La segmentation hiérarchique est une méthode pour produire des partitions qui représentent une même image de manière de moins en moins fine. En même temps, elle sert d'entrée à la recherche d'une partition optimale, qui combine des extraits des diverses partitions en divers endroits. Le traitement hiérarchique des images est un domaine émergent en vision par ordinateur, et en particulier dans la communauté qui étudie les images hyperspectrales et les SIG, du fait de son capacité à structurer des données hyper-dimensionnelles. Le chapitre 1 porte sur les deux concepts fondamentaux de tresse et de treillis énergétique. La tresse est une notion plus riche que celle de hiérarchie de partitions, en ce qu'elle incorpore, en plus, des partitions qui ne sont pas emboîtées les unes dans les autres, tout en s'appuyant globalement sur une hiérarchie. Le treillis énergétique est une structure mixte qui regroupe une tresse avec une énergie, et permet d'y définir des éléments maximaux et minimaux. Lorsqu'on se donne une énergie, trouver la partition formée de classes de la tresse (ou de la hiérarchie) qui minimise cette énergie est un problème insoluble, de par sa complexité combinatoriale. Nous donnons les deux conditions de h-croissance et de croissance d'échelle, qui garantissent l'existence, l'unicité et la monotonie des solutions, et conduisent à un algorithme qui les détermine en deux passes de lecture des données. Le chapitre 2 reste dans le cadre précédent, mais étudie plus spécifiquement l'optimisation sous contrainte. Il débouche sur trois généralisations du modèle Lagrangien. Le chapitre 3 applique l'optimisation par treillis énergétique au cas de figure où l'énergie est introduite par une « vérité terrain », c'est à dire par un jeu de dessins manuel, que les partitions optimales doivent serrer au plus près. Enfin, le chapitre 4 passe des treillis énergétiques à ceux des courbes de Jordan dans le plan euclidien, qui définissent un modèle continu de segmentations hiérarchiques. Il permet entre autres de composer les hiérarchies avec diverses fonctions numériques / Hierarchical segmentation has been a model which both identifies with the construct of extracting a tree structured model of the image, while also interpreting it as an optimization problem of the optimal scale selection. Hierarchical processing is an emerging field of problems in computer vision and hyper-spectral image processing community, on account of its ability to structure high-dimensional data. Chapter 1 discusses two important concepts of Braids and Energetic lattices. Braids of partitions is a richer hierarchical partition model that provides multiple locally non-nested partitioning, while being globally a hierarchical partitioning of the space. The problem of optimization on hierarchies and further braids are non-tractable due the combinatorial nature of the problem. We provide conditions, of h-increasingness, scale-increasingness on the energy defined on partitions, to extract unique and monotonically ordered minimal partitions. Furthermore these conditions are found to be coherent with the Braid structure to perform constrained optimization on hierarchies, and more generally Braids. Chapter 2 demonstrates the Energetic lattice, and how it generalizes the Lagrangian formulation of the constrained optimization problem on hierarchies. Finally in Chapter 3 we apply the method of optimization using energetic lattices to the problem of extraction of segmentations from a hierarchy, that are proximal to a ground truth set. Chapter 4 we show how one moves from the energetic lattice on hierarchies and braids, to a numerical lattice of Jordan Curves which define a continous model of hierarchical segmentation. This model enables also to compose different functions and hierarchies

Design of antenna arrays for present day requirements has to take into account both mechanical and electrical aspects. Mechanical aspects demand the antennas to have low profile, non-protruding structures, structures compatible to aerodynamic require­ments and so on. Electrical aspects may introduce several constraints either due to. technical reasons or due to readability conditions in practice. Thus, arrays of modern requirements may not fall into the category of linear or planar arrays. Further, due to the nearby environment, the elements will generate complicated individual patterns. These issues necessitate the analysis and synthesis of antenna arrays which are arbi­trary as far as the orientation, position or the element pattern are concerned. Such arrays which may be called arbitrary arrays are being investigated in this thesis. These investigations have been discussed as different aspects as indicated below: Radiation Characteristics of Arbitrary Arrays Radiation fields of an arbitrarily oriented dipole are obtained. Such fields are plotted for typical cases. Further, methods for transforming the electromagnetic fields are discussed. Having obtained the field due to an arbitrary element, the fields due to an arbitrary array are obtained. Factors controlling the radiation fields, like, the curvature in the array and element pattern are investigated. Radiation patterns of circular and cylindrical arrays are plotted. Synthesis of a Side Lobe Topography Requirements of a narrow beam pattern generated by an antenna array are identified. A problem of synthesizing such a pattern using an arbitrary array is formulated. The envelope of the side lobe region which may be called, the side lobe topography (sit), is included in the computation of the covariance matrix. This problem which has been formulated as a problem of minimizing a quadratic function subjected to a system of linear constraints is solved by the method of Lagrangian multipliers. An iterative procedure is used to satisfy all the requirements of the pattern synthesis. The procedure has been validated by synthesizing linear arrays and is used to synthesize circular and parabolic arrays. Patterns with tapered sit, Taylor-like sit have been synthesized. Asymmetric patterns are also synthesized. Role of sit is brought out. Shaped Beam Synthesis Synthesis of shaped broad beams is discussed. Amplitude constraints are formulated. Phase distribution is linked with the phase centre. Quadratic problems thus formu­lated are solved by the Lagrangian method of undetermined multipliers. An iterative procedure is made use of to synthesize flat topped beams as well as cosecant squared-patterns using linear arrays as well as circular arrays. Reasonable excitation dynamic has been obtained. Optimum phase centres obtained by trial and error are made use of. Effects of the Frequency and Excitation on the Synthesized Patterns In general, synthesized patterns can be sensitive towards any specific parameter either excitation or to frequency or any such parameter. Several methods can be used to observe these issues. In this thesis, these effects are also studied. Using a specific array configuration, to synthesize a specified radiation pattern, frequency is changed by 10% from the design frequency and the pattern is computed. Similarly, excitation phase distribution is rounded to the nearest available phase distribution using a digital phase shifter (say 8 bit) and the resulting pattern is computed. Further, excitation dynamic is also controlled by boosting the amplitudes of the array elements which are less than the permissible (i.e. the maximum excitation/allowed dynamic). Effects of these variations are also recorded. It appears that reasonable patterns can be obtained, in spite of significant variations in these parameters in most of the cases. Reconfigurable Arbitrary Arrays It would be very useful if a single array configuration can be used for different ap- plications. This may be either for the different phases of a single application or for different applications that may be required at different times. Attempts are made to synthesize a variety of patterns from a single array. Such arrays which may be called as reconfigurable arrays can be of much use. Obviously, the excitations are different for different patterns. Both narrow beams, as well as shaped broad beams, with different side lobe topographies have been synthesized using a single array.

Objectif: Lors de l’ablation de tumeurs cérébrales, la navigation chirurgicale est basée sur les examens IRM pré-opératoires. Or, la déformation per-opératoire du cerveau, appelée brain-shift, affecte cette navigation. Dans cette thèse, une méthode de compensation du brain-shift intégrable dans un processus clinique est présentée.Méthode: Avant la chirurgie, un modèle biomécanique patient-spécifique est construit à partir des images pré-opératoires. Il intègre la géométrie des tissus mous mais également des vaisseaux. Pendant l’opération, des acquisitions échographiques localisées sont réalisées directement en contact avec le cerveau. Les modalités mode B et Doppler sont enregistrées simultanément, permettant respectivement l’extraction des vaisseaux et de l’empreinte de la sonde. Une simulation biomécanique est ensuite jouée pour compenser le brain-shift. Différentes contraintes sont appliquées au modèle de cerveau afin de modéliser les contacts avec la dure-mère, recaler les vaisseaux pré- et per-opératoires et contraindre la surface corticale avec l’empreinte de la sonde. Lors de la résection de tumeurs profondes, la trajectoire chirurgicale est également contrainte au sein de la cavité réséquée afin de retrouver les déformations latérales induites par l’écartement des tissus. Les images IRM pré-opératoires ont finalement mises à jour suivant le champ de déformation du modèle biomécanique.Résultats: La méthode a été évaluée quantitativement à partir de données synthétiques et cliniques de cinq patients. De plus, l’alignement des images a également été apprécié qualitativement, au regard des attentes des neurochirurgiens. Des résultats très satisfaisants, de l’ordre de 2 mm d’erreur, sont obtenus à l’ouverture de la dure-mère et dans le cas de résection de tumeurs en surface. Lors de la résection de tumeurs profondes, si la trajectoire chirurgicale permet de retrouver une grande partie des déformations induites par l’écartement des tissus, plusieurs limitations dues au fait que cette rétraction ne soit pas effectivement simulée sont montrées.Conclusion: Cette thèse propose une nouvelle méthode de compensation du brain-shit efficace et intégrable au bloc opératoire. Elle aborde de plus le sujet peu traité de la résection, en particulier de tumeurs profondes. Elle présente ainsi une étape supplémentaire vers un système optimal en neurochirurgie assistée par ordinateur. / Purpose: During brain tumor surgery, planning and guidance are based on preoperative MR exams. The intraoperative deformation of the brain, called brain-shift, however affect the accuracy of the procedure. In this thesis, a brain-shift compensation method integrable in a surgical workflow is presented.Method: Prior to surgery, a patient-specific biomechanical model is built frompreoperative images. The geometry of the tissues and blood vessels is integrated. Intraoperatively, navigated ultrasound images are performed directly in contact with the brain. B-mode and Doppler modalities are recorded simultaneously, enabling the extraction of the blood vessels and probe footprint, respectively. A biomechanical simulation is then executed in order to compensate for brain-shift. Several constraints are imposed to the biomechanical model in order to simulate the contacts with the dura mater, register the pre- and intraoperative vascular trees and constrain the cortical surface with the probe footprint. During deep tumors resection, the surgical trajectory is also constrained to remain inside the cavity induced by the resected tissues in order to capture the lateral deformations issued from tissues retraction. Preoperative MR images are finally updated following the deformation field of the biomechanical model.Results: The method was evaluated quantitatively using synthetic and clinical data. In addition, the alignment of the images was qualitatively assessed with respect to surgeons expectations. Satisfactory results, with errors in the magnitude of 2 mm, are obtained after the opening of the dura mater and for the resection of tumors close to the cortical surface. During the resection of deep tumors, while the surgical trajectory enable to capture most of the deformations induced by tissues retraction, several limitations reflects the fact that this retraction is not actually simulated.Conclusion: A new efficient brain-shift compensation method that is integrable in an operating room is proposed in this thesis. The few studied topic of the resection, and more specifically of deep tumors, is also addressed. This manuscript thus present an additional step towards an optimal system in computer assisted neurosurgery.