# Single-Shot Compressed Imaging via Random Phase Modulation
###### Abstract  
Compressed sensing (CS) provides an innovative framework for signal sampling, which enables accurate recovery of the sparse or compressible signal from a small set of linear measurements far fewer than the Nyquist rate in traditional signal processing. In compressed sensing, random modulation plays a key role, which can spread out the signal information more or less evenly across all locations. There are many modulation techniques, such as amplitude modulation, frequency modulation, phase modulation, spectrum modulation, and so on. Among these modulation techniques, phase modulation is vital due to the efficiency and convenience of modulation. In this paper, we review both the theoretical and application of compressed sensing and several compressed imaging systems using random phase modulation. First, we review the fundamentals of compressed sensing, dividing it into three parts: sparse representation, incoherent measurement, and nonlinear reconstruction algorithm. We then show how phase modulation can be applied to compressed sensing and compressed imaging, where the presentation can be divided into six main parts, corresponding to different aspects of phase modulation applied in compressed sensing or compressed imaging: (1) Fundamentals of compressed sensing. (2) Principles of phase modulation. (3) Single-shot compressed imaging with spatial-domain single random phase mask (CI-SSRPM). (4) Single-shot compressed imaging with a random convolution using a double random phase mask (CI-DRPM). (5) Single-shot compressed imaging with Fourier-domain single random phase mask (CI-FSRPM). (6) Single-shot compressed imaging with double random phase encoding (CI-DRPE). compressed sensing; compressed imaging; random phase modulation; random convolution; double random phase encoding 20222022022022022022022020220202022020220202202022020220202022020220202202022020220202022020202202022020202202  
recover from fewer measurements less than the Nyquist rate. In classical CS, the random Gaussian matrix (RGM) is the most popular measurement matrix due to its theoretical guarantees. However, the drawback of RGM is that it is impractical or too expensive for physical implementation, and other random modulation techniques must be introduced to realize random modulation. CS theory is not only available to traditional imaging applications but also available in more complex cases, e.g., infrared imaging and terahertz imaging. The introduction of CS theory can help enhance the performance of various imaging systems in terms of high-quality reconstruction, low cost, small size and etc. From the aspect of physical realization, various modulation techniques existed, such as amplitude modulation, frequency modulation, phase modulation, spectrum modulation, and so on. Among these modulation techniques, phase modulation plays a vital role in the efficiency and convenience of phase modulation. In 2009, [PERSON] [3] proposed a compressed imaging system by random convolution and that the convolution with a specific-designed random pulse followed by random subsampling was considered as a universal compressed sampling operator, which was equivalent to a random phase modulation in the Fourier-domain of the well-known 4-f system. In 2010, [PERSON] et al. [4] proposed a single-exposure compressed imaging system by combining the well-known optical scheme of double random phase encoding (DRPE), which could be applied to overcome the resolution limit caused by both diffraction and geometrical limitation. In 2014, [PERSON] et al. proposed a quasi-monochromatic thermo optical phase modulation compressed sensing imaging system, which overcomes the problem of collecting too much data due to high-resolution requirements in the imaging process [5]. In 2017, [PERSON] et al. [6] proposed a single exposure compressed imaging system based on the conventional optical 4-f architecture with an RPM placed in the Fourier plane and a random coded aperture in the image plane for subsampling. In 2021, [PERSON] et al.