From 94196cadf096cb14c1cab04e3cb1baa69e8c21a9 Mon Sep 17 00:00:00 2001 From: mdesco Date: Mon, 18 Nov 2024 20:11:00 -0500 Subject: [PATCH] tp4 --- static/courses/imn359/tp04.tex | 259 ++++++++++++++++++++++++++++ static/courses/imn359/tp04_2023.pdf | Bin 118289 -> 0 bytes static/courses/imn359/tp04_data.zip | Bin 0 -> 1029953 bytes 3 files changed, 259 insertions(+) create mode 100644 static/courses/imn359/tp04.tex delete mode 100644 static/courses/imn359/tp04_2023.pdf create mode 100644 static/courses/imn359/tp04_data.zip diff --git a/static/courses/imn359/tp04.tex b/static/courses/imn359/tp04.tex new file mode 100644 index 0000000..9a5716e --- /dev/null +++ b/static/courses/imn359/tp04.tex @@ -0,0 +1,259 @@ +\documentclass{article} +%%%%%%%%%%%%% Definitions %%%%%%%%%%%%%%%%%% + +%%%% Largeur et tailles des marges +\oddsidemargin 0in +\evensidemargin 0in +\textwidth 6.5in + +%%%%% Pour les figures, graphiques et math (amssymb) +\usepackage{color,graphicx,float,epsfig,amssymb} + +%%% Pour avoir les accents %%%%%%%%%%%%%%%%% +\usepackage[latin1]{inputenc} +\usepackage[cyr]{aeguill} +\usepackage[francais]{babel} +%%%%%%%%%% + + +%%%%%%%%%%%%% Document %%%%%%%%%%%%%%%%%%%%% +\title{TP 4 - Au delà de Fourier \\ + (DCT, DCT locale, multirésolution et ondelettes)} +\author{Maxime Descoteaux} +\date{\today} +\begin{document} +\maketitle +Vous devez rédiger un rapport +et me remettre un zip avec votre code python. +Commentez le code et assurez-vous que je puisse +reproduire vos résultats et figures. +Séparez votre code en différents fichiers pour faciliter la +lecture. Des points seront attribués pour la qualité du document +et ses figures, et la qualité du code python (10 points). + + +L'approximation linéaire garde que les $M$ coefficients de plus +basses fréquences dans l'espace de Fourier, c'est-à-dire les $M$ +premiers coefficients. Une approximation nonlinéaire +garde que les $M$ coefficients les plus élévés de la décomposition ou +transformée. Dans ce devoir, quand vous comparez des transformées, +assurez-vous de toujours prendre les mêmes $M$. + +\vspace{1cm} +\begin{enumerate} + +\item {\bf Appproximations 1D. [10 points]} + À partir du \emph{piece regular} ou d'un signal 1D de votre choix, + reproduisez les + figures de mon code \emph{locDCT\_demo.py}, disponible dans la Demo08. + \begin{enumerate} + \item Illustrez votre signal. Combien de points dans votre signal?\\ + (Si vous choississez un signal à vous, prenez un signal de taille + ``raisonable'' et assurez-vous qu'il se découpe bien.) + \item Décrivez votre choix de taille de segment et nombres de + coefficients à couper $M$. + \item Faites des approximations linéaires et non-linéaires de votre + signal + \item Rapporter les erreures quadratiques moyennes de toutes les + approximations dans un tableau. + \end{enumerate} +\vspace{4mm} + + +%\vspace{1cm} +%\begin{enumerate} +% \item {\bf L'approximations multirésolution de Shannon.} +% \\ +% %% Watch out avec la definition des conditions +% \begin{enumerate} +% \item On défini l'espace +% $\mathbf{V}_j$ comme l'ensemble des fonctions dont la transformée +% de Fourier a un support limité dans l'interval $[-2^{j} \pi, +% 2^{j} \pi]$. +% Démontrez que $\{\mathbf{V_j}\}_{j \in \mathbb{Z}}$ est une approximation +% multirésolution avec la fonction d'échelle suivante: +% $$ +% \phi(t) = \frac{\sin \pi t}{\pi t} +% $$ + +% Il vous faut donc vérifier les 5 propriétés de la définition de +% multirésolution donnée en classe. \\\\ +% (Indice) Le théorème d'échantillonnage vous sera utile pour montrer +% que $\{\phi(t - n)\}_{n \in \mathbb{Z}}$ forme une base orthonormale de +% $V_0$. + + +% \item Trouvez les filtres de décomposition passe-bas et passe-haut, +% $h$ et $g$, associés à cette multirésolution. + +% %\item Montrez que la fonction $\phi$ suit la relation suivante: +% %$$ +% %\phi(x) = \phi(2x) + \sum_{k \in \mathbb{Z}} \frac{2(-1)^k}{(2k+1)\pi} +% %\phi(2x - 2k - 1). +% %$$ + +% % \item Trouvez une expression pour l'ondelette $\psi$ associée à la +% % fonction d'échelle $\phi$. + +% \end{enumerate} + +% \vspace{1cm} +% \item {\bf Démontrez le théorème de reconstruction de Haar. } + +% Supposons que +% $$ +% f = f_0 + w_0 + w_1 + ... + w_{j-1} +% $$ +% avec +% $$ +% f_0(x) = \sum_{k\in \mathbb{Z}} a_k^0 \phi(x-k) \in V_0 \quad +% \textrm{et} \quad +% w_{j'}(x) = \sum_{k\in\mathbb{Z}} b_k^{j'} \psi(2^{j'}x - k) \in W_{j'} +% $$ +% pour $0 \leq j' < j$. Alors, +% $$ +% f(x) = \sum_{k\in\mathbb{Z}} a_l^j \phi(2^jx - l) \in V_j +% $$ +% o\`u les $a_l^{j'}$ sont obtenus récursivement pour $j' = 1$, ensuite +% $j' = 2$, jusqu'à $j' = j$ avec la relation suivante: +% $$ +% a_l^{j'} = \left \{ +% \begin{array}{ll} +% a_k^{j'-1} + b_k^{j'-1} & \textrm{si $l = 2k$ est pair} \\ +% a_k^{j'-1} - b_k^{j'-1} & \textrm{si $l = 2k+1$ est impair} +% \end{array} \right . +% $$ +% \vspace{1cm} +% \item {\bf Splines cubiques 1D}. +% Faites la décompositions en ondelettes du signal 'piece-regular' +% à partir des splines cubiques. Utilisez les valeurs du filtre miroir +% conjugué $h$ dans la table 7.1 donnée en classe. +% Je vous donne les fonctions suivantes pour vous aider: +% (\emph{subsampling}, \emph{upsampling}, \emph{reverse}, \emph{cconv}) +% \\ + +% i) Montrez les détails et les +% approximations, $W_j$ et $V_j$, intermédiaires.\\ + + +% ii) Montrez l'orthogonalité de la transformation d'ondelettes +% (conservation d'énergie). Est-ce que l'énergie est parfaitement +% conservée? Pourquoi? \\ + +% iii) Faites la reconstruction en montrant les reconstructions +% partielles du signal.\\ + +% iv) Calculez l'erreur sur le signal reconstruit. \\ + +% v) Pour les échelles $2^j$, $4 \leq j \leq 8$, visualisez la forme +% des ondelettes provenant de votre analyse. +% \vspace{1cm} + + +% \item {\bf Ondelettes de Daubechies 2D.} +% À partir des coefficients du filtre h de la Table 7.2 et de la même +% image que pour la question précédente, implémentez +% la décomposition/reconstruction 2D en ondelettes de Daubechies avec +% $p=1, 2, 4$ et 10 moments nuls. \\ + +% i) Faites les décompositions et reconstructions en montrant les +% détails, approximations et reconstructions partielles intermédiares +% (semblable à la question 4). +% Que remarquez-vous? Plus +% particulièrement, notez vos observations sur les régions régulières +% et aux contours de l'image.\\ + +% ii) Faites un tableau des erreurs de reconstruction de l'image, +% dépendant de l'ondelette utilisée. Que remarquez-vous?\\ + +% iii) Pour chaque $p$, générez la forme des ondelettes 2D pour les +% orientations diagonal, horizontal, vertical. Les fonctions mesh ou +% surf vous serons utile. Que remarquez-vous?\\ +% \vspace{1cm} + +\item {\bf Approximations dans les bases de Fourier et d'ondelettes} + sur une image de votre choix. \\ + \begin{enumerate} +\item [5 pts] a) {\bf Approximation de Fourier.} L'approximation linéaire garde + que les $M$ coefficients de plus basses fréquences dans l'espace + de Fourier. Faites l'approximation linéaire dans Fourier gardant que + $M$ coefficients, pour 3 $M$ différents et rapportez le SNR des + images approximées ainsi que les erreurs relatives d'approximation + sur les \emph{plot}. (Vous avez déjà fait cela dans le TP3) + +\item [10 pts] b) {\bf Approximation avec cosinus discrets et des cosinus discrets + locaux.} En premier temps, faites l'approximation linéaire avec + une base de cosinus discrets (DCT) et une base de cosinus discrets + locaux (locDCT), gardant que les $M$ coefficients de plus basses + fréquences, pour 3 $M$ différents et rapportez le SNR des images + approximées ainsi que les erreurs relatives d'approximation sur les + \emph{plot}. Utilisez les + fonctions \emph{dct2.m} et \emph{idct2.m}, données en classe (vous + devez étendre ce que j'ai déjà fait en 1D en 2D). + +\item [10 pts] c) {\bf Approximation d'ondelettes de Haar périodiques}. À partir + de votre ondelette de Haar, faites + l'approximation linéaire sur cette base en utilisant 3 $M$ + différents. Rapportez le SNR des images approximées et les erreurs relatives + d'approximation sur les + \emph{plot}. + %Vous pouvez faire la version algébrique (comme en + %classe) ou celle avec les convolutions circulaires. + +%% \item [BONUS - 20 pts] d) {\bf Approximation d'ondelettes de Daubechies}. Utilisez une +%% des ondelettes de Daubechies (e.g. [h, g] $=$ +%% compute\_wavelet\_filter('Daubechies', 4) pour l'ondelette de +%% Daubechies de support 4). Faites l'approximation linéaire pour 3 $M$ +%% différents. Rapportez le SNR des images approximées et les erreurs +%% relatives d'approximation. Pour celle-ci, vous n'aurez pas le choix +%% que d'implémenter la version avec la convolution. + +\item [10 pts] d) {\bf Approximations nonlinéaires.} Une approximation nonlinéaire + garde que les $M$ coefficients les plus élévés de la décomposition + dans une base. Faites les approximations nonlinéaires + avec les bases des parties (a) à (d). + +\item [10 pts] e) Faites un {\bf tableau des SNR et des erreurs + d'approximations linéaires et nonlinéaires}. Que remarquez-vous? + +%% \item [BONUS - 10 pts] f) Tracez la courbe des erreurs +%% d'approximation en fonction de $M$. Pour un graphe optimal, mettez +%% sur l'axe x, $\log_{10}(M/N)$ et sur l'axe y, $\log_{10}( |f - f_M|^2 / +%% |f|^2)$, ou $N$ est le nombre de pixel dans l'image, $M$ le nombre +%% de coefficients, $f$ l'image originale et $f_M$ l'image approximée. \\ +%% Utilisez plus de 3 $M$ pour tracer cette courbe. Pensez à +%% votre affaire! Il y a moyen de faire ça très simplement et rapidement. \\ +%% (indice: Conservation d'énergie) + +\end{enumerate} + + \begin{figure}[!b] + \begin{center} + \includegraphics[height=12cm]{image.eps} + \caption{4 images classiques en traitement d'images à approximer + nonlinéairement.} + \end{center} +\end{figure} + + + + +\newpage + + +\item {\bf Approximations sur des images de différentes complexités}. \\ + +\begin{enumerate} +\item [20 pts] i) Faites les approximations nonlinéaires pour les 4 images de la + figure 1 (regular3, phantom, lena et le mandrill). Que + remarquez-vous? Les approximations dépendent-elles de la + ''complexité'' de l'image? Expliquez. + +\item [5 pts] ii) Selon vous, quelle est la meilleure base? + Pourquoi? + +\end{enumerate} +\end{enumerate} + + +\end{document} diff --git a/static/courses/imn359/tp04_2023.pdf b/static/courses/imn359/tp04_2023.pdf deleted file mode 100644 index 87e9f913078a35af6fb432723912125502131138..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 118289 zcmdSA1yo$kwl3PZYXSr(KyU)VU4sQD!KH%*Y1{%qI=DlC-~o~lTpD-i;10pv2?TGv zfu?!A_xb<(?mqY2clLO1yfNNeV~rl&YZXxRB z6F-xyl_S2C6h5DZt&5$PJ(GZ_xERwvFMK{_2PZFE4<;x?%CR5h^70{YsOdYzfuKS2B@yJ34BWnjX%Q%Xlk2) zd<@BVra1`T))Bp&ZG_P#i{@%qS7zh^ao^I?`1J!J(>vU1PJN>M&z6C&;+scf*&I~- z@mj~;n56lf(8&43!MdIG*>Mz#Qi4JWq|uFdvdw0c`O=ijl;~J z!h!k+t#rZMT7_(DLUdyx^mH>C#_uR;-cp7??Xcm3plM?NISZLU{OXEc+lah43^V!| zVc%jIzMbnfpX!bsq_c5(!?3Vy*`EX3~&~9a6lFycD+B zZ)Vx{!W}>Hi`uSJfqP;e>kob_$|nD+mD@$!+PdW3MTEGdca=D~x$*9?ab-tVCq1n!f1Y|iWEHZEgUQTI>vC7-sj~Q16tC{?O9Xt7IIYvsn_cGg`P8C zM~tX5@{jnxJ{ORdxM{JW3GX#FW&^%7)OEB?LU4_58Bd1;yxQhM#IKNgKERX47YV^= 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