Solution of Sparse High-Dimensional Linear Inverse problems with Application to Analysis of Dynamic Contrast Enhanced Imaging Data

该项目的动机是对动态对比度增强（DCE）成像数据的分析。DCE成像提供了一种无创的肿瘤血管生成测量方法，在癌症检测和表征方面具有很大的潜力。它为评估和优化新的治疗策略以及长期评估抗血管生成治疗的治疗效果提供了非常有用的工具。目前的项目将大大有利于：a)通过分析医学图像取代昂贵的侵入性检查，从而降低医疗保健成本；通过更好地监测药物效果，缩短中风患者的住院时间；b)医疗实践，因为用于分析DCE成像数据的开创性新方法的发展，将为癌症检测和表征以及对抗血管生成治疗的治疗效果进行纵向评估提供非侵入性工具，从而有可能改善临床结果；c)医学研究，因为用于分析DCE成像数据的软件将免费提供给所有检查这些数据的人，因此将有助于设计新的方法；D)其他类型的医学成像技术，因为该建议产生的方法将有助于理解存在噪声的稀疏高维函数的重建；许多科学领域，如地球物理学和天文学，都依赖于噪声逆问题的解。目前的项目是一个整合应用和理论的整体努力。在数学上，该问题简化为基于离散测量的矩阵变量拉普拉斯卷积方程的噪声解。我们将使用为恢复稀疏表示而设计的非常现代的技术，这些技术在图像和信号分析以及其他应用中取得了许多成功的发展。然而，由于这些方法大多依赖于非常严格的假设，只有少数方法被用于求解逆病态问题。此外，最近，新的数据类型的获取带来了新的线性逆问题的类型，其中感兴趣的函数本身是向量或矩阵值。到目前为止，新的挑战是通过单独恢复解决方案的每个组件来解决的。然而，最近在稀疏矩阵估计领域的发展允许更连贯的解决问题。此外，在许多应用中，算子本身是未知的，只能从数据中估计。虽然算子的不确定性常常被忽略，但我们打算考虑算子的不确定性。本提案的目标是将基于惩罚优化或指数权重的技术扩展到求解稀疏病态高维线性逆问题，其中算子可能不完全已知并且感兴趣的函数是矩阵变量。我们计划在提出的新方法下建立坚实的理论基础，为其实现开发计算算法，并将新构建的算法应用于拉普拉斯卷积方程的求解，随后应用于正在进行的REMISCAN研究中获得的DCE成像数据的分析。

The project is motivated by analysis of Dynamic Contrast Enhanced (DCE) imaging data. DCE imaging provides a noninvasive measure of tumor angiogenesis and has great potential for cancer detection and characterization. It offers an extremely useful tool for the evaluation and optimization of new therapeutic strategies as well as for long-term evaluation of therapeutic impacts of anti-angiogenic treatments. The current project will be greatly beneficial for a) reducing health care costs by replacing expensive and invasive tests by analysis of medical images, and by shortening hospital stays for stroke patients due to better monitoring of drug effectiveness;b) the medical practice, since development of novel path-breaking methodologies for analysis of DCE imaging data will potentially improve clinical outcomes by providing non-invasive tools for cancer detection and characterization and for longitudinal evaluation of therapeutic impact of anti-angiogenic treatments; c) medical research, since the software for analysis of DCE imaging data will be freely available to everyone who carries out examination of such data and, thus, will contribute to design of new methodologies; d) other types of medical imaging techniques, since methodologies resulted from the proposal will contribute to understanding of reconstruction of sparse high-dimensional functions in the presence of noise; ande) various fields of science such as geophysics and astronomy, which rely on solution of noisy inverse problems. The current project presents an integral effort of merging applications and theory. Mathematically, the problem reduces to solution of a noisy version of a matrix-variate Laplace convolution equation based on discrete measurements. We shall use very modern techniques designed for recovery of sparse representations that led to many successful developments in image and signal analysis and other applications. However, since majority of those methods rely on very stringent assumptions, only few of them have been adopted for solution of inverse ill-posed problems. In addition, recently, acquisition of new types of data brought to light new types of the linear inverse problems where the function of interest is itself vector or matrix-valued. So far, the new challenge has been addressed by separate recovery of each component of the solution. However, recent developments in the area of sparse matrix estimation allow for much more coherent solution of the problem. Furthermore, in many applications, the operator itself is unknown and is estimated from data. Although often the uncertainty in the operator is ignored, we are planning to account for operator uncertainty. The goal of the present proposal is to extend techniques based on penalized optimization or exponential weights to solution of sparse ill-posed high-dimensional linear inverse problems where the operator may be not be known exactly and the function of interest is matrix-variate. We are planning to put a solid theoretical foundation under the proposed new methodologies, develop computational algorithms for their implementation, and apply the newly constructed algorithms to solution of Laplace convolution equation and, subsequently, to analysis of DCE imaging data obtained in ongoing REMISCAN studies.