses détails 74
3.6.5 La décomposition traitée sur un exemple 76
3.6.6 Les deuxécueils de l’analyse multi-résolution 78
3.7 Le matching pursuit 79
3.8 La Proper Orthogonal Decomposition » 83
4 La notion d’entropie et les algorithmes d’analyse du signal afférents 87
4.1 La notion d’entropie dans le cadre Hilbertien 87
4.1.1 Dimension théorique d’une information 89
4.2 Le splitting lemma (lemme de décomposition) dans le cadre dyadique 90
4.3 L’algorithme Split and Merge » de V. Wickerhauser 93
4.3.1 Le cadre des signaux digitaux 93
4.3.2 Le cas des images digitales 96
4.3.3 Implémentation algorithmique en 1D 96
4.3.4 La généralisation au cadre p-adique 101
4.4 La décomposition temps-fréquences adaptative 106
5 L’outil Fourier, combiné avec l’algorithmique Hilbertienne 113
5.1 Les transformations de type Radon et leur inversion 114
5.1.1 Aspects théoriques : deux transformations mathématiques 114
5.1.2 Aspects pratiques : l’imagerie médicale 116
5.1.3 Aspects numériques en 2D 118
5.2 Face au problème de l’extrapolation des informations 123
6 Quelques aspects du traitement des signaux : le filtrage 129
6.1 Filtres digitaux ; réalisabilité, causalité, stationnarité 129
6.2 Le design d’un filtre digital stationnaire à partir de sa fonction de transfert 131
6.3 Notion de z-transformée d’un filtre digital 134
6.4 Quelques rappels d’algèbre concernant C(X) et IR(X) 136
6.5 Filtres A.R, filtres A.R.M.A 138
6.6 Les filtres analogiques 139
6.6.1 Signaux analogiques : le point de vue distribution » 139
6.6.2 Filtres analogiques causaux, filtres rationnels 141
6.6.3 Filtres analogiques rationnels de Butterworth, de Tchebychev, etc. 142
6.6.4 Le passage de l’analogique au digital 147
7 Quelques notions de filtrage stochastique des processus discrets 153
7.1 Densité spectrale de puissance d’un processus stationnaire 153
7.2 Action d’un filtre digital sur un processus discret 154
7.3 Aspects pratiques du calcul de l’estimation spectrale stochastique 155
7.4 Filtrage de Wiener et déconvolution 159
7.4.1 Le filtrage de Wiener des processus discrets 159
7.4.2 La déconvolution des processus stationnaires discrets 160
7.4.3 Le filtrage de Wiener-Hopf 162

Ondelettes et traitement du signal et d’image 32

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