CAREER: Nonconvex Optimization and Identifiability with Applications to Medical Imaging

在物理和生物科学中产生的现代大规模数据集往往表现出不符合现有方法框架的复杂特征，使所收集数据中的信息无法得到充分利用。在医学成像中，计算机断层扫描（CT）和正电子发射断层扫描（PET）扫描通常用复杂的模型来表示，超出了大多数可用计算方法的范围，因为现有的方法和理论分析大多局限于更简单的优化问题类别。由于CT和PET扫描对患者的辐射剂量较小，因此这些成像设备获得的数据的更好模型将在辐射风险和获得精确图像以进行有效诊断和治疗的好处之间取得更好的权衡。本研究项目将研究复杂优化问题的广泛框架，适用于医学成像以及物理和生物科学领域的一系列问题，为这些领域出现的许多问题提供具体的方法和保障。通过与医学成像研究人员的合作，开发的工具将用于解决CT和PET成像中的特定图像重建问题，这些研究人员将提供实际的扫描数据，目标是为这些流行的临床工具提供更高的诊断准确性。在这个项目下开发的方法和代码都将公开提供。在整个过程中，研究者将指导对高维统计、优化和医学成像交叉领域工作感兴趣的学生，并将通过新课程和新合作增加这些领域的互动和交流。在许多现代应用领域中出现的统计问题往往表现出一系列具有挑战性的特征，包括非凸性和不可微性，这对高维优化和理论分析提出了重大挑战。该研究将探索复杂的非凸优化和可识别性问题，为应用研究人员在实践中面临的广泛问题提供方法和理论。该研究将研究和开发适应稀疏或低秩优化、原始/对偶方法、交替最小化或交替下降等技术的算法，目的是实现有效的经验性能和广泛的理论收敛保证。由此产生的方法将适用于解决医学成像中的具体问题，其中必须将噪声侧信息纳入重建图像中，并且图像表示被成像设备建模的附加参数混淆。

Modern large-scale data sets arising in the physical and biological sciences often exhibit complex features that do not fit into the framework of existing methodologies, preventing the information in the gathered data from being fully utilized. In medical imaging, computed tomography (CT) scans and positron emission tomography (PET) scans are often best represented with models that are complex, beyond the scope of most available computational approaches, as existing methodology and theoretical analysis are mostly restricted to simpler classes of optimization problems. Since CT and PET scans come with a cost of a small radiation dose to the patient, better models for the data obtained by these imaging devices would result in a better tradeoff between the risk due to radiation and the benefit of obtaining a precise image for effective diagnosis and treatment. This research project will study a broad framework for complex optimization problems, applicable to medical imaging and across a range of problems in the physical and biological sciences, providing concrete methods and guarantees for many problems arising in these fields. The developed tools will be implemented on specific image reconstruction problems in CT and PET imaging, through collaborations with medical imaging researchers who will provide actual scan data, with the goal of enabling greater diagnostic accuracy for these popular clinical tools. Methods and code developed under this project will all be made publicly available. Throughout, the investigator will mentor students interested in working at the intersection of high-dimensional statistics, optimization, and medical imaging, and will increase interaction and communication across these fields through new courses and new collaborations.Statistical problems arising in many modern applied fields often exhibit a range of challenging features, including non-convexity and non-differentiability, that pose significant challenges for high dimensional optimization and theoretical analysis. The proposed research will explore complex non-convex optimization and identifiability problems to develop methodology and theory for a broad range of problems facing applied researchers in practice. The research will study and develop algorithms that adapt techniques such as sparse or low-rank optimization, primal/dual methods, and alternating minimization or alternating descent, with the aim of achieving efficient empirical performance and broad theoretical convergence guarantees. The resulting methods will be adapted to address concrete problems in medical imaging, where noisy side information must be incorporated into the reconstructed image, and where the image representation is confounded by additional parameters modeling the imaging device.