# Internship 2016 - Master MVA (2015-2016) > (Fr) Ce [dépôt git](https://en.wikipedia.org/wiki/Git) contient les sources LaTeX et Python, notamment, des recherches effectuées pour mon stage de recherche de fin de master 2. Pendant ma dernière année de formation à l'[ENS Cachan](http://www.ens-cachan.fr/), en 2015-2016, j'ai suivi le master MVA. > - [Informations sur le master MVA](http://www.math.ens-cachan.fr/version-francaise/formations/master-mva/) et [aussi en anglais ?](www.cmla.ens-cachan.fr/version-anglaise/academics/mva-master-degree-227777.kjsp). > - [Plus d'informations sur le site de l'Université Paris-Saclay](http://www.universite-paris-saclay.fr/fr/formation/master/m2-mathematiques-vision-apprentissage). ## Informations sur le stage / Information about my internship - *Où / where* : [équipe LIB / BIG team](http://bigwww.epfl.ch/), [EPFL](http://www.epfl.ch/), [Lausanne](https://fr.wikipedia.org/wiki/Lausanne) en [Suisse](https://fr.wikipedia.org/wiki/Suisse) / in [Switzerland](https://en.wikipedia.org/wiki/Switzerland) - *Quand / when* : avril / April 2016 → août / August 2016, - *Avec qui / with whom* : [Julien Fageot](http://bigwww.epfl.ch/fageot/) et / and [Michael Unser](http://bigwww.epfl.ch/unser/), - *Thèmes / themes* : splines, ondelettes / wavelets, espaces et normes de Sobolev / Sobolev spaces and norms, TV-L1 and L1 minimization, théories des opérateurs / operators theory, etc. - *Sujet / topic* : **« A theoretical study of steerable convolution operators, and possible applications to sparse processes for 2D images »**. ---- ### Abstract [du rapport](https://goo.gl/xPzw4A) / Abstract of [my report](https://goo.gl/xPzw4A) > This report sums up and presents the research I did between April and August 2016, > in theoretical function analysis and operator theory. > We focus mainly on two aspects: convolution operators in any dimension; and steerable convolution operators in two dimensions, for images, and their possible applications. > Most of our results are valid regardless of the dimensions, so we try to keep a general setting as long as we can, and then we restrict to operators on $2$D images for the study of steerable operators. > > We start by recalling the common notations > used in signal processing, splines theory and functional analysis, > and then by recalling the main properties of a fundamental tool for this domain, the Fourier transform F. > We present the main goals of our research, in order to motivate the interests of such a theoretical study of convolution operators from a practical point of view. > We chose to follow a very didactic approach, and so we redefine "from scratch" > the theory of functional operators, along with the most important results. > Our operators can have structural as well as geometric properties, > namely linearity or continuity, > and translation-, scaling-, rotation-invariance, unity > -- all these properties being already well-known -- or steerability. > > However, we study extensively the links between all these properties, and we present many theorems of characterizations of different properties. > To the best of our knowledge, this document is the first attempt to summarize all these results, > and some of our latest characterizations seem to be new results. > > After a very broad section on operators, we focus on steerable convolution operators $G$, mainly in $2D$, > as they appear to be the natural framework for shape- and contour-detection operators for images. > Our main results consist on characterizations of steerable convolution operators, first written as a sum of modulated and iterated real Riesz transforms. > Then adding the $\gamma$-scale-invariance gives a nicer form, as a composition of a fractional Laplacian $(-\Delta)^{\gamma/2}$, some directional derivatives $D_{\alpha_i}$ and a "nice" part $\nice{G}$, > and another form as a composition of elementary blocks that all have the same form $G_{\lambda, \alpha}$. > This last form is very appealing for implementation, as it is enough to program the elementary block, and to compose it to obtain every $2$D steerable gamma-scale-invariant (gSI) convolution operator, > and has a strong theoretical interpretation: a $G$ gSI and steerable of order n_G gets > decomposed as a product of elementary blocks, all 1-SI > steerable of order 1 or 2. > > We conclude by presenting the results of some experiments on $2$D stochastic processes, > in order to illustrate the effects of our elementary blocks as well as more complicated operators. > We highlight some properties on the examples, like their trade-of between the directionality of $D_{\alpha_i}$ and the isotropy of $(-\Delta)^{\gamma/2}$. > Our operators could also be used to develop new splines (\ie new sampling schemes), and new Green's functions (\ie new denoising and data recovery algorithms), > but we did not have the time to fully study these aspects. ---- ## Sous-dossiers / Folders : ### Productions - [biblio](./biblio): réferences / bibliographic references, - [report](./report): rapport de stage / internship report (LaTeX source et/and PDF) (*almost done*). Le document se trouve ici / the document is here : https://goo.gl/xPzw4A - [slides](./slides): slides pour la présentation finale / final oral presentation (LaTeX source et/and PDF) (*not yet*). Le document se trouve ici / the document is here : https://goo.gl/vm8WPF ### Code - [src](./src): programmes et scripts / source code and scripts (*not much!*), - [fig](./fig): figures et/and illustrations. ---- ## À propos / about ### Copyright / Copyleft - (Fr) Tout le contenu est ma propriété, ©, 2016, [Lilian J. Besson](http://perso.crans.org/besson/) (sauf mentions contraires). - (En) All this content is my property, ©, 2016, [Lilian J. Besson](http://perso.crans.org/besson/) (except if stated otherwise). ### Licence / License - (Fr) Ce projet entier est placé sous [la licence libre MIT](http://lbesson.mit-license.org/). - (En) This entire project is publicly released under [the MIT open-source license](http://lbesson.mit-license.org/).