Single-pixel imaging (SPI) uses a single-pixel detector instead of a detector array with a lot of pixels in traditional imaging techniques to realize two-dimensional or even multi-dimensional imaging. For SPI using compressed sensing, the target to be imaged is illuminated by a series of patterns with spatial resolution, and then the reflected or transmitted intensity is compressively sampled by the single-pixel detector to reconstruct the target image while breaking the limitation of the Nyquist sampling theorem. Recently, in the area of signal processing using compressed sensing, many measurement matrices as well as reconstruction algorithms have been proposed. It is necessary to explore the application of these methods in SPI. Therefore, this paper reviews the concept of compressive sensing SPI and summarizes the main measurement matrices and reconstruction algorithms in compressive sensing. Further, the performance of their applications in SPI through simulations and experiments is explored in detail, and then their advantages and disadvantages are summarized. Finally, the prospect of compressive sensing with SPI is discussed.

Utilizing a complex-number basis for single-pixel imaging (SPI) a spatial wavefront in high quality is still an open issue. To solve this issue, here, we proposed a method called OAM-basis spatial wavefront single-pixel imaging (WSPI) via compressed sensing (CS) for reconstructing a spatial wavefront in high quality. In there, we generate the orbital angular momentum (OAM) beams as the complex-number basis for the measurement, and propose to use the TVAL3 CS algorithm for the wavefront reconstruction, for the first time to the best of our knowledge. The reconstruction method is appropriate for WSPI using a complex-number basis. Both simulations and experiments show that our method is superior to the existing second-order correlation method. It provides a promising complex-number-basis-based WSPI technique for high-quality wavefront reconstruction. Thus, it may find applications involving optical communication, optical detection and imaging, optical information processing and encryption, and so on.