Search Results (12)

We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of the equation. Noteworthy, our results are still valid for fractional powers of the negative Laplacian operator. We base our methods on the theory of operator-valued Fourier multipliers on vector-valued Lebesgue spaces of periodic functions. Full article

From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As a generalization of SLQ, interesting questions are, “how about the situation in the differential game?”, “will the same phenomenon appear in SLQ?”. This paper will provide the answers. In this paper, we consider a closed-loop two-person zero-sum LQ stochastic differential game with random coefficients (SDG, for short) and generalize the results of Lü–Wang–Zhang into the stochastic differential game case. Under some regularity assumptions, we establish the equivalence between the existence of the robust optimal feedback control strategy operators and the solvability of the corresponding backward stochastic Riccati equations, which leads to the existence of the closed-loop saddle points. On the other hand, the problem is not closed-loop solvable if the solution of the corresponding backward stochastic Riccati equations does not have the needed regularity. Full article

High-dimensional covariance matrix estimation is one of the fundamental and important problems in multivariate analysis and has a wide range of applications in many fields. In practice, it is common that a covariance matrix is composed of a low-rank matrix and a sparse matrix. In this paper we estimate the covariance matrix by solving a constrained $L_q$-type regularized optimization problem. We establish the first-order optimality conditions for this problem by using proximal mapping and the subspace method. The proposed stationary point degenerates to the first-order stationary points of the unconstrained $L_q$ regularized sparse or low-rank optimization problems. A smoothing alternating updating method is proposed to find an estimator for the covariance matrix. We establish the convergence of the proposed calculation method. The numerical simulation results show the effectiveness of the proposed approach for high-dimensional covariance estimation. Full article

L_q (0 < q ≤ 1) regularization has been confirmed effective when applied to sparse SAR imaging. However, the inaccuracies caused by motion errors in the observation model will lead to various degradations and defocus in the reconstructed image. For high-resolution and light-small SAR systems, the range-variant motion errors will decrease the accuracy of range cell migration correction (RCMC), and residual range cell migration (RCM) will exceed multiple range resolution cells and degrade the image quality substantially. Aiming at this problem, in this paper, a novel azimuth-range decoupled sparse SAR imaging method with coarse-to-fine range-variant motion errors and residual RCM correction method is proposed. First, a one-step motion compensation (MOCO) operator is proposed using the inertial navigation systems (INS)/global positioning systems (GPS) information, which can significantly reduce the residual RCM and improve the reconstruction accuracy. Second, a fine high-order phase-error correction method is performed to correct the range and cross-range-varying phase errors using a joint imaging and phase-error estimation scheme, which will further improve the image focusing quality. Experimental results indicate the effectiveness of the proposed method. Full article

Three-dimensional (3D) synthetic aperture radar (SAR) imaging provides complete 3D spatial information, which has been used in environmental monitoring in recent years. Compared with matched filtering (MF) algorithms, the regularization technique can improve image quality. However, due to the substantial computational cost, the existing observation-matrix-based sparse imaging algorithm is difficult to apply to large-scene and 3D reconstructions. Therefore, in this paper, novel 3D sparse reconstruction algorithms with generalized $L_q$-regularization are proposed. First, we combine majorization–minimization (MM) and $L_1$ regularization (MM-$L_1$) to improve SAR image quality. Next, we combine MM and $L_{1/2}$ regularization (MM-$L_{1/2}$) to achieve high-quality 3D images. Then, we present the algorithm which combines MM and $L_0$ regularization (MM-$L_0$) to obtain 3D images. Finally, we present a generalized MM-$L_q$ algorithm (GMM-$L_q$) for sparse SAR imaging problems with arbitrary $q\left(0\le q\le 1\right)$ values. The proposed algorithm can improve the performance of 3D SAR images, compared with existing regularization techniques, and effectively reduce the amount of calculation needed. Additionally, the reconstructed complex image retains the phase information, which makes the reconstructed SAR image still suitable for interferometry applications. Simulation and experimental results verify the effectiveness of the algorithms. Full article

High spatial quality (HQ) optical remote sensing images are very useful for target detection, target recognition and image classification. Due to the influence of imaging equipment accuracy and atmospheric environment, HQ images are difficult to acquire, while low spatial quality (LQ) remote sensing images are very easy to acquire. Hence, denoising and super-resolution (SR) reconstruction technology are the most important solutions to improve the quality of remote sensing images very effectively, which can lower the cost as much as possible. Most existing methods usually only employ denoising or SR technology to obtain HQ images. However, due to the complex structure and the large noise of remote sensing images, the quality of the remote sensing image obtained only by denoising method or SR method cannot meet the actual needs. To address these problems, a method of reconstructing HQ remote sensing images based on Generative Adversarial Network (GAN) named “Restoration Generative Adversarial Network with ResNet and DenseNet” (RRDGAN) is proposed, which can acquire better quality images by incorporating denoising and SR into a unified framework. The generative network is implemented by fusing Residual Neural Network (ResNet) and Dense Convolutional Network (DenseNet) in order to consider denoising and SR problems at the same time. Then, total variation (TV) regularization is used to furthermore enhance the edge details, and the idea of Relativistic GAN is explored to make the whole network converge better. Our RRDGAN is implemented in wavelet transform (WT) domain, since different frequency parts could be handled separately in the wavelet domain. The experimental results on three different remote sensing datasets shows the feasibility of our proposed method in acquiring remote sensing images. Full article

We propose regularization methods for linear models based on the $L_q$-likelihood, which is a generalization of the log-likelihood using a power function. Regularization methods are popular for the estimation in the normal linear model. However, heavy-tailed errors are also important in statistics and machine learning. We assume q-normal distributions as the errors in linear models. A q-normal distribution is heavy-tailed, which is defined using a power function, not the exponential function. We find that the proposed methods for linear models with q-normal errors coincide with the ordinary regularization methods that are applied to the normal linear model. The proposed methods can be computed using existing packages because they are penalized least squares methods. We examine the proposed methods using numerical experiments, showing that the methods perform well, even when the error is heavy-tailed. The numerical experiments also illustrate that our methods work well in model selection and generalization, especially when the error is slightly heavy-tailed. Full article

Sparse signal processing has already been introduced to synthetic aperture radar (SAR), which shows potential in improving imaging performance based on raw data or a complex image. In this paper, the relationship between a raw data-based sparse SAR imaging method (RD-SIM) and a complex image-based sparse SAR imaging method (CI-SIM) is compared and analyzed in detail, which is important to select appropriate algorithms in different cases. It is found that they are equivalent when the raw data is fully sampled. Both of them can effectively suppress noise and sidelobes, and hence improve the image performance compared with a matched filtering (MF) method. In addition, the target-to-background ratio (TBR) or azimuth ambiguity-to-signal ratio (AASR) performance indicators of RD-SIM are superior to those of CI-SIM in down-sampling data-based imaging, nonuniform displace phase center sampling, and sparse SAR imaging model-based azimuth ambiguity suppression. Full article

High-speed remote transmission and large-capacity data storage are difficult issues in signals acquisition of rotating machines condition monitoring. To address these concerns, a novel multichannel signals reconstruction approach based on tunable Q-factor wavelet transform-morphological component analysis (TQWT-MCA) and sparse Bayesian iteration algorithm combined with step-impulse dictionary is proposed under the frame of compressed sensing (CS). To begin with, to prevent the periodical impulses loss and effectively separate periodical impulses from the external noise and additive interference components, the TQWT-MCA method is introduced to divide the raw vibration signal into low-resonance component (LRC, i.e., periodical impulses) and high-resonance component (HRC), thus, the periodical impulses are preserved effectively. Then, according to the amplitude range of generated LRC, the step-impulse dictionary atom is designed to match the physical structure of periodical impulses. Furthermore, the periodical impulses and HRC are reconstructed by the sparse Bayesian iteration combined with step-impulse dictionary, respectively, finally, the final reconstructed raw signals are obtained by adding the LRC and HRC, meanwhile, the fidelity of the final reconstructed signals is tested by the envelop spectrum and error analysis, respectively. In this work, the proposed algorithm is applied to simulated signal and engineering multichannel signals of a gearbox with multiple faults. Experimental results demonstrate that the proposed approach significantly improves the reconstructive accuracy compared with the state-of-the-art methods such as non-convex Lq (q = 0.5) regularization, spatiotemporal sparse Bayesian learning (SSBL) and L1-norm, etc. Additionally, the processing time, i.e., speed of storage and transmission has increased dramatically, more importantly, the fault characteristics of the gearbox with multiple faults are detected and saved, i.e., the bearing outer race fault frequency at 170.7 Hz and its harmonics at 341.3 Hz, ball fault frequency at 7.344 Hz and its harmonics at 15.0 Hz, and the gear fault frequency at 23.36 Hz and its harmonics at 47.42 Hz are identified in the envelope spectrum. Full article

The periodical transient impulses caused by localized faults are sensitive and important characteristic information for rotating machinery fault diagnosis. However, it is very difficult to accurately extract transient impulses at the incipient fault stage because the fault impulse features are rather weak and always corrupted by heavy background noise. In this paper, a new transient impulse extraction methodology is proposed based on impulse-step dictionary and re-weighted minimizing nonconvex penalty Lq regular (R-WMNPLq, q = 0.5) for the incipient fault diagnosis of rolling bearings. Prior to the sparse representation, the original vibration signal is preprocessed by the variational mode decomposition (VMD) technique. Due to the physical mechanism of periodic double impacts, including step-like and impulse-like impacts, an impulse-step impact dictionary atom could be designed to match the natural waveform structure of vibration signals. On the other hand, the traditional sparse reconstruction approaches such as orthogonal matching pursuit (OMP), L1-norm regularization treat all vibration signal values equally and thus ignore the fact that the vibration peak value may have more useful information about periodical transient impulses and should be preserved at a larger weight value. Therefore, penalty and smoothing parameters are introduced on the reconstructed model to guarantee the reasonable distribution consistence of peak vibration values. Lastly, the proposed technique is applied to accelerated lifetime testing of rolling bearings, where it achieves a more noticeable and higher diagnostic accuracy compared with OMP, L1-norm regularization and traditional spectral Kurtogram (SK) method. Full article

In vivo fluorescence molecular tomography (FMT) has been a popular functional imaging modality in research labs in the past two decades. One of the major difficulties of FMT lies in the ill-posed and ill-conditioned nature of the inverse problem in reconstructing the distribution of fluorophores inside objects. The popular regularization methods based on L2, L1 and total variation (TV ) norms have been applied in FMT reconstructions. The non-convex Lq(0 < q < 1) semi-norm and Log function have also been studied recently. In this paper, we adopt a uniform optimization transfer framework for these regularization methods in FMT and compare their individual, as well as the combined effects on both small, localized targets, such as tumors in the early stage, and large targets, such as liver. Numerical simulation studies and phantom experiments have been carried out, and we found that Lq with q near 1/2 performs the best in reconstructing small targets, while joint L2 and Log performs the best for large targets. Full article