Frame Theory and Phase Retrieval

帧为从线性测量中重建信号提供了一种连续、线性和稳定的方法。然而，在许多情况下，物理限制会导致丢失这些测量的重要方面。这对哪些帧可以用于信号分析施加了限制，并且仅基于部分信息需要不同的重建算法。例如，位相恢复应用于X射线结晶学和相干衍射成像，在这些领域，科学家只能识别信号的每一次线性测量的幅度(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，会出现不同的情况。在这种情况下，任何强度高于该范围的测量都会使传感器饱和，然后传感器会输出最大值。在这两种情况下，我们都可以将测量过程施加的物理限制形式化为将非线性运算符应用于线性测量序列。虽然这些非线性算子非常简单，但线性度的损失会给高维信号重建带来极大的困难，在存在误差的情况下，高维信号重建会变得更加混乱。这类重建场景在许多情况下都会自然出现，不同学科的研究人员已经为特定的应用开发了解决方案。这使得解决这类逆问题的数学基础特别重要，并导致了特别致力于相位恢复数学的重要研究。研究人员和他们的学生正在研究一种独特的方法，通过使用来自框架理论、概率和Banach空间几何的技术组合来扩展相位恢复和饱和恢复的数学理论。相位恢复和饱和恢复都需要帧的冗余，并且在基础上是不可能的。该项目的一个组成部分涉及确定使用帧或融合帧进行相位恢复或饱和恢复所需的确切冗余量。第二部分考虑将相位恢复场景推广到Banach格子空间的设置，其中目标是根据其绝对值来识别子空间中的矢量。这种联系允许Banach格中已有的技术来证明关于相位恢复的新定理，并在Banach格理论本身中开辟了一条新的探索路线。不仅使相位恢复成为可能，而且使相位恢复在误差情况下保持稳定，这一点非常重要。对于进行稳定相位恢复的高维空间，构造框架的最好的已知方法是随机构造，其以高概率达到一定的稳定界。构造具有一定稳定界的稳定相恢复的连续框架要容易得多，但离散框架更适合于计算。因此，该项目的重要部分包括：(1)确定何时可以对连续帧进行采样以构造具有给定帧界限的帧，该帧以给定的稳定界进行相位恢复；(2)使用概率方法来确定何时可以对连续帧进行随机采样以以高概率获得这样的帧。该项目的最后一个部分介绍了流形上向量丛的相位恢复。也就是说，不是从帧系数的大小恢复单个向量，而是使用连续移动的帧来恢复向量束的一部分，直到达到等价关系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.