Hyperspectral image restoration using framelet-regularized low-rank nonnegative matrix factorization

Hyperspectral image (HSI) restoration is a process to remove a mixture of various kinds of noise, which is a key preprocessing step to improve the performance of subsequent applications. Since the HSI has a large correlation between spectral bands and abundant geometric features in the spatial domain, thus the low-rank prior and spatial structure prior can be introduced to HSI restoration. However, many existing approaches usually directly use the nuclear norm and total variation (TV) regularization to depict the spectral-spatial priors of HSI, which inevitably requires singular value decomposition (SVD) computation and causes staircase artifacts in the image, respectively. To overcome these limitations, in this work, we propose a novel HSI restoration method named framelet-regularized low-rank nonnegative matrix factorization (F-LRNMF), in which the low-rank nonnegative matrix factorization is developed to describe that the HSI lies in a low-rank subspace. Furthermore, to decrease the staircase artifacts caused by TV regularization that directly applies to HSI, we use framelet regularization to constrain the factor whose size is much less than HSI itself. The framelet regularization can effectively preserve the details and geometric features of the restored HSI in the spatial domain. An efficient block successive upper-bound minimization (BSUM) algorithm is designed to solve the proposed optimization model. Meanwhile, we theoretically analyze that the algorithm can converge to the set of coordinate-wise minimizers. Experiments under various cases of simulated and real HSI data demonstrate the effectiveness of the proposed model and the efficiency of the numerical algorithm in terms of both quantitative and qualitative assessments.