论文标题
关于来自样品的唯一性与Gabor阶段检索中的稳定性之间的联系
On the connection between uniqueness from samples and stability in Gabor phase retrieval
论文作者
论文摘要
Gabor相的检索是仅从其Gabor变换的幅度重建信号的问题。先前的发现表明，离散问题的唯一解决性（从晶格上的测量值中恢复）与连续问题的稳定性（从$ \ mathbb {r}^2 $的开放子集中恢复）之间可能存在联系。在本文中，我们通过证明无法建立这种链接来缩小这一差距。更确切地说，我们建立了函数的存在，这些功能从样本中破坏了独特性，而不会影响连续问题的稳定性。此外，我们证明了小说的结果是，从样本中唯一恢复的反示例在$ l^2（\ Mathbb {r}）$中是密集的。最后，我们建立了一个直观的论点，即有关相位检索中不稳定性方向与与小特征值相关的某些laplacian特征函数之间的联系。
Gabor phase retrieval is the problem of reconstructing a signal from only the magnitudes of its Gabor transform. Previous findings suggest a possible link between unique solvability of the discrete problem (recovery from measurements on a lattice) and stability of the continuous problem (recovery from measurements on an open subset of $\mathbb{R}^2$). In this paper, we close this gap by proving that such a link cannot be made. More precisely, we establish the existence of functions which break uniqueness from samples without affecting stability of the continuous problem. Furthermore, we prove the novel result that counterexamples to unique recovery from samples are dense in $L^2(\mathbb{R})$. Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.
