## 1 Introduction
However, this method based on Tucker decomposition requires more intermediate variables to be estimated, which requires more computational effort and usually results in a slower fusion rate. A fusion method based on canonical polyadic decomposition (CPD) is proposed in [30], which decomposes the target HR-HSI into the sum of vector products of three dimensions. It effectively reduces the computational effort of tensor decomposition-based methods. However, this approach assumes that the ranks of the three dimensions are the same (which is not valid in some cases), so this usually leads to undesirable fusion results. In [31], the advantages of Tucker decomposition and CP decomposition are combined, and a fusion method based on tensor ring factorization is proposed. This method views the target HR-HSI as a matrix product of three dimensions and achieves hyperspectral image super-resolution by estimating the spectral dimension matrix from LR-HSI and the two matrices of spatial dimension from HR-MSI. This fusion method usually requires the estimation of three matrices, which also has a significant computation and can lead to poor timeliness of fusion. Although the recently proposed tensor factorization-based method can effectively preserve the three-dimensional characteristics of hyperspectral images, the tensor factorization-based method also has to be operated by converting to a two-dimensional matrix, and the fusion quality is not substantially improved. Compared with the tensor factorization-based method, the matrix factorization-based method has better timeliness. This paper proposes a fast and stable fusion method based on matrix factorization to ensure the fusion quality and shorten the fusion time. This method first solves the fuzzy estimation of the spectral basis through a simple singular value decomposition method. Then, through the Moore-Penrose inversion operation, it calculates the fuzzy estimate of the spectral coefficient matrix and then obtains the optimized spectral base and coefficient matrix estimate through a multiplication iterative optimization process. Finally, the exact HR-HSI estimate is obtained by doing another multiplicative iterative optimization operation on the fuzzy version of the HR-HSI obtained in the previous step. Overall, this paper presents a fast and stable hyperspectral multispectral image fusion technique. The significant contributions of this work are as follows. * It is the first time that the Moore-Penrose inverse has been used to determine the solution of the spectral coefficient matrix, which simplifies the solution process of the fusion model, dramatically reduces the computational workload, and shortens the fusion time. * A multiplicative iterative optimization process is used to optimize the spectral basis and coefficient matrices, enabling further enhancement of the fused results and improving the quality of the fusion. Previous fusion methods stop optimizing the fusion results after obtaining the spectral basis and coefficient matrices. In contrast, the method proposed in this paper continues to optimize the rough HR-HSI estimates after fusion to further improve the fusion quality after obtaining the spectral basis and coefficient matrices. The rest of this paper is organized as follows. Section 2 details the proposed method. Section 3 gives the experimental settings. The results and analysis are presented in Section 4, and the conclusion is given in Section 5.