Quaternions et Algèbres Géométriques, de nouveaux outils pour les images numériques couleur. (Quaternions and Geometric Algebras, new tools for digital colour images)
چکیده
The main subject of this PhD thesis is colour image processing. The first methods dealing with these images consisted in applying existing greyscale processing alorithms on each of the three colour components. Colour processing has improved using perceptual colour spaces but also by considering colours as vectors. In this work, we follow the idea of colour modelization and we propose to encode their vectorial information into mathematical models in order to manipulate them globally and geometrically. Three formalisms are presented to cope with colour : complex numbers, quaternions and geometric algebras (also called Clifford algebras). New colour tools are proposed to analyse the digital spectrum embedded in each of these formalisms and the definition of Fourier transforms. We give the main advantages and drawbacks of each model, namely : impossibility for the complex numbers to represent whole colour vectors that needs at least three components to be described properly ; distinction between objects and operations on objects (projections, rotations, . . . ) with geometric algebras whereas it is not possible with quaternions. We then showed that the quaternionic Fourier transform analyse colours with a direction whereas the Clifford G3 Fourier transform has not got any direction to analyse the colour so it treats every colour channel independently. Eventually one of the main applications is the definition of a spatial colour edge detector filter using these formalisms. Discipline : traitement du signal et des images. Mots clés : Analyse d’images, espace numérique couleur, quaternions, algèbres géométriques, transformations de Fourier, filtrage spatial et fréquentiel. Patrice Denis Laboratoire Signal Image Communication Bât. SP2MI Téléport 2, Bd Marie et Pierre Curie BP 30179 86962 Futuroscope Cedex