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The Fourier coded aperture transform hyperspectral imaging system (FCTS), which we proposed in our previous work, has excellent physical interpretability compared to other encoding methods. Due to the excellent energy concentration characteristic of Fourier encoding, it can achieve high-quality spectral imaging even under undersampling conditions. However, since the grayscale encoding method limits the system's sampling speed, we propose a binary sampling Fourier-encoded spectral imaging method. Besides, to address the significant errors of reconstructed spectral image at low sampling rates, we construct a spectral image enhancement network, which introduces spatial and channel attention modules to successively enhance the spatial and spectral information of the reconstructed spectral images. During the spectral image refinement and enhancement stage, band grouping is performed to alleviate the difficulty in feature extraction and make the training process more stable. The effectiveness of our proposed method has been validated on both public datasets and our own datasets. When the sampling times are reduced by an order of magnitude, on the three public datasets, the PSNR improved by 10.79 dB, the SSIM increased by 0.35, and the SAM decreased by 12.1° on average. On our own datasets, the PSNR improved by 12 dB, the SSIM increased by 0.24, and the SAM decreased by 6.5°. • A Fourier coded aperture transform rapid spectral imaging system. • An end-to-end spectral image enhancement network. • Adopting undersampling and binarizing the masks to achieve rapid spectral imaging. • Binarizing the masks to enable the use of high-speed DMD during sampling process. • Introducing spatial attention and channel attention to enhance the spatial and spectral information. [ABSTRACT FROM AUTHOR]