Pan-sharpening is an important means to improve the spatial resolution of multispectral (MS) images. Although a large number of pan-sharpening methods have been developed, improving the spatial resolution of MS while effectively maintaining its spectral information has not been well solved so far, and it has also been taken as a criterion to measure whether the sharpened product can meet the practical needs. The back-projection (BP) method iteratively injects spectral information backwards into the sharpened results in a post-processing manner, which can effectively improve the generally unsatisfied spectral consistency problem in pan-sharpening methods. Although BP has received some attention in recent years in pan-sharpening research, the existing related work is basically limited to the direct utilization of the BP process and lacks a more in-depth intrinsic integration with pan-sharpening. In this paper, we analyze the current problems of improving the spectral consistency based on BP in pan-sharpening, and the main innovative works carried out on this basis include the following: (1) We introduce the spatial consistency condition and propose the spatial–spectral BP (SSBP) method, which takes into account both spatial and spectral consistency conditions, to improve the spectral quality while effectively solving the problem of spatial distortion in the results. (2) The proposed SSBP method is analyzed theoretically, and the convergence condition of SSBP and a more relaxed convergence condition for a specific BP type, degradation transpose BP, are given and proved theoretically. (3) Fast computation of BP and SSBP is investigated, and non-iterative fast BP (FBP) and fast SSBP algorithms (FSSBP) methods are given in a closed-form solution with significant improvement in computational efficiency. Experimental comparisons with combinations formed by seven different BP-related post-processing methods and up to 18 typical base methods show that the proposed methods are generally applicable to the optimization of the spatial–spectral quality of various sharpening methods. The fast method improves the computational speed by at least 27.5 times compared to the iterative version while maintaining the evaluation metrics well.

The enormous amount of data that are generated by hyperspectral remote sensing images (HSI) combined with the spatial channel’s limited and fragile bandwidth creates serious transmission, storage, and application challenges. HSI reconstruction based on compressed sensing has become a frontier area, and its effectiveness depends heavily on the exploitation and sparse representation of HSI information correlation. In this paper, we propose a low-rank sparse constrained HSI reconstruction model (LRCoSM) that is based on joint spatial-spectral HSI sparseness. In the spectral dimension, a spectral domain sparsity measure and the representation of the joint spectral dimensional plane are proposed for the first time. A Gaussian mixture model (GMM) that is based on unsupervised adaptive parameter learning of external datasets is used to cluster similar patches of joint spectral plane features, capturing the correlation of HSI spectral dimensional non-local structure image patches while performing low-rank decomposition of clustered similar patches to extract feature information, effectively improving the ability of low-rank approximate sparse representation of spectral dimensional similar patches. In the spatial dimension, local-nonlocal HSI similarity is explored to refine sparse prior constraints. Spectral and spatial dimension sparse constraints improve HSI reconstruction quality. Experimental results that are based on various sampling rates on four publicly available datasets show that the proposed algorithm can obtain high-quality reconstructed PSNR and FSIM values and effectively maintain the spectral curves for few-band datasets compared with six currently popular reconstruction algorithms, and the proposed algorithm has strong robustness and generalization ability at different sampling rates and on other datasets.