Robust Lp-Norm Inversion for High-Resolution Fluid Contents from Nuclear Magnetic Resonance Measurements

Fluid content computed from nuclear magnetic resonance (NMR) has proved to be an accurate and reliable tool for petrophysical property estimation. To overcome the limitations of conventional NMR measurements, high spatial resolution NMR (HSR-NMR) has been introduced to achieve the desired resolution for cores of any size. However, inversion of fluid contents from HSR-NMR data suffers from nonreliable measurements at the ends of the cores due to the heterogeneities of the magnetic fields caused by the relatively small size of the coil. A robust Lp-norm inversion algorithm, developed for geophysical inverse problems, has been implemented and applied on the inversion of NMR measurements. The estimated fluid content from Lp inversion matches well with the kerogen content in the cores both visually and quantitively. The resolution of the inverted fluid contents is as high as 1 inch. Further testing on the raw data with large derivations demonstrated that reliable results can only be achieved by using Lp inversion with low p’s values within the range of (1, 1.1].

Sparsity Enhanced Beamforming in The Presence of Coherent Signals

In order to find the directions of coherent signals, a sparsity enhanced beam-forming method is proposed. Unlike the conventional minimum variance distortless response (MVDR) method, the minimum variance in the proposed method corresponds to the orthogonal relationship between the noise subspace and the sparse representation of the received signal vector, whereas the distortless response corresponds to the nonorthogonal relationship between the signal subspace and the sparse representation of the received signal vector. The proposed sparsity enhanced MVDR (SEMVDR) method is carried out by the iterative reweighted Lp-norm constraint minimization. for direction finding of coherent signals. Simulation results are shown that SEMVDR has better performance than the existing algorithms, such as MVDR and MUSIC, when coherent signals are present.

Hyperspectral Image Denoising Based on Nonconvex Low-Rank Tensor Approximation and lp Norm Regularization

A new nonconvex smooth rank approximation model is proposed to deal with HSI mixed noise in this paper. The low-rank matrix with Laplace function regularization is used to approximate the nuclear norm, and its performance is superior to the nuclear norm regularization. A new phase congruency lp norm model is proposed to constrain the spatial structure information of hyperspectral images, to solve the phenomenon of “artificial artifact” in the process of hyperspectral image denoising. This model not only makes use of the low-rank characteristic of the hyperspectral image accurately, but also combines the structural information of all bands and the local information of the neighborhood, and then based on the Alternating Direction Method of Multipliers (ADMM), an optimization method for solving the model is proposed. The results of simulation and real data experiments show that the proposed method is more effective than the competcing state-of-the-art denoising methods.

Convergence in Lp Norm

This chapter looks in detail at proofs of the weak law of large numbers (convergence in probability) using the technique of establishing convergence in Lp‐norm. The extension to a proof of almost‐sure convergence is given, and then special results for martingale differences, mixingales, and approximable processes. These results are proved in array notation to allow general forms of heterogeneity.

A Total Lp-Norm Optimization for Bearing-Only Source Localization in Impulsive Noise with SαS Distribution

This paper considers the problem of robust bearing-only source localization in impulsive noise with symmetric α-stable distribution based on the Lp-norm minimization criterion. The existing Iteratively Reweighted Pseudolinear Least-Squares (IRPLS) method can be used to solve the least LP-norm optimization problem. However, the IRPLS algorithm cannot reduce the bias attributed to the correlation between system matrices and noise vectors. To reduce this kind of bias, a Total Lp-norm Optimization (TLPO) method is proposed by minimizing the errors in all elements of system matrix and data vector based on the minimum dispersion criterion. Subsequently, an equivalent form of TLPO is obtained, and two algorithms are developed to solve the TLPO problem by using Iterative Generalized Eigenvalue Decomposition (IGED) and Generalized Lagrange Multiplier (GLM), respectively. Numerical examples demonstrate the performance advantage of the IGED and GLM algorithms over the IRPLS algorithm.