Quasinorm

Abstract: Owing to the limitations of the imaging principle as well as the properties of imaging systems, infrared images often have some drawbacks, including low resolution, a lack of detail, and indistinct edges. Therefore, it is essential to improve infrared image quality. Considering the information of neighbors, a description of sparse edges, and by avoiding staircase artifacts, a new super-resolution reconstruction (SRR) method is proposed for infrared images, which is based on fractional order total variation (FTV) with quaternion total variation and the L p quasinorm. Our proposed method improves the sparsity exploitation of FTV, and efficiently preserves image structures. Furthermore, we adopt the plug-and-play alternating direction method of multipliers (ADMM) and the fast Fourier transform (FFT) theory for the proposed method to improve the efficiency and robustness of our algorithm; in addition, an accelerated step is adopted. Our experimental results show that the proposed method leads to excellent performances in terms of an objective evaluation and the subjective visual effect.

Abstract: The aim of this paper is to study a nonlinear stationary Stokes problem with mixed boundary conditions that describes the ice velocity and pressure fields of grounded glaciers under Glen's flow law. Using convex analysis arguments, we prove the existence and the uniqueness of a weak solution. A finite element method is applied with approximation spaces that satisfy the inf-sup condition, and a priori error estimates are established by using a quasinorm technique. Several algorithms (including Newton's method) are proposed to solve the nonlinearity of the Stokes problem and are proved to be convergent. Our results are supported by numerical convergence studies.

Abstract: The quantitative explanation of the potential field data of 3-D geological structures remains one of the most challenging issues in modern geophysical inversion. Obtaining a stable solution that can simultaneously resolve complicated geological structures is a critical inverse problem in the geophysics field. I have developed a new method for determining a 3D petrophysical property distribution, which produces a corresponding potential field anomaly. In contrast with the tradition inverse algorithm, my inversion method proposes a new model norm, which incorporates two important weighting functions. One is the L0 quasi norm (enforcing sparse constraints), and the other is depth weighting that counteracts the influence of source depth on the resulting potential field data of the solution. Sparseness constraints are imposed by using the L0 quasi-norm on model parameters. To solve the representation problem, a L0 quasi-norm minimization model with different smooth approximations is proposed. Hence, the data space (N) method, which is much smaller than model space (M), combined with the gradient projected method and the model space combined with the modified Newton method for L0 quasi-norm sparse constraints leads to a computationally efficient method by using an N × N system versus an M × M one, because N << M. Tests on synthetic data and real data sets demonstrate the stability and validity of the L0 quasi-norm spare norms inversion method. With the aim of obtaining the blocky results, the inversion method with the L0 quasi-norm sparse constraints method performs better than the traditional L2 norm (standard Tikhonov regularization). It can obtain the focus and sparse results easily. Then, the Bouguer anomaly survey data of the Salt Dome, offshore Louisiana is considered as a real case study. The real inversion result shows that the inclusion the L0 quasi-norm sparse constraints leads to a simpler and better resolved solution, and the density distribution is obtained in this area to reveal its geological structure. These results confirm the validity of the L0 quasi-norm sparse constraint method and indicate its application for other potential field data inversions and the exploration of geological structures. This article is protected by copyright. All rights reserved