The phase-shifting digital holography (PSDH) is a typical problem in holography. It is traditionally conducted by the measurements from multiple shots. The four- and three-shot approaches to PSDH are commonly utilized for the imaging of static objects. Unlike this, the imaging of dynamic objects requires that PSDH should be conducted by single-shot measurements. Motivated by this, we are interested in the single-shot phase retrieval (PR) problem for periodic signals. The reference wave we use for such a PR model is the plane reference wave, and the measurements are the single-shot intensities. A single-shot PR algorithm is established. Additionally, it is observed that what is required for the four- and three-shot approaches are essentially the intensity differences. From the perspective of intensity difference, the stability of the algorithm is systematically investigated. In particular, it is found that the stability depends on the correlation between the wave vector and sampling set. Based on the wave vector, the choice scheme for sampling set is designed such that the recovery is well-conditioned in the noisy setting. The simulation of image recovery confirms the correctness of related results.

Graph signal processing (GSP) is an emerging field in the signal processing community. Novel GSP-based transforms, such as graph Fourier transform and graph wavelet filter banks, have been successfully utilized in image processing and pattern recognition. As a rapidly developing research area, graph signal processing aims to extend classical signal processing techniques to signals with irregular underlying structures. One of the hot topics in GSP is to develop multi-scale transforms such that novel GSP-based techniques can be applied in image processing or other related areas. For designing graph signal multi-scale frameworks, downsampling operations that ensuring multi-level downsampling should be specifically constructed. Among the existing downsampling methods in graph signal processing, the state-of-the-art method was constructed based on the maximum spanning tree (MST). However, when using this method for multi-level downsampling of graph signals defined on unweighted densely connected graphs, such as social network data, the sampling rates are not close to [Formula: see text]. This phenomenon is summarized as a new problem and called downsampling unbalance problem in this paper. Due to the unbalance, MST-based downsampling method cannot be applied to construct graph signal multi-scale transforms. In this paper, we propose a novel and efficient method to detect and reduce the downsampling unbalance generated by the MST-based method. For any given graph signal, we apply the graph density to construct a measurement of the downsampling unbalance generated by the MST-based method. If a graph signal has large unbalance possibility, the multi-level downsampling is conducted after the MST is improved. The experimental results on synthetic and real-world social network data show that downsampling unbalance can be efficiently detected and then reduced by our method.