Early detection, diagnosis, and suitable treatment are known to significantly improve the chance of survival for breast cancer (BC) patients. To date, the most cost effective method for screening and early detection is screen-film mammography, which is also the only tool that has demonstrated its ability to reduce BC mortality. Full-field digital mammography (FFDM) is an extension of screen-film mammography that eliminates the need for film-processing because the images are detected electronically from their inception. Tomosynthesis is an emerging technology in digital mammography built on the FFDM framework, which offers an alternative to conventional two-dimensional mammography. Tomosynthesis produces three-dimensional (volumetric) images of the breast that may be superior to planar imaging due to improved visualization. In this work preliminary tomosynthesis data derived from cadaver breasts are analyzed, which includes volume data acquired from various reconstruction techniques as well as the planar projection data. The noise and power spectra characteristics analyses are the focus of this study. Understanding the noise characteristics is significant in the study of radiological images and in the evaluation of the imaging system, so that its degrading effect on the image can be minimized, if possible and lead to better diagnosis and optimal computer aided diagnosis schemes. Likewise, the power spectra behavior of the data are analyzed, so that statistical methods developed for digitized film images or FFDM images may be applied directly or modified accordingly for tomosynthesis applications. The work shows that, in general, the power spectra for three of the reconstruction techniques are very similar to the spectra of planar FFDM data as well as digitized film; projection data analysis follows the same trend. To a good approximation the Fourier power spectra obey an inverse power law, which indicates a degree of self-similarity. The noise analysis indicates that the noise and signal are dependent and the dependency is a function of the reconstruction technique. New approaches for the analysis of signal dependent noise were developed specifically for this work based on both the linear wavelet expansion and on nonlinear order statistics. These methods were tested on simulated data that closely follow the statistics of mammograms prior to the real-data applications. The noise analysis methods are general and have applications beyond mammography.

Dans ce travail de thèse, nous nous intéressons à certaines propriétés d'une classe de processus stochastiques à accroissements stationnaires et autosimilaires. Ces processus sont représentés par des intégrales multiples de Wiener-Itô. Dans le premier chapitre, nous étudions les propriétés géométriques des trajectoires de ce type de processus. En particulier, nous obtenons un développement en ondelettes presque-sûr. Celui-ci permet alors de trouver une borne supérieure pour le module de continuité uniforme, une borne supérieure pour le comportement asymptotique du processus et un résultat presque-sûr concernant les coefficients ponctuel et local de Hölder. De plus, nous obtenons des bornes inférieures et supérieures pour les dimensions de Hausdorff du graphe et de l'image des versions multidimensionnelles anisotropes de la classe de processus considérée. Dans le deuxième et le troisième chapitre de cette thèse, nous nous intéressons au calcul différentiel stochastique relatif au processus de Rosenblatt. A l'aide de la théorie des distributions de Hida, nous définissons une intégrale stochastique par rapport au processus de Rosenblatt. Nous obtenons une formule d'Itô pour certaines fonctionnelles du processus de Rosenblatt. Nous calculons explicitement la variance de l'intégrale stochastique par rapport au processus de Rosenblatt pour une classe spécifique d'intégrandes aléatoires. Enfin, nous comparons l'intégrale introduite avec d'autres définitions utilisées dans la littérature et procédons à une étude fine des termes résiduels faisant le lien entre ces différentes définitions. / In this PhD thesis, we are concerned with some properties of a class of self-similar stochastic processes with stationary increments. These processes are represented by multiple Wiener-Itô integrals. In the first chapter, we study geometric properties of the sample path of this type of processes. Specifically, we obtain an almost sure wavelet expansion which, in turn, allows us to compute an upper bound for the uniform modulus of continuity, an upper bound for the asymptotic growth at infinity of the processes and the almost sure values of the pointwise and local Hölder exponents at any points. Moreover, we obtain lower and upper bounds for the Hausdorff dimensions of the graph and the image of multidimensional anisotropic versions of the class of processes previously considered. In the second and in the third chapters, we are interested in the stochastic calculus with respect to the Rosenblatt process. Using Hida distributions theory, we define a stochastic integral with respect to the Rosenblatt process. We obtain an Itô formula for some functional of the Rosenblatt process. We compute explicitly the variance of the stochastic integral with respect to the Rosenblatt process for a specific class of stochastic integrands. At last, we compare the considered integral with other definitions used in the literature and provide a careful analysis of the residual terms linking the different definitions of integrals.