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Synchrotron radiation (SR) X-ray tomography is a main technique in realizing imaging of an object section by section. We apply this technique in recovering shale microstructures which becomes a hot topic recently in non-conventional oil and gas exploration. We first set the experimental setup using SR sources at BL13W1 beamline at Shanghai Synchrotron Radiation Facility to obtain the measured data of the shale sample, and then we establish the -norm constrained regularization model to reconstruct the microstructure of the shale. A fast convergent gradient descent method is developed to realize this purpose. It indicates from our test results that the new method possesses higher resolution of imaging than the conventional results given by the commercial software. [ABSTRACT FROM AUTHOR]

• This paper propose a new image denoising model by coupling the rotation matrix and the weighted matrix into the TVp quasinorm regularization. • Since the proposed model is nonconvex and non-Lipschitz, we employe the ADMM to solve it with the help of the half-quadratic scheme. • The convergence of the proposed numerical methods is also discussed. • Numerical comparisons show that the proposed model leaded to considerable performance gains when testing on several denoising tasks. Image denoising problem still remains an active research field in the image processing. To improve the denoising quality, it is very important to describe the local structure of the image in the proposed model. This fact motivates us to introduce an adaptive weighted TVp regularization-based denoising model, where the rotation matrix and the weighted matrix depend on the local structure of the image. Specially, these two matrices can enhance the diffusion of the responding Euler-Lagrangian equation along with the tangential direction of the edge. This procedure offers more control over the regularization and then allows more denoising in smooth regions and less denoising when processing edge regions. In addition, since the proposed model is nonsmooth and non-Lipschitz, we employ the alternating direction method of multipliers (ADMM) to solve it with the help of using the half-quadratic scheme to solve the related ℓ 2 − ℓ p subproblem. In particular, we also provide the convergence analysis of the used numerical methods. Some numerical comparisons show that the proposed model leads to considerable performance gains when tested on several denoising tasks. [ABSTRACT FROM AUTHOR]