Recently, many variational models involving high order derivatives have been widely used in image processing, because they can reduce staircase effects during noise elimination. However, it is very challenging to construct efficient algorithms to obtain the minimizers of original high order functionals. In this paper, we propose a new linearized augmented Lagrangian method for Euler's elastica image denoising model. We detail the procedures of finding the saddle-points of the augmented Lagrangian functional. Instead of solving associated linear systems by FFT or linear iterative methods (e.g., the Gauss-Seidel method), we adopt a linearized strategy to get an iteration sequence so as to reduce computational cost. In addition, we give some simple complexity analysis for the proposed method. Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method, and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.

With the help of endmember spectral library, sparse unmixing techniques have been successfully applied to hyperspectral image interpretation. The inclusion of spatial information in the sparse unmixing significantly improves the resulting fractional abundances. However, most existing spatial sparse unmixing algorithms are sensitive to noise and produce unstable solutions. To alleviate this drawback, a new robust double spatial regularization sparse unmixing (RDSRSU) method is proposed, which simultaneously exploits the spatial structure information from hyperspectral images and estimated abundance maps to mitigate the negative influence of noise on unmixing, so as to achieve robust sparse unmixing. To this end, a precalculated spatial weighting factor is introduced to maintain the original spatial information of the hyperspectral image. Meanwhile, the total variation spatial regularizer is used to capture the piecewise smooth structure of each abundance map. The experimental results, conducted by two sets of simulated data, as well as Cuprite and Mangrove real hyperspectral data, uncover that the proposed RDSRSU algorithm can offer better antinoise ability and obtain more accurate results over those gave by other advanced sparse unmixing algorithms.

An adaptive shrinkage threshold for image denoising in shearlet domain is presented. The shearlet transform not only provides the mean to detect orientations and to lead to sparse representations, but is moreover equipped with a rich mathematical structure similar to wavelets. Finally, a simulation is carried out to show the effectiveness of the new estimator. Experimental results show that the new estimator achieves state-of-art performance in terms of peak signal-to-noise ratio (PSNR) and visual quality.

Sparse unmixing has emerged as a powerful technique for addressing the presence of mixed pixels in hyperspectral images. A commonly employed approach involves integrating spectral information and spatial features within a sparse unmixing framework, with the ultimate goal of improving interpretability for remote sensing applications. In practical situations, hyperspectral images frequently encounter various forms of noise contamination, posing challenges the differentiation of target signals from mixed noise and complicating the interpretation of mixed pixels. To tackle this issue, we introduce a new spatial filtering-based sparse unmixing (SFSU) algorithm. The proposed SFSU model combines filter-guided dual spatial weighting factors and a regularization term to minimize abundance errors within the unmixing framework. By means of this integration, the model facilitates the analysis of spatial information in the image from multiple perspectives, with the objective of mitigating the impact of different noise types on the unmixing outcomes. In the first part of the SFSU model, the original spatial information of the hyperspectral image is preserved by utilizing a pre-computed filter-guided spatial weighting factor, thereby augmenting the sparsity of the abundance estimation process. Subsequently, a local neighborhood spatial weighting factor is employed to achieve segmented smoothing in the abundance maps. In the regularization term for reducing abundance errors, multiple filters are introduced to improve the algorithm’s robustness to various types of noise by using the difference between the filtered and denoised abundance map and the expected abundance map as a constraint. Extensive experiments conducted on both simulated and real hyperspectral datasets demonstrate the exceptional performance of the SFSU algorithm in effectively addressing complex mixed noise pollution. Moreover, the proposed spatial filtering strategy has been proven to significantly improve the accuracy of unmixing.

By given a sparse signal in a high dimensional space, compressive sensing system, which combines with sampling and compression, can reconstruct that signal accurately and efficiently from fewer linear measurements much less than its actual dimension using sparse priors of signal. Currently, researchers always use orthogonal wavelet to represent the images. But the wavelet only has single scaling function and can not simultaneously satisfy the orthogonality, high vanishing moments, compact support, symmetry characteristic and regularity. Developed from the theory of wavelet, multi-wavelet transform, which can simultaneously satisfy the five characteristics, provides a great potential to obtain high-performance coding. According to the three main steps (Sparse representation, measurement matrix, reconstruction algorithm) of compressive sensing image reconstruction, this paper proposes a compressive sensing image reconstruction based on sparse representation of the image in multi-wavelet transform domain while using Orthogonal Matching Pursuit iterative as the reconstruction algorithm. The experimental results show that the reconstructed image has batter vision quality and a good performance on PSNR. Meanwhile, the algorithm of reconstruction gets a faster convergence rate.