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                <title xml:lang="en">Transformations rigides sur les images numériques 2D : analyse combinatoire et topologique</title>
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                <authority type="supervisor">Michel Couprie</authority>
                <authority type="jury">Éric Andres [Président]</authority>
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                <authority type="jury">Jean-Pierre Reveillès</authority>
                <authority type="jury">Yukiko Kenmochi</authority>
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                <term xml:lang="en">Topological invariance</term>
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                <term xml:lang="en">Combinatorial structure</term>
                <term xml:lang="en">Digital image</term>
                <term xml:lang="en">Discrete rigid transformation</term>
                <term xml:lang="fr">Invariance topologique</term>
                <term xml:lang="fr">Algorithmique discrète</term>
                <term xml:lang="fr">Structure combinatoire</term>
                <term xml:lang="fr">Image numérique</term>
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              <p>In this thesis, we study rigid transformations in the context of computer imagery. In particular, we develop a fully discrete framework for handling such transformations. Rigid transformations, initially defined in the continuous domain, are involved in a wide range of digital image processing applications. In this context, the induced digital rigid transformations present different geometrical and topological properties with respect to their continuous analogues. In order to overcome the issues raised by these differences, we propose to formulate rigid transformations on digital images in a fully discrete framework. In this framework, Euclidean rigid transformations producing the same digital rigid transformation are put in the same equivalence class. Moreover, the relationship between these classes can be modeled as a graph structure. We prove that this graph has a polynomial space complexity with respect to the size of the considered image, and presents useful structural properties. In particular, it allows us to generate incrementally all digital rigid transformations without numerical approximation. This structure constitutes a theoretical tool to investigate the relationships between geometry and topology in the context of digital images. It is also interesting from the methodological point of view, as we illustrate by its use for assessing the topological behavior of images under rigid transformations</p>
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              <p>Dans cette thèse, nous étudions les transformations rigides dans le contexte de l'imagerie numérique. En particulier, nous développons un cadre purement discret pour traiter ces transformations. Les transformations rigides, initialement définies dans le domaine continu, sont impliquées dans de nombreuses applications de traitement d'images numériques. Dans ce contexte, les transformations rigides digitales induites présentent des propriétés géométriques et topologiques différentes par rapport à leurs analogues continues. Afin de s'affranchir des problèmes inhérents à ces différences, nous proposons de formuler ces transformations rigides dans un cadre purement discret. Dans ce cadre, les transformations rigides sont regroupées en classes correspondant chacune à une transformation digitale donnée. De plus, les relations entre ces classes de transformations peuvent être modélisées par une structure de graphe. Nous prouvons que ce graphe présente une complexité spatiale polynômiale par rapport à la taille de l'image. Il présente également des propriétés structurelles intéressantes. En particulier, il permet de générer de manière progressive toute transformation rigide digitale, et ce sans approximation numérique. Cette structure constitue un outil théorique pour l'étude des relations entre la géométrie et la topologie dans le contexte de l'imagerie numérique. Elle présente aussi un intérêt méthodologique, comme l'illustre son utilisation pour l'évaluation du comportement topologique des images sous des transformations rigides</p>
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