Frame Theory and Phase Retrieval

框架理论和相位检索

• 批准号: 2154931
• 负责人: Daniel Freeman
• 金额: $ 33.27万
• 依托单位: Saint Louis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154931&HistoricalAwards=false
• 关键词: Frame; Theory; Phase; Retrieval

Frames give a continuous, linear, and stable method for reconstructing a signal from linear measurements. However, there are many situations where physical limitations result in the loss of important aspects of those measurements. This imposes constraints on which frames can be used for the analysis of a signal, and different algorithms are required for reconstruction based only on partial information. For example, phase retrieval is applied in X-ray crystallography and coherent diffraction imaging where scientists are only able to identify the magnitude (or intensity) of each linear measurement of a signal. A different scenario occurs when using sensors with a fixed range, such as a pixel in a digital camera. In this case, any measurement with an intensity above the range saturates the sensor which then outputs the maximum value. In both situations, we can formalize the physical limitations imposed by the measurement process as applying a non-linear operator to a sequence of linear measurements. Although these non-linear operators are very simple, the loss of linearity can cause significant difficulty for signal reconstruction in high dimensions which becomes further confounded in the presence of error. These kinds of reconstruction scenarios arise naturally in many circumstances, and researchers in a variety of disciplines have developed solutions for specific applications. This makes the mathematical foundation for solving these types of inverse questions particularly important, and has led to significant research being devoted to the mathematics of phase retrieval in particular. The investigators and their students are working on a unique approach to expanding the mathematical theory of phase retrieval and saturation recovery by using a combination of techniques from frame theory, probability, and the geometry of Banach spaces. Both phase retrieval and saturation recovery require the redundancy of a frame, and are not possible with a basis. One component of the project concerns identifying the exact amount of redundancy which is necessary to do phase retrieval or saturation recovery using a frame or fusion frame. The second component considers a generalization of the phase retrieval scenario to the setting of subspaces of Banach lattices where the goal is to identify a vector in a subspace from its absolute value. This connection allows for established techniques in Banach lattices to prove new theorems about phase retrieval, and also opens a new line of inquiry in the theory of Banach lattices itself. It is not only important for phase retrieval to be possible, but for phase retrieval to be stable under error. The best known methods for constructing frames for high dimensional spaces which do stable phase retrieval are random constructions which achieve a certain stability bound with high probability. It is much easier to construct continuous frames which do stable phase retrieval with a certain stability bound, but discrete frames are better suited for computations. Because of this, important parts of the project involve both (1) determining when a continuous frame may be sampled to construct a frame with given frame bounds which does phase retrieval with a given stability bound and (2) using probabilistic methods to determine when a continuous frame may be randomly sampled to achieve such a frame with high probability. The final component of the project introduces phase retrieval for vector bundles over manifolds. That is, instead of recovering a single vector up to a phase factor from the magnitude of its frame coefficients, the goal is to use a continuously moving frame to recover a section of a vector bundle up to an equivalence relation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

框架提供了一种连续，线性和稳定的方法，用于从线性测量中重建信号。但是，在许多情况下，物理限制会导致这些测量值的重要方面丧失。 这构成了可以使用哪些帧用于分析信号的约束，并且仅基于部分信息的重建需要不同的算法。例如，相位检索应用于X射线晶体学和相干衍射成像中，其中科学家只能识别信号的每个线性测量的大小（或强度）。当使用具有固定范围的传感器（例如数码相机中的像素）时，会发生不同的情况。 在这种情况下，任何强度高于范围的测量值都使传感器饱和，然后输出最大值。 在这两种情况下，我们都可以将测量过程施加的物理局限性形式化，因为将非线性操作员应用于一系列线性测量序列。 尽管这些非线性运算符非常简单，但线性丧失可能会在高维度中引起信号重建的严重困难，在存在误差的情况下，这会进一步混淆。 在许多情况下，这些类型的重建方案自然出现，各种学科的研究人员为特定应用开发了解决方案。 这使得解决这类类型的逆问题的基础特别重要，并导致了大量研究专门研究相位检索的数学。 研究人员及其学生正在通过使用框架理论，概率和Banach空间几何形状的技术组合来扩展相位检索和饱和恢复的数学理论的独特方法。 相位检索和饱和恢复都需要框架的冗余，并且在基础上是不可能的。 项目的一个组成部分是识别使用框架或融合框架进行相位检索或饱和恢复所必需的确切冗余量。 第二个组件认为相位检索方案的概括为Banach Lattices的子空间的设置，在该设置中，目标是从其绝对值中识别子空间中的向量。 这种连接允许Banach Lattices中建立的技术证明有关相位检索的新定理，并在Banach Lattices本身的理论中开辟了新的询问线。 相位检索的可能性不仅重要，而且对于相位检索在误差下保持稳定。 为高维空间构建框架的最著名方法是稳定相位检索的随机构造，它们具有高概率的一定稳定性结合。 构造连续框架的连续框架可以使用一定的稳定性结合进行稳定的相位检索，但是离散帧更适合计算。 因此，项目的重要部分涉及（1）确定何时可以采样连续框架以构建具有给定的框架边界的帧，该帧界限确实具有给定稳定性结合的相位检索，以及（2）使用概率方法来确定何时可以随机采样连续框架以实现具有高概率的帧。该项目的最终组成部分引入了矢量束的相位检索。 也就是说，目的不是将单个向量从其框架系数的大小中恢复到一个相位因素，而是使用不断移动的框架来恢复矢量捆绑包的一部分，直至等效关系。该奖项反映了NSF的法定任务，并认为通过基金会的知识优点和广泛的criperia criperia criperia criperia criperia criperia criperia rection supporation the奖项。

项目成果

Discretizing L norms and frame theory

离散 L 范数和框架理论

• DOI: 10.1016/j.jmaa.2022.126846
• 发表时间: 2023
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Freeman, Daniel;Ghoreishi, Dorsa
• 通讯作者: Ghoreishi, Dorsa