This paper proposes a two-phase scheme for removing salt-and-pepper impulse noise. In the first phase, an adaptive median filter is used to identify pixels which are likely to be contaminated by noise (noise candidates). In the second phase, the image is restored using a specialized regularization method that applies only to those selected noise candidates. In terms of edge preservation and noise suppression, our restored images show a significant improvement compared to those restored by using just nonlinear filters or regularization methods only. Our scheme can remove salt-and-pepper-noise with a noise level as high as 90%.

The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the $c$-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group $SO(3)$, and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with $SO(3)$-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.

Real images usually have two layers, namely, cartoons (the piecewise smooth part of the image) and textures (the oscillating pattern part of the image). Both these two layers have sparse approximations under some tight frame systems such as framelet, translation invariant wavelet, curvelet, and local DCTs. In this paper, we solve image inpainting problems by using two separate tight frame systems which can sparsely represent cartoons and textures respectively. Different from existing schemes in the literature which are either analysis-based or synthesis-based sparsity priors, our minimization formulation balances these two priors. We also derive iterative algorithms to find their solutions and prove their convergence. Numerical simulation examples are given to demonstrate the applicability and usefulness of our proposed algorithms in image inpainting.

Monitoring changes within the land surface and open water bodies is critical for natural resource management, conservation, and environmental policy. While the use of satellite imagery for these purposes is common, fine-scale change detection can be a technical challenge. Difficulties arise from variable atmospheric conditions and the problem of assigning pixels to individual objects. We examined the degree to which two machine learning approaches can better characterize change detection in the context of a current conservation challenge, artisanal small-scale gold mining (ASGM). We obtained Sentinel-2 imagery and consulted with domain experts to construct an open-source labeled land-cover change dataset. The focus of this dataset is the Madre de Dios (MDD) region in Peru, a hotspot of ASGM activity. We also generated datasets of active ASGM areas in other countries (Venezuela, Indonesia, and Myanmar) for out-of-sample testing. With these labeled data, we utilized a supervised (E-ReCNN) and semi-supervised (SVM-STV) approach to study binary and multi-class change within mining ponds in the MDD region. Additionally, we tested how the inclusion of multiple channels, histogram matching, and La*b* color metrics improved the performance of the models and reduced the influence of atmospheric effects. Empirical results show that the supervised E-ReCNN method on 6-Channel histogram-matched images generated the most accurate detection of change not only in the focal region (Kappa: 0.92 (± 0.04), Jaccard: 0.88 (± 0.07), F1: 0.88 (± 0.05)) but also in the out-of-sample prediction regions (Kappa: 0.90 (± 0.03), Jaccard: 0.84 (± 0.04), and F1: 0.77 (± 0.04)). While semi-supervised methods did not perform as accurately on 6- or 10-channel imagery, histogram matching and the inclusion of La*b* metrics generated accurate results with low memory and resource costs. These results show that E-ReCNN is capable of accurately detecting specific and object-oriented environmental changes related to ASGM. E-ReCNN is scalable to areas outside the focal area and is a method of change detection that can be extended to other forms of land-use modification.