The 3-D fluorescence microscope has become the method of choice in biological sciences for living cells study. However, the data acquired with conventional3-D fluorescence microscopy are not quantitatively significant because of distortions induced by the optical acquisition process. Reliable measurements need the correction of theses distortions. Knowing the instrument impulse response, also known as the PSF, one can consider the backward process of convolution induced by the microscope, known as "deconvolution". However, when the system response is not invariant in the observation field, the classical algorithms can introduce large errors in the results. In this thesis we propose a new approach, which can be easily adapted to any classical deconvolution algorithm, direct or iterative, for bypassing the non-invariance PSF problem, without any modification to the later. Based on the hypothesis that the minimal error in a restored image using non-invariance assumption is located near the used PSF position, the EMMA (Evolutive Merging Masks Algorithm) blends multiple deconvolutions in the invariance assumption using a specific merging mask set. In order to obtain sufficient number of measured PSF at various depths for a better restoration using EMMA (or any other depth-variant deconvolution algorithm) we propose a 3D PSF interpolation algorithm based on the image moments theory using Zernike polynomials as decomposition base. The known PSF are decomposed into Zernike moments set and each moment's variation is fitted into a polynomial function, the resulting functions are then used to interpolate the needed PSF's Zernike moments set to reconstruct the interpolated PSF.

La reconstruction 3D par coupes sériées en microscopie optique est un moyen efficace pour étudier des spécimens biologiques fluorescents. Dans un tel système, la formation d'une image peut être représentée comme une convolution linéaire d'un objet avec une réponse impulsionnelle optique de l'instrument (PSF). Pour une étude quantitative, une estimation de l'objet doit être calculée en utilisant la déconvolution qui est le phénomène inverse de la convolution. Plusieurs algorithmes de déconvolution ont été développés en se basant sur des modèles statistiques ou par inversion directe, mais ces algorithmes se basent sur la supposition de l'invariance spatiale de la PSF pour simplifier et accélérer le processus. Dans certaines configurations optiques la PSF 3D change significativement en profondeur et ignorer ces changements implique des erreurs quantitatives dans l'estimation. Nous proposons un algorithme (EMMA) qui se base sur une hypothèse où l'erreur minimale sur l'estimation par un algorithme ne tenant pas compte de la non-invariance, se situe aux alentours de la position (profondeur) de la PSF utilisée. EMMA utilise des PSF à différentes positions et fusionne les différentes estimations en utilisant des masques d'interpolation linéaires adaptatifs aux positions des PSF utilisées. Pour obtenir des PSF à différentes profondeurs, un algorithme d'interpolation de PSF a également été développé. La méthode consiste à décomposer les PSF mesurées en utilisant les moments de Zernike pseudo-3D, puis les variations de chaque moment sont approximés par une fonction polynomiale. Ces fonctions polynomiales sont utilisées pour interpoler des PSF aux profondeurs voulues. / The 3-D fluorescence microscope has become the method of choice in biological sciences for living cells study. However, the data acquired with conventional3-D fluorescence microscopy are not quantitatively significant because of distortions induced by the optical acquisition process. Reliable measurements need the correction of theses distortions. Knowing the instrument impulse response, also known as the PSF, one can consider the backward process of convolution induced by the microscope, known as "deconvolution". However, when the system response is not invariant in the observation field, the classical algorithms can introduce large errors in the results. In this thesis we propose a new approach, which can be easily adapted to any classical deconvolution algorithm, direct or iterative, for bypassing the non-invariance PSF problem, without any modification to the later. Based on the hypothesis that the minimal error in a restored image using non-invariance assumption is located near the used PSF position, the EMMA (Evolutive Merging Masks Algorithm) blends multiple deconvolutions in the invariance assumption using a specific merging mask set. In order to obtain sufficient number of measured PSF at various depths for a better restoration using EMMA (or any other depth-variant deconvolution algorithm) we propose a 3D PSF interpolation algorithm based on the image moments theory using Zernike polynomials as decomposition base. The known PSF are decomposed into Zernike moments set and each moment's variation is fitted into a polynomial function, the resulting functions are then used to interpolate the needed PSF's Zernike moments set to reconstruct the interpolated PSF.