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

稀疏高维线性反问题的求解及其在动态对比度增强成像数据分析中的应用

• 批准号: 1407475
• 负责人: Marianna Pensky
• 金额: $ 12万
• 依托单位: The University of Central Florida Board of Trustees
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1407475&HistoricalAwards=false
• 关键词: Solution; Sparse; Dimensional; Linear; Inverse

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.

项目 是由动态对比增强（DCE）成像数据的分析激发的。 DCE成像提供了一种非侵入性的肿瘤血管生成的措施，并有很大的潜力，为癌症的检测和表征。它为评价和优化新的治疗策略以及长期评价抗血管生成治疗的治疗效果提供了一个非常有用的工具。目前的项目将大大有利于a）通过分析医学图像取代昂贵和侵入性的测试来降低医疗保健成本，并通过更好地监测药物有效性来缩短中风患者的住院时间; B）医疗实践，由于用于分析DCE成像数据的新的开创性方法的开发将通过提供非用于癌症检测和表征以及用于抗血管生成治疗的治疗影响的纵向评估的侵入性工具; 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对这种数据进行检查的每个人，因此将有助于设计新的方法; d）其他类型的医学成像技术，因为由该提议产生的方法将有助于理解在存在噪声的情况下稀疏高维函数的重建;以及）诸如物理学和天文学的各种科学领域，其依赖于噪声逆问题的解决方案。目前的项目提出了合并应用和理论的整体努力。在数学上，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决方案。我们将使用非常现代的技术，旨在恢复稀疏表示，导致许多成功的发展，在图像和信号分析和其他应用。然而，由于大多数这些方法依赖于非常严格的假设，只有少数已被采用的逆不适定问题的解决方案。 此外，最近，新类型的数据采集带来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。到目前为止，新的挑战已经通过单独回收解决方案的每个组件来解决。然而，稀疏矩阵估计领域的最新发展使得该问题的解决方案更加一致。此外，在许多应用中，算子本身是未知的，并且是从数据中估计的。虽然通常忽略了运营商的不确定性，但我们计划考虑运营商的不确定性。本建议的目标是扩展技术的基础上惩罚优化或指数权重的解决方案稀疏不适定的高维线性逆问题的运营商可能是不知道确切和感兴趣的功能是矩阵变量。我们正计划把一个坚实的理论基础下提出的新方法，开发计算算法的实施，并应用新构建的算法解决的拉普拉斯卷积方程，随后，在正在进行的REMISCAN研究中获得的DCE成像数据的分析。