In this paper, we consider the generalized phase retrieval from affine measurements. This problem aims to recover signals $\mathit{x}\in {\mathbb{F}}^d$ from the magnitude of the affine transformations $y_j={\Vert M_j^{⁎}\mathit{x}+{\mathit{b}}_j\Vert }_2^2,j=1,\dots ,m$, where $M_j\in {\mathbb{F}}^{d\times r},{\mathit{b}}_j\in {\mathbb{F}}^r,\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ and we call it generalized affine phase retrieval. We first develop a framework for generalized affine phase retrieval with presenting several necessary and sufficient conditions for ${\{(M_j,{\mathit{b}}_j)\}}_{j=1}^m$ having generalized affine phase retrieval property. Next, we focus on the minimal measurement number problem and establish some results for it. Particularly, we show if ${\{(M_j,{\mathit{b}}_j)\}}_{j=1}^m\subset {\mathbb{F}}^{d\times r}\times {\mathbb{F}}^r$ has generalized affine phase retrieval property, then $m\ge d+\lfloor d/r\rfloor$ for $\mathbb{F}=\mathbb{R}$ ($m\ge 2d+\lfloor d/r\rfloor$ for $\mathbb{F}=\mathbb{C}$). We also show that the lower bounds are tight provided $r|d$. These results imply that one can reduce the measurement number by raising r, i.e. the rank of $M_j$. This highlights a notable difference between generalized affine phase retrieval and generalized phase retrieval. Furthermore, using tools of algebraic geometry, we show that $m\ge 2d$ (resp. $m\ge 4d-1$) generic measurements $\mathcal{A}={\{(M_j,b_j)\}}_{j=1}^m$ have the generalized phase retrieval property for $\mathbb{F}=\mathbb{R}$ (resp. $\mathbb{F}=\mathbb{C}$).

在本文中，我们考虑从仿射测量中进行广义相位检索。这个问题旨在恢复信号$\mathit{X}\in {\mathbb{F}}^d$ 从仿射变换的幅度 ${是}_j={\Vert {米}_j^{⁎}\mathit{X}+{\mathit{乙}}_j\Vert }_2^2,j=1,\dots ,米$， 在哪里 ${米}_j\in {\mathbb{F}}^{d\times r},{\mathit{乙}}_j\in {\mathbb{F}}^r,\mathbb{F}\in \{\mathbb{电阻},\mathbb{C}\}$我们称之为广义仿射相位检索。我们首先开发了一个广义仿射相位检索框架，并提出了几个必要和充分条件${\{({米}_j,{\mathit{乙}}_j)\}}_{j=1}^{米}$具有广义仿射相位检索特性。接下来，我们关注最小测量数问题并为其建立一些结果。特别地，我们展示了如果${\{({米}_j,{\mathit{乙}}_j)\}}_{j=1}^{米}\subset {\mathbb{F}}^{d\times r}\times {\mathbb{F}}^r$ 具有广义仿射相位检索性质，则 $米\ge d+\lfloor d/r\rfloor$ 为了 $\mathbb{F}=\mathbb{电阻}$ ($米\ge 2d+\lfloor d/r\rfloor$ 为了 $\mathbb{F}=\mathbb{C}$）。我们还表明下限是严格的$r|d$. 这些结果意味着可以通过提高r来减少测量数，即${米}_j$. 这突出了广义仿射相位检索和广义相位检索之间的显着差异。此外，使用代数几何工具，我们表明$米\ge 2d$ （分别 $米\ge 4d-1$) 一般测量 $\mathcal{一种}={\{({米}_j,{乙}_j)\}}_{j=1}^{米}$ 具有广义相位检索特性 $\mathbb{F}=\mathbb{电阻}$ （分别 $\mathbb{F}=\mathbb{C}$）。