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增强(DCE)成像数据的分析。DCE成像提供了一种非侵入性的肿瘤血管生成的测量方法，在肿瘤的检测和定性方面具有巨大的潜力。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一个极其有用的工具。目前的项目将极大地有利于a)通过医学图像分析取代昂贵的侵入性测试，并由于更好地监测药物有效性而缩短中风患者的住院时间，从而降低医疗保健成本；b)医疗实践，因为开发用于分析DCE成像数据的新的开创性方法将通过为癌症检测和表征以及抗血管生成治疗的治疗效果的纵向评估提供非侵入性工具，潜在地改善临床结果；C)医学研究，因为对这种数据进行检查的每个人都可以免费获得用于分析DCE成像数据的软件，从而将有助于设计新的方法；d)其他类型的医学成像技术，因为该提议产生的方法将有助于理解在存在噪声的情况下重建稀疏的高维函数；以及)各种科学领域，如地球物理学和天文学，它们依赖于解决有噪声的反问题。目前的项目是一项融合应用和理论的整体努力。在数学上，该问题归结为基于离散测量的矩阵变量拉普拉斯卷积方程的噪声版本的解。我们将使用为恢复稀疏表示而设计的非常现代的技术，这些技术导致了图像和信号分析以及其他应用程序的许多成功发展。然而，由于这些方法大多依赖于非常严格的假设，因此很少被用于求解逆不适定问题。此外，最近，新类型数据的获取揭示了新类型的线性反问题，其中感兴趣的函数本身是向量或矩阵值的。到目前为止，已经通过单独回收解决方案的每个组成部分来应对新的挑战。然而，稀疏矩阵估计领域的最新发展允许更连贯地解决该问题。此外，在许多应用中，运算符本身是未知的，并且是根据数据估计的。虽然运营商的不确定性经常被忽略，但我们计划考虑运营商的不确定性。本方案的目标是将基于惩罚优化或指数加权的技术扩展到稀疏不适定高维线性反问题的求解中，其中算子可能不是精确已知的，并且感兴趣的函数是矩阵变量的。我们计划在提出的新方法下奠定坚实的理论基础，开发实现这些方法的计算算法，并将新构建的算法应用于拉普拉斯卷积方程的求解，并随后用于分析正在进行的REMISCAN研究中获得的DCE成像数据。

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.