Inverse problems in 3D image processing and medical imaging

这项研究计划的目标是开发新的3D图像处理方法，并将它们应用于已经建立了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上：1)更好地对数据形成过程进行建模--由于实际限制，标准成像技术通常依赖非常接近的数据形成模型，这可能会导致结果图像的显著恶化。开发更准确的、通常是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(另见第3点)。2)在图像估计方法中引入适当的先验信息--当显式地考虑被成像数据量的显著特征时，成像技术的稳健性和效率得到显著提高。我们的目标将是制定这些特征的适当数学公式，以改进估计图像质量、估计器的实施简易性和成像程序的数字成本之间的折衷。3)开发适用于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。非常适合这种情况的算法确实存在于最常见的情况；然而，一些数值分析方法很少在图像处理领域及其适应性中使用。利用上述领域的现有知识和我们3D成像问题的具体特点，将开发出新的、高效的数值方法。我们的方法将应用于两个正在进行的合作项目：(A)减少X射线CT中的射束硬化伪影；(B)用于检测乳腺肿瘤的微波成像。

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.