During the acquisition process, hyperspectral images (HSIs) are inevitably contaminated by mixed noise, which seriously affects the image quality. To improve the image quality, HSI denoising is a critical preprocessing step. In HSI denoising tasks, the method based on low-rank prior has achieved satisfying results. Among numerous denoising methods, the tensor nuclear norm (TNN), based on the tensor singular value decomposition (t-SVD), is employed to describe the low-rank prior approximately. Its calculation can be sped up by the fast Fourier transform (FFT). However, TNN is computed by the Fourier transform, which lacks the function of locating frequency. Besides, it only describes the low-rankness of the spectral correlations and ignores the spatial dimensions’ information. In this paper, to overcome the above deficiencies, we use the basis redundancy of the framelet and the low-rank characteristics of HSI in three modes. We propose the framelet-based tensor fibered rank as a new representation of the tensor rank, and the framelet-based three-modal tensor nuclear norm (F-3MTNN) as its convex relaxation. Meanwhile, the F-3MTNN is the new regularization of the denoising model. It can explore the low-rank characteristics of HSI along three modes that are more flexible and comprehensive. Moreover, we design an efficient algorithm via the alternating direction method of multipliers (ADMM). Finally, the numerical results of several experiments have shown the superior denoising performance of the proposed F-3MTNN model.

Farmland carbon pools are pivotal to the terrestrial carbon cycle. Studying soil organic carbon density (SOCD) spatial heterogeneity is crucial for uncovering regional carbon balance mechanisms and optimizing sequestration strategies. Focusing on county-scale SOCD heterogeneity, this study aids sustainable farmland management, carbon sequestration, and spatial carbon pool modeling. Based on the 2020 farmland soil organic carbon (SOC) analysis survey data. The SOCD and spatial pattern of farmland in Chengcheng County, Shaanxi Province were estimated and analysed by combining GIS and GPS technologies. Optimal Parameter Geodetector (OPGD) was used to detect spatial heterogeneity and quantitatively attribute the influencing factors of SOCD in farmland in Chengcheng. The results showed that the SOCD of farmland in Chengcheng, ranged from 1.10 to 1.79 kg m⁻², with an average of 1.39 kg m⁻². The SOC stock was 151.96 × 10⁴ t. Both values were lower than the national average and much lower than those in European and American countries. Between the north-west and south-east of the plough layer, Chengcheng’s farmland’s SOCD clearly distinguishes two distinct sections. The average farmland SOCD in the north-west is lower than that in the south-east by 0.27 kg m⁻². The explanatory power of temperature, population density and DEM all exceeded 0.3, and the explanatory power of the interaction between them exceeded 0.6. This implies that the combined influences of nature and human activity are responsible for the spatial patterns of SOCD in farming at the county scale in the Weibei dry loess plateau.