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The hyperspectral snapshot compressive imaging (SCI) system encodes three-dimensional hyperspectral images into a single two-dimensional snapshot measurement and then decodes the underlying 3-D hyperspectral images by solving the compressive sensing reconstruction problem. Practical applications of SCI imaging systems require fast and high-quality reconstruction. To meet this requirement, we propose a novel encoder and decoder network with dense back-projection joint attention for hyperspectral SCI. The main contributions of our network lie in two aspects. First, we propose a dense back-projection module and deploy it in an encoder with five scales. It computes the back-projection between each scale and all its preceding scales, thereby fusing complementary information between different scales for efficient reconstruction. Second, we design a spatial-spectral attention module and deploy it in the decoder to boost reconstruction quality. By exploiting a cascade of spatial-spectral attention, it can efficiently capture spatial and spectral correlations in hyperspectral images with a low volume of parameters. In addition, a compound loss, including the reconstruction loss, and the spatial–spectral total variation loss, is designed to guide network learning in an end-to-end manner. Intensive experiments on simulation and real data show that our method has obvious advantages over multiple state-of-the-art methods, achieving a significant improvement in reconstruction quality and a substantial reduction in running time.