Internship 2016 - Master MVA (2015-2016)

(Fr) Ce dépôt git contient les sources (LaTeX et Python, notamment) des travaux effectués pour mon stage de recherche de fin de master 2. Pendant ma dernière année de formation à l'ENS Cachan, en 2015-2016, j'ai suivi le master MVA.

Informations sur le stage / Information about my internship

  • Où / where : équipe LIB / BIG team, EPFL, Lausanne en Suisse / in Switzerland
  • Quand / when : avril / April 2016 → août / August 2016,
  • Avec qui / with whom : Julien Fageot et / and Michael 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 ».
  • Note finale / final grade : I got 20/20 for my internship (as given by my supervisors) and 18.43/20 for my average grade for my Master degree. Sweet !

Sous-dossiers / Folders :

  • biblio: réferences / bibliographic references,
  • report: rapport de stage / internship report (LaTeX source et/and PDF) (completely done). Le document se trouve ici / the document is here :
  • slides: slides pour la présentation finale / final oral presentation (LaTeX source et/and PDF) (completely done). Le document se trouve ici / the document is here :


Abstract du rapport / Abstract of my report

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 2D 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 an invertible part G0, 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 2D 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 2D 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.

À propos / about

Merci / Thank you

  • Merci grandement à / Thanks a lot to : Michael Unser & Julien Fageot (advisors), Virginie Uhlmann (collaborator), Thibault Groueix ("desktop-mate")
  • Merci à tous les membres du LIB à l'EPFL (Laboratoire d'Imagerie Biomédicale) / Thanks to all the members of the BIG team at EPFL (Biomedical Imaging Group)
  • Merci spécial à / Special thanks to : Gabriel Peyré
  • (Fr) Tout le contenu de ce dépôt git est ma propriété, © 2016, Lilian Besson (sauf mentions contraires).
  • (En) All the content in this git repository is my property, © 2016, Lilian Besson (except if stated otherwise).

Licence / License :

Creative Commons License - CC-BY 4.0