论文标题
从三个测量值中取回相位的独特性
Uniqueness of phase retrieval from three measurements
论文作者
论文摘要
在本文中，我们考虑了一个问题的问题，即在$ l^2（r）$上找到一个尽可能小的运算师$（t_j）_ {j \ in J} $，它可以进行相位检索：每个$φ$都是独特地确定的（最多可通过phaseless apher phase castion $）$（|t_jjφ|t_jφ|）_ {这个问题出现在应用科学领域的各个领域，通常操作员遵守进一步的限制。这里特别感兴趣的是所谓的{\ em编码衍射paterns}，其中运算符的形式为$t_jφ= \ Mathcal {f} m_j或在这里，我们明确构建了三个实价蒙版$ m_1，m_2，m_3 \ in l^\ infty（r）$，以便相关的编码衍射模式可以进行相位检索。这意味着三个自我接合运算符$t_jφ= \ MATHCAL {f} [m_j \ MATHCAL {f}^{ - 1}φ] $也可以进行相位检索。然后，证明使用复杂的分析。然后，我们证明这些操作员在有限尺寸设置中的某些天然类似物并不总是会导致相同的唯一性结果，这是由于采样效果。
In this paper we consider the question of finding an as small as possible family of operators $(T_j)_{j\in J}$ on $L^2(R)$ that does phase retrieval: every $φ$ is uniquely determined (up to a constant phase factor) by the phaseless data $(|T_jφ|)_{j\in J}$. This problem arises in various fields of applied sciences where usually the operators obey further restrictions. Of particular interest here are so-called {\em coded diffraction paterns} where the operators are of the form $T_jφ=\mathcal{F}m_jφ$, $\mathcal{F}$ the Fourier transform and $m_j\in L^\infty(R)$ are "masks". Here we explicitely construct three real-valued masks $m_1,m_2,m_3\in L^\infty(R)$ so that the associated coded diffraction patterns do phase retrieval. This implies that the three self-adjoint operators $T_jφ=\mathcal{F}[m_j\mathcal{F}^{-1}φ]$ also do phase retrieval. The proof uses complex analysis.We then show that some natural analogues of these operators in the finite dimensional setting do not always lead to the same uniqueness result due to an undersampling effect.
