Frame Theory and Phase Retrieval

框架理论和相位检索

基本信息

  • 批准号:
    2154931
  • 负责人:
  • 金额:
    $ 33.27万
  • 依托单位:
  • 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

项目摘要

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,用于从线性测量值重建信号。然而,在许多情况下,物理限制导致这些测量的重要方面的损失。 这对哪些帧可以用于信号分析施加了约束,并且仅基于部分信息的重建需要不同的算法。例如,相位恢复应用于X射线晶体学和相干衍射成像,其中科学家只能识别信号的每个线性测量的幅度(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,会发生不同的情况。 在这种情况下,强度高于范围的任何测量都会使传感器饱和,然后输出最大值。 在这两种情况下,我们可以将测量过程所施加的物理限制形式化为将非线性算子应用于线性测量序列。 虽然这些非线性算子非常简单,但线性度的损失可能导致高维信号重构的显著困难,这在存在误差的情况下变得进一步混乱。 这些类型的重建场景在许多情况下自然出现,并且各种学科的研究人员已经为特定应用开发了解决方案。 这使得解决这些类型的逆问题的数学基础特别重要,并导致了显着的研究正在致力于数学相位恢复特别。 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。 相位恢复和饱和度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冗余的确切数量,这是必要的相位恢复或饱和度恢复使用帧或融合帧。 第二个组件认为,泛化的相位恢复方案的设置的Banach格的子空间,其目标是确定一个向量的子空间中的绝对值。 这种联系允许在Banach格中建立的技术来证明关于相位恢复的新定理,并且还在Banach格理论本身中开辟了一条新的调查路线。 不仅相位恢复是可能的,而且相位恢复在误差下是稳定的是重要的。 用于构造用于进行稳定相位恢复的高维空间的框架的最公知的方法是随机构造,其以高概率实现一定的稳定性界限。 在一定的稳定性范围内进行稳定相位恢复的连续框架容易得多,而离散框架更适合于计算。 因此,该项目的重要部分涉及(1)确定何时可以对连续帧进行采样以构建具有给定帧边界的帧,该帧以给定的稳定性边界进行相位恢复,以及(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实现具有高概率的帧。该项目的最后一个组成部分介绍了相位检索流形上的向量丛。 也就是说,我们的目标是使用一个连续移动的坐标系来恢复矢量束的一部分,直到等价关系。这个奖项反映了NSF的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Discretizing L norms and frame theory
离散 L 范数和框架理论
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Daniel Freeman其他文献

Barkour: Benchmarking Animal-level Agility with Quadruped Robots
Barkour:用四足机器人对动物级敏捷性进行基准测试
  • DOI:
    10.48550/arxiv.2305.14654
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Ken Caluwaerts;Atil Iscen;J. Kew;Wenhao Yu;Tingnan Zhang;Daniel Freeman;Kuang;Lisa Lee;Stefano Saliceti;Vincent Zhuang;Nathan Batchelor;Steven Bohez;Federico Casarini;José Enrique Chen;O. Cortes;Erwin Coumans;Adil Dostmohamed;Gabriel Dulac;Alejandro Escontrela;Erik Frey;Roland Hafner;Deepali Jain;Bauyrjan Jyenis;Yuheng Kuang;Edward Lee;Linda Luu;Ofir Nachum;Kenneth Oslund;Jason Powell;D. Reyes;Francesco Romano;Feresteh Sadeghi;R. Sloat;B. Tabanpour;Daniel Zheng;Michael Neunert;R. Hadsell;N. Heess;F. Nori;J. Seto;Carolina Parada;Vikas Sindhwani;Vincent Vanhoucke;Jie Tan
  • 通讯作者:
    Jie Tan
Annual Research Review: Immersive virtual reality and digital applied gaming interventions for the treatment of mental health problems in children and young people: the need for rigorous treatment development and clinical evaluation
年度研究回顾:沉浸式虚拟现实和数字应用游戏干预治疗儿童和青少年心理健康问题:需要严格的治疗开发和临床评估
  • DOI:
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Brynjar Halldorsson;Claire Hill;Polly Waite;Kate Partridge;Daniel Freeman;C. Creswell
  • 通讯作者:
    C. Creswell
S0033291719003155jra 1..10
S0033291719003155jra 1..10
  • DOI:
  • 发表时间:
    2019
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Daniel Freeman;B. S. Loe;D. Kingdon;Helen Startup;Andrew Molodynski;Laina Rosebrock;Poppy Brown;Bryony Sheaves;Felicity Waite;Jessica C. Bird
  • 通讯作者:
    Jessica C. Bird
Correction: Testing the combination of Feeling Safe and peer counselling against formulation-based cognitive behaviour therapy to promote psychological wellbeing in people with persecutory delusions: study protocol for a randomized controlled trial (the Feeling Safe-NL Trial)
  • DOI:
    10.1186/s13063-023-07750-x
  • 发表时间:
    2023-12-18
  • 期刊:
  • 影响因子:
    2.000
  • 作者:
    Eva Tolmeijer;Felicity Waite;Louise Isham;Laura Bringmann;Robin Timmers;Arjan van den Berg;Hanneke Schuurmans;Anton B. P. Staring;Paul de Bont;Rob van Grunsven;Gert Stulp;Ben Wijnen;Mark van der Gaag;Daniel Freeman;David van den Berg
  • 通讯作者:
    David van den Berg
A validation study to trigger nicotine craving in virtual reality
在虚拟现实中触发尼古丁渴望的验证研究

Daniel Freeman的其他文献

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{{ truncateString('Daniel Freeman', 18)}}的其他基金

Immersive Virtual Reality Cognitive Treatment (VRCT) for persecutory delusions.
针对被害妄想的沉浸式虚拟现实认知治疗(VRCT)。
  • 批准号:
    MR/P02629X/1
  • 财政年份:
    2017
  • 资助金额:
    $ 33.27万
  • 项目类别:
    Research Grant
Topics in the geometry of Banach spaces
Banach 空间几何主题
  • 批准号:
    1332255
  • 财政年份:
    2012
  • 资助金额:
    $ 33.27万
  • 项目类别:
    Standard Grant
Understanding and treating persecutory delusions: an interventionist-causal model approach
理解和治疗被害妄想:干预因果模型方法
  • 批准号:
    G0902308/2
  • 财政年份:
    2011
  • 资助金额:
    $ 33.27万
  • 项目类别:
    Fellowship
Topics in the geometry of Banach spaces
Banach 空间几何主题
  • 批准号:
    1139143
  • 财政年份:
    2010
  • 资助金额:
    $ 33.27万
  • 项目类别:
    Standard Grant
Topics in the geometry of Banach spaces
Banach 空间几何主题
  • 批准号:
    1001929
  • 财政年份:
    2010
  • 资助金额:
    $ 33.27万
  • 项目类别:
    Standard Grant
Understanding and treating persecutory delusions: an interventionist-causal model approach
理解和治疗被害妄想:干预因果模型方法
  • 批准号:
    G0902308/1
  • 财政年份:
    2010
  • 资助金额:
    $ 33.27万
  • 项目类别:
    Fellowship

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