Inverse problems in 3D image processing and medical imaging

该研究计划的目标是开发新的3D图像处理方法，并将其应用于已经建立特定合作的特定项目。从方法论的角度来看，重点将放在以下三点：1）更好地建模的数据形成过程-由于实际的限制，标准的成像技术往往依赖于非常近似的数据形成模型，可能会导致显着恶化的结果图像。开发更精确的、一般为非线性的成像模型将减少这种失真，但可能需要开发特定的算法（也见第3点）。2)在图像估计方法中引入足够的先验-当明确考虑被成像的量的显著特征时，获得成像技术的鲁棒性和效率的显著改善。我们的目标将是开发适当的数学制定这样的特点，以提高估计图像的质量，易于实现的估计和成像过程的数值成本之间的妥协。3)开发适合于3D成像方法的特定约束的算法- 3D成像问题导致非常大的优化问题。最常见的情况下，确实存在非常适合这种情况的算法;然而，一些数值分析方法很少用于图像处理领域及其适应。新的，有效的数值方法将开发利用现有的知识，从上述领域和我们的三维成像问题的具体特点。我们的方法将被应用到两个正在进行的合作项目：（a）减少束硬化伪影在X射线CT;（B）微波成像检测乳腺肿瘤。

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.