Inverse problems in 3D image processing and medical imaging

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The goal of this research proposal is to develop new 3D image processing methods, and to apply them to specific projects for which specific collaborations have already been established. From a methodological standpoint, the focus will be on the following three points: 1) Better modeling of the data formation process - Because of practical constraints, standard imaging techniques often rely on very approximate data formation models that may induce significant deterioration of the resulting image. Development of more accurate, generally non-linear image formation models will reduce such distortions, but may require the development of specific algorithms (see also point 3). 2) Introduction of adequate priors in the image estimation method - Significant improvement of robustness and efficiency of imaging techniques is obtained when the salient characteristics of the quantity being imaged are explicitly accounted for. Our goal will be to develop appropriate mathematetical  formulation of such characteristics in order to improve the compromise between the quality of the estimated image, the ease of implementation of the estimator and the numerical cost of the imaging procedure. 3) Development of algorithms adapted to the specific constraints of 3D imaging methods - 3D imaging problems result in very large optimization problems. Algorithms well suited to this situation do exist for the most common situations; however, some numerical analysis methods are seldom used in the image processing field and their adaptation. New, efficient numerical methods will be developed using available knowledge from the aforementioned fields and specific characteristics of our 3D imaging problems. Our methods will be applied to two ongoing collaborative projects: (a) reduction of beam-hardening artifacts in X-ray CT; (b) microwave imaging for detection of breast tumors.