Search Results (13)

This study presents an image inpainting model based on an energy functional that incorporates the $L^p$ norm of the fractional Laplacian operator as a regularization term and the $H^{-1}$ norm as a fidelity term. Using the properties of the fractional Laplacian operator, the $L^p$ norm is employed with an adjustable parameter p to enhance the operator’s ability to restore fine details in various types of images. The replacement of the conventional $L^2$ norm with the $H^{-1}$ norm enables better preservation of global structures in denoising and restoration tasks. This paper introduces a diffusion partial differential equation by adding an intermediate term and provides a theoretical proof of the existence and uniqueness of its solution in Sobolev spaces. Furthermore, it demonstrates that the solution converges to the minimizer of the energy functional as time approaches infinity. Numerical experiments that compare the proposed method with traditional and deep learning models validate its effectiveness in image inpainting tasks. Full article

In this study, we introduce a novel model, the Combined Model, composed of a conditional denoising diffusion model (SR3) and an enhanced residual network (EResNet), for reconstructing high-resolution turbulent flow fields from low-resolution flow data. The SR3 model is adept at learning the distribution of flow fields. The EResNet architecture incorporates a long skip connection extending from the input directly to the output. This modification ensures the preservation of essential features learned by the SR3, while simultaneously enhancing the accuracy of the flow field. Additionally, we incorporated physical gradient constraints into the loss function of EResNet to ensure that the flow fields reconstructed by the Combined Model are consistent with the direct numerical simulation (DNS) data. Consequently, the high-resolution flow fields reconstructed by the Combined Model exhibit high conformity with the DNS results in terms of flow distribution, details, and accuracy. To validate the effectiveness of the model, experiments were conducted on two-dimensional flow around a square cylinder at a Reynolds number (Re) of 100 and turbulent channel flow at Re = 4000. The results demonstrate that the Combined Model can reconstruct both high-resolution laminar and turbulent flow fields from low-resolution data. Comparisons with a super-resolution convolutional neural network (SRCNN) and an enhanced super-resolution generative adversarial network (ESRGAN) demonstrate that while all three models perform admirably in reconstructing laminar flows, the Combined Model excels in capturing more details in turbulent flows, aligning the statistical outcomes more closely with the DNS results. Furthermore, in terms of L2 norm error, the Combined Model achieves an order of magnitude lower error compared to SRCNN and ESRGAN. Experimentation also revealed that SR3 possesses the capability to learn the distribution of flow fields. This work opens new avenues for high-fidelity flow field reconstruction using deep learning methods. Full article

The matrix nuclear norm minimization problem has been extensively researched in recent years due to its widespread applications in control design, signal and image restoration, machine learning, big data problems, and more. One popular model is nuclear norm minimization with the $l_2$-norm fidelity term, but it is only effective for those problems with Gaussian noise. A nuclear norm minimization problem with the $l_1$-norm fidelity term has been studied in this paper, which can deal with the problems with not only non-Gaussian noise but also Gaussian noise or their mixture. Moreover, it also keeps the efficiency for the noiseless case. Given the nonsmooth proposed model, we transform it into a separated form by introducing an auxiliary variable and solve it by the semi-proximal alternating direction method of multipliers (sPADMM). Furthermore, we first attempt to solve its dual problem by sPADMM. Then, the convergence guarantees for the aforementioned algorithms are given. Finally, some numerical studies are dedicated to show the robustness of the proposed model and the effectiveness of the presented algorithms. Full article

In this paper, we present a novel image denoising algorithm, specifically designed to effectively restore both the edges and texture of images. This is achieved through the use of an innovative model known as the overlapping group sparse fractional-order total variation regularization model (OGS-FOTVR). The OGS-FOTVR model ingeniously combines the benefits of the fractional-order (FO) variation domain with an overlapping group sparsity measure, which acts as its regularization component. This is further enhanced by the inclusion of the well-established L2-norm, which serves as the fidelity term. To simplify the model, we employ the alternating direction method of multipliers (ADMM), which breaks down the model into a series of more manageable sub-problems. Each of these sub-problems can then be addressed individually. However, the sub-problem involving the overlapping group sparse FO regularization presents a high level of complexity. To address this, we construct an alternative function for this sub-problem, utilizing the mean inequality principle. Subsequently, we employ the majorize-minimization (MM) algorithm to solve it. Empirical results strongly support the effectiveness of the OGS-FOTVR model, demonstrating its ability to accurately recover texture and edge information in images. Notably, the model performs better than several advanced variational alternatives, as indicated by superior performance metrics across three image datasets, PSNR, and SSIM. Full article

Pansharpening refers to the fusion of a panchromatic (PAN) and a multispectral (MS) image aimed at generating a high-quality outcome over the same area. This particular image fusion problem has been widely studied, but until recently, it has been challenging to balance the spatial and spectral fidelity in fused images. The spectral distortion is widespread in the component substitution-based approaches due to the variation in the intensity distribution of spatial components. We lightened the idea using the total variation optimization to improve upon a novel GIHS-TV framework for pansharpening. The framework drew the high spatial fidelity from the GIHS scheme and implemented it with a simpler variational expression. An improved L1-TV constraint to the new spatial–spectral information was introduced to the GIHS-TV framework, along with its fast implementation. The objective function was solved by the Iteratively Reweighted Norm (IRN) method. The experimental results on the “PAirMax” dataset clearly indicated that GIHS-TV could effectively reduce the spectral distortion in the process of component substitution. Our method has achieved excellent results in visual effects and evaluation metrics. Full article

In this work, we investigate the application of Deep Learning in Portfolio selection in a Markowitz mean-variance framework. We refer to a $l_1$ regularized multi-period model; the choice of the $l_1$ norm aims at producing sparse solutions. A crucial issue is the choice of the regularization parameter, which must realize a trade-off between fidelity to data and regularization. We propose an algorithm based on neural networks for the automatic selection of the regularization parameter. Once the neural network training is completed, an estimate of the regularization parameter can be computed via forward propagation. Numerical experiments and comparisons performed on real data validate the approach. Full article

In X-ray tomography image reconstruction, one of the most successful approaches involves a statistical approach with $l_2$ norm for fidelity function and some regularization function with $l_p$ norm, $1<p<2$ . Among them stands out, both for its results and the computational performance, a technique that involves the alternating minimization of an objective function with $l_2$ norm for fidelity and a regularization term that uses discrete gradient transform (DGT) sparse transformation minimized by total variation (TV). This work proposes an improvement to the reconstruction process by adding a bilateral edge-preserving (BEP) regularization term to the objective function. BEP is a noise reduction method and has the purpose of adaptively eliminating noise in the initial phase of reconstruction. The addition of BEP improves optimization of the fidelity term and, as a consequence, improves the result of DGT minimization by total variation. For reconstructions with a limited number of projections (low-dose reconstruction), the proposed method can achieve higher peak signal-to-noise ratio (PSNR) and structural similarity index measurement (SSIM) results because it can better control the noise in the initial processing phase. Full article

We present a nonintrusive approach for combining high-fidelity simulations using Finite-Volume (FV) methods with Proper Orthogonal Decomposition (POD) and Galerkin Reduced-Order Modeling (ROM) methodology. By nonintrusive we here imply an approach that does not need specific knowledge about the high-fidelity Computational Fluid Dynamics (CFD) solver other than the velocity and pressure results given on an element mesh representing the related discrete interpolation spaces. The key step in the presented approach is the projection of the FV results onto suitable finite-element (FE) spaces and then use of classical POD Galerkin ROM framework. We do a numerical investigation of aerodynamic flow around an airfoil cross-section (NACA64) at low Reynolds number and compare the ROM results obtained with high-fidelity FV-generated snapshots against similar high-fidelity results obtained with FE using Taylor–Hood velocity and pressure spaces. Our results show that we achieve relative errors in the range of 1–10% in both $H_1$ -seminorm of the computed velocities and in the $L_2$ -norm of the computed pressure with reasonably few ROM modes. Similar accuracy was obtained for lift and drag. Full article

Images may be corrupted by salt and pepper impulse noise during image acquisitions or transmissions. Although promising denoising performances have been recently obtained with sparse representations, how to restore high-quality images remains challenging and open. In this work, image sparsity is enhanced with a fast multiclass dictionary learning, and then both the sparsity regularization and robust data fidelity are formulated as minimizations of L₀-L₀ norms for salt and pepper impulse noise removal. Additionally, a numerical algorithm of modified alternating direction minimization is derived to solve the proposed denoising model. Experimental results demonstrate that the proposed method outperforms the compared state-of-the-art ones on preserving image details and achieving higher objective evaluation criteria. Full article

It is a primary challenge in the fault diagnosis community of the gearbox to extract the weak fault features under heavy background noise and nonstationary conditions. For this purpose, a novel weak fault detection approach based on majorization–minimization (MM) and asymmetric convex penalty regularization (ACPR) is proposed in this paper. The proposed objective cost function (OCF) consisting of a signal-fidelity term, and two parameterized penalty terms (i.e., one is an asymmetric nonconvex penalty regularization term, and another is a symmetric nonconvex penalty regularization term).To begin with, the asymmetric and symmetric penalty functions are established on the basis of an L1-norm model, then, according to the splitting idea, the majorizer of the symmetric function and the majorizer of the asymmetric function are respectively calculated via the MM algorithm. Finally, the MM is re-introduced to solve the proposed OCF. As examples, the effectiveness and reliability of the proposed method is verified through simulated data and gearbox experimental real data. Meanwhile, a comparison with the state of-the-art methods is illustrated, including nonconvex penalty regularization (NCPR) and L1-norm fused lasso optimization (LFLO) techniques, the results indicate that the gear chipping characteristic frequency 13.22 Hz and its harmonic (2f, 3f, 4f and 5f) can be identified clearly, which highlights the superiority of the proposed approach. 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

Gaofen-4 is China’s first geosynchronous orbit high-definition optical imaging satellite with extremely high temporal resolution. The features of staring imaging and high temporal resolution enable the super-resolution of multiple images of the same scene. In this paper, we propose a super-resolution (SR) technique to reconstruct a higher-resolution image from multiple low-resolution (LR) satellite images. The method first performs image registration in both the spatial and range domains. Then the point spread function (PSF) of LR images is parameterized by a Gaussian function and estimated by a blind deconvolution algorithm based on the maximum a posteriori (MAP). Finally, the high-resolution (HR) image is reconstructed by a MAP-based SR algorithm. The MAP cost function includes a data fidelity term and a regularized term. The data fidelity term is in the L₂ norm, and the regularized term employs the Huber-Markov prior which can reduce the noise and artifacts while preserving the image edges. Experiments with real Gaofen-4 images show that the reconstructed images are sharper and contain more details than Google Earth ones. Full article

Single-image blind deblurring for imaging sensors in the Internet of Things (IoT) is a challenging ill-conditioned inverse problem, which requires regularization techniques to stabilize the image restoration process. The purpose is to recover the underlying blur kernel and latent sharp image from only one blurred image. Under many degraded imaging conditions, the blur kernel could be considered not only spatially sparse, but also piecewise smooth with the support of a continuous curve. By taking advantage of the hybrid sparse properties of the blur kernel, a hybrid regularization method is proposed in this paper to robustly and accurately estimate the blur kernel. The effectiveness of the proposed blur kernel estimation method is enhanced by incorporating both the $L_1$ -norm of kernel intensity and the squared $L_2$ -norm of the intensity derivative. Once the accurate estimation of the blur kernel is obtained, the original blind deblurring can be simplified to the direct deconvolution of blurred images. To guarantee robust non-blind deconvolution, a variational image restoration model is presented based on the $L_1$ -norm data-fidelity term and the total generalized variation (TGV) regularizer of second-order. All non-smooth optimization problems related to blur kernel estimation and non-blind deconvolution are effectively handled by using the alternating direction method of multipliers (ADMM)-based numerical methods. Comprehensive experiments on both synthetic and realistic datasets have been implemented to compare the proposed method with several state-of-the-art methods. The experimental comparisons have illustrated the satisfactory imaging performance of the proposed method in terms of quantitative and qualitative evaluations. Full article