Fast algorithm with theoretical guarantees for constrained low-tubal-rank tensor recovery in hyperspectral images denoising

Hyperspectral images (HSIs) are unavoidably degraded by mixed noise, including Gaussian noise and sparse noise. In this paper, we consider a constrained tubal rank and sparsity model (CTSD) to tackle the HSIs mixed noise removal, which characterizes the clean HSIs via the low-tubal-rank constraint and the sparse noise via the l₀ and l_∞ norm constraints, respectively. Due to the strong non-convexity, the CTSD model is challenging to solve. To tackle the CTSD, we develop the proximal alternating minimization (PAM) algorithm via the exact tensor singular value decomposition (t-SVD) and establish the global convergence under mild assumptions. Since the t-SVD is computationally expensive, especially for large scale images, we further design an efficient inexact PAM algorithm via an inexact t-SVD. The inexact PAM enjoys two advantages: (1) The computational complexity for SVDs of the inexact PAM (O(rn₁n₂n₃)) is about twofold faster than that of the exact PAM (O(min(n₁,n₂)n₁n₂n₃)) for r≪min(n₁,n₂); (2) The accuracy of the inexact PAM is theoretically guaranteed. Extensive experiments on HSIs denoising demonstrate that the exact and inexact methods both outperform comparing methods in quantitative evaluation metrics and visual effects, and the inexact PAM can compromise between the accuracy and efficiency for large scale HSIs.