论文标题
迈向晶体学期检索问题的数学理论
Toward a mathematical theory of the crystallographic phase retrieval problem
论文作者
论文摘要
由X射线晶体学技术确定生物分子的原子结构的动机，我们研究了晶体学期检索问题，可以说是领先且最难的阶段检索设置。这个问题需要从其傅立叶幅度恢复长度n的k-sparse信号，或从其周期性自动相关中等效地恢复。具体而言，这项工作着重于独特性的基本问题：最大稀疏度K/N是什么，允许在信号及其傅立叶幅度之间进行唯一的映射，直到固有的对称性。我们设计了一种系统性计算技术，以确认任何特定对（K，n）的唯一性，并建立以下猜想：傅立叶幅度可以决定一个通用信号，直到k <= n/2。基于组理论考虑和另一种计算技术，我们制定了第二个猜想：如果k <n/2，那么对于任何信号，晶体学相检索问题的一组解决方案在所有信号的集合中均具有给定的傅立叶级的所有信号。共同，这些猜想构成了建立晶体学阶段检索问题的数学理论的首次尝试。
Motivated by the X-ray crystallography technology to determine the atomic structure of biological molecules, we study the crystallographic phase retrieval problem, arguably the leading and hardest phase retrieval setup. This problem entails recovering a K-sparse signal of length N from its Fourier magnitude or, equivalently, from its periodic auto-correlation. Specifically, this work focuses on the fundamental question of uniqueness: what is the maximal sparsity level K/N that allows unique mapping between a signal and its Fourier magnitude, up to intrinsic symmetries. We design a systemic computational technique to affirm uniqueness for any specific pair (K,N), and establish the following conjecture: the Fourier magnitude determines a generic signal uniquely, up to intrinsic symmetries, as long as K<=N/2. Based on group-theoretic considerations and an additional computational technique, we formulate a second conjecture: if K<N/2, then for any signal the set of solutions to the crystallographic phase retrieval problem has measure zero in the set of all signals with a given Fourier magnitude. Together, these conjectures constitute the first attempt to establish a mathematical theory for the crystallographic phase retrieval problem.
