A spatial signal is defined by its evaluations on the whole domain. In this paper, we consider stable reconstruction of real-valued signals with finite rate of innovation (FRI), up to a sign, from their magnitude measurements on the whole domain or their phaseless samples on a discrete subset. FRI signals appear in many engineering applications such as magnetic resonance spectrum, ultra wide-band communication and electrocardiogram. For an FRI signal, we introduce an undirected graph to describe its topological structure, establish the equivalence between its graph connectivity and its phase retrievability by point evaluation measurements on the whole domain, apply the graph connected component decomposition to find its unique landscape decomposition and the set of FRI signals that have the same magnitude measurements. We construct discrete sets with finite density so that magnitude measurements of an FRI signal on the whole domain are determined by its phaseless samples taken on those discrete subsets, and we show that the corresponding phaseless sampling procedure has bi-Lipschitz property with respect to a new induced metric on the signal space and the standard \(\ell ^{p}\)-metric on the sampling data set. In this paper, we also propose an algorithm with linear complexity to reconstruct an FRI signal from its (un)corrupted phaseless samples on the above sampling set without restriction on the noise level and apriori information whether the original FRI signal is phase retrievable. The algorithm is theoretically guaranteed to be stable, and numerically demonstrated to approximate the original FRI signal in magnitude measurements.

空间信号是通过对整个域的评估来定义的。在本文中，我们将考虑对有限范围的实值信号进行稳定的重构，其中有限创新率（FRI）可以从整个域上的幅度测量值或离散子集上的无相位采样值提高到一个符号。FRI信号出现在许多工程应用中，例如磁共振频谱，超宽带通信和心电图。对于FRI信号，我们引入了无向图来描述其拓扑结构，通过对整个域进行点评估测量来建立其图连通性和其相位可恢复性之间的等价关系，应用图连接分量分解来找到其独特的景观分解和具有相同幅度测量值的一组FRI信号。\（\ ell ^ {p} \） -采样数据集上的指标。在本文中，我们还提出了一种具有线性复杂度的算法，可以从上述采样集的（未）损坏的无相位采样中重建FRI信号，而不受噪声水平和先验信息是否原始FRI信号可相位恢复的限制。从理论上保证该算法是稳定的，并通过数值证明可以在幅度测量中近似原始FRI信号。