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Single-pixel imaging (SPI) recently has made significant progress, however, it is still an open issue to adaptively tradeoff its imaging fidelity and speed. Here, we present an adaptive sampling method for SPI to reduce the sampling rate for improving the imaging speed adaptively, sacrificing as little as possible of its imaging fidelity. The proposed method only requires setting an error band to realize adaptive under-Nyquist sampling, based upon the fact that the coefficient matrix consisted of the frequency components obtained by the orthogonal transforms is sparse. First, an appropriately elaborated sampling path, which can be divided into multiple segments, can be set according to this coefficient matrix measured by a single-pixel detector. Then the variance of each segment is calculated and expressed in logarithmic form for the real-time polynomial curve fitting. When the variation of the curve's slope is small enough and reaches an error band, the sampling is adaptively stopped. Finally, the object image can be reconstructed via the adaptive sampling method following the elaborated sampling path for improving the imaging speed adaptively. The effectiveness of the adaptive sampling SPIs based on the orthogonal transforms, such as Hadamard transform, discrete cosine transform, Fourier transform, and Krawtchouk moments transform, is theoretically and experimentally demonstrated. It should be noted the proposed method can be also applied in SPIs using other orthogonal transforms, in which only an error band is set by the users. • An adaptive under-Nyquist sampling method for single-pixel imaging. • The sparser the target, the higher the fidelity of the image. • The proposed method is not limited to a specific orthogonal transform. • The proposed method can be suitable for SPI with high-frequency noise. [ABSTRACT FROM AUTHOR]

• Simultaneously amplitude and phase (i.e. spatial wavefront) imaging of a biological sample via DMD-based single-pixel phase-shifting interferometric techniques is demonstrated. • The experimental performances of those techniques for wavefront imaging of a biological sample are quantitatively and comprehensively compared, when different orthogonal bases, different reference strategies, and different modulation methods (i.e. binary grating, Lee, and super-pixel methods) are employed there. • Experimental results suggest that the grating-phase-shifting-based SWI using Hadamard basis (G-HSWI) performs best, in which one can use the checkerboard reference for the maximum field of view or select the peripheral reference for a better imaging resolution. • 2fps dynamic wavefront imaging of the sample at a 10 % downsampling rate with a resolution of 128 × 128 pixels can be realized by G-HSWI. Single-pixel wavefront imaging (SWI) is an innovative technique for simultaneously measuring the amplitude and phase of an unknown field, which is particularly important for wavefront imaging in invisible wavelengths and under low light conditions. However, most of the existing SWI techniques are still in the technical development stage. Here, we provide a comprehensive analysis and comparison of the SWI methods using DMD-based single-pixel multi-step phase-shifting interferometric techniques. Through experimental demonstrations, we first compare the performances of the techniques using the Hadamard basis, Fourier basis, and discrete cosine transform (DCT) basis. Further, the performances of the Hadamard basis-based SWI (HSWI) technique using phase-shifting interferometry with the different reference strategies are investigated while the binary grating, Lee, and super-pixel methods are applied to control the DMD for realizing the necessary complex amplitude modulations there. Experimental results suggest the grating-phase-shifting-based HSWI performs best, in which one can use the checkerboard reference for the maximum field of view or select the peripheral reference for a better imaging resolution; and it can do 2 fps wavefront imaging of the sample at 10 % downsampling rate with a resolution of 128 × 128 pixels. Our work provides researchers with a guideline to choose a suitable SWI technique for their applications. [ABSTRACT FROM AUTHOR]

• To guarantee almost the same wavefront imaging quality as Fourier single-pixel imaging (FSPI) does while the imaging speed can be accelerated several folds, a discrete cosine transform single-pixel imaging (DCT-SPI) technique for measuring the spatial wavefront of an unknown field was proposed. • The DCT-SPI technique for measuring different wavefronts as well as its application for quantitative phase imaging of a dragonfly wing was experimentally demonstrated, and its performance was compared with that of FSPI in detail. • Experimental results suggest that the imaging speed of DCT-SPI can be several times faster than that of FSPI, and it can obtain the phase and amplitude of a wavefront with a higher imaging resolution under the RAM capacity limit of the DMD, as compared to the FSPI technique. • The results also suggest that DCT-SPI can be used for measuring the phase and amplitude of a dragonfly wing, which suggests it can be applied in the field of quantitative phase imaging with the merits mentioned above. In this work, a checkerboard pattern is used to divide an unknown field into the signal and reference parts, and based on the interference between them, we propose and demonstrate a discrete cosine transform single-pixel imaging (DCT-SPI) technique for measuring the spatial wavefront of an unknown field. Here, the discrete cosine transform (DCT) basis was used for generating the modulation patterns to acquire the spectrum of the wavefront by the phase-shifting technique, and the wavefront to be measured can be retrieved by applying a second-order correlation calculation. We first demonstrate the proposed DCT-SPI can be used for measuring the spatial wavefronts simulated by an amplitude SLM(A-SLM). As compared to the phase-shifting Fourier single-pixel imaging (FSPI) technique, the imaging speed of DCT-SPI is four times faster. In addition, under the RAM capacity limit of a digital micromirror device (DMD), DCT-SPI can realize a higher resolution. Further, we experimentally demonstrate that DCT-SPI can also be used for measuring the phase and amplitude of a dragonfly wing, which suggests it can be applied in the field of quantitative phase imaging with a higher speed and imaging resolution as compared with FSPI. [ABSTRACT FROM AUTHOR]