Noise remains a persistent challenge in medical image segmentation, with existing mathematical and computer science-based algorithms still leaving substantial room for improvement. This paper proposes an active contour model combining frequency domain information (ACFDI) to enhance segmentation accuracy and efficiency for noisy vessel images. The innovation lies in leveraging Fourier transform-derived frequency domain information: low-frequency region information serves as the initial contour, eliminating the need for traditional initial contour selection and addressing sensitivity issues, while high-frequency region information is incorporated as weight coefficients in the energy functional to accelerate functional evolution and improve segmentation precision. Comparative experiments with other level set and deep learning models on noisy vessel images validate the ACFDI model's superiority and stability.

In this paper, we analyze the local discontinuous Galerkin (LDG) method with generalized numerical fluxes to study the superconvergent properties of one-dimensional linearized KdV equations. Compared with traditional upwind and alternating fluxes, a slower error growth of the LDG solution using generalized numerical fluxes can be obtained for long time simulations. By establishing five energy identities and properties of correction functions with the appropriate numerical initial condition, we derive the supercloseness between the LDG solution and the interpolation function. The errors of the numerical fluxes as well as the cell averages achieve the (2k+ 1)th-order super-convergence. In addition, we prove that the superconvergent rates of the function and derivative values at the interior generalized Radau points are k + 2 and k + 1, respectively. An extension to mixed boundary conditions is given, for which we present the generalized skew-symmetry property and propose an appropriate conservation property for the numerical initial condition. Numerical experiments are shown to demonstrate the theoretical results, including cases with other boundary conditions and nonlinear KdV equations.

A Hessian-recovery-based C0 finite element framework is proposed for second-order elliptic equations in non-divergence form. The construction is based on a direct approximation of the strong non-divergence operator: the Hessian D2u is replaced by a recovered Hessian Hhuh, so that A : D2u is approximated by A : Hhuh. The resulting discretizations include a nodal formulation and a Galerkin-type formulation for general Lagrange finite element spaces, as well as a biorthogonal Petrov-Galerkin formulation for linear elements. The analysis focuses on the recovered nodal matrix and identifies two verifiable algebraic solvability mechanisms. The first is a globally monotone regime leading to a discrete maximum principle, and the second is a localized Schur-complement criterion for sign-violating rows. A uniform inverse bound and a condition-number estimate are derived in the globally monotone case. Residual consistency estimates are obtained from the Hessian recovery error. In the globally monotone regime, these estimates combine with the uniform inverse bound to give a nodal L-error estimate for the nodal formulation. Numerical experiments with nonsmooth and discontinuous coefficients support the predicted algebraic diagnostics and show the accuracy of the proposed recovered-residual discretizations. A Monge-Ampere type test further illustrates the use of the recovered Hessian in a Newton iteration for a fully nonlinear problem.

Multiple kernel clustering (MKC) enhances clustering performance by integrating diverse data views, yet existing methods often struggle with kernel redundancy and suboptimal optimization. The recently proposed method localized simple multiple kernel k-means (LSMKKM) achieves remarkable achievements by using a novel Min-Max optimization framework. Current methods improve on this by incorporating additional prior knowledge. However, its optimization strategy is incompatible with matrix regularization. By reformulating the Min-Max optimization framework, we introduce a tailored reduced gradient descent algorithm that partitions optimization variables and transforms constraints into a linear programming format, enabling seamless integration of matrix and vector regularizations. We first propose a novel LSMKKM with representative kernels regularization (LSMKKM-RKR) that incorporates representative kernel regularization to address kernel redundancy via subset selection. This approach maximizes kernel diversity while enhancing clustering accuracy. Furthermore, we improve LSMKKM-RKR by proposing LSMKKM with representative kernels regularization and matrix-induced regularization (LSMKKM-RKMR) where kernel correlation and dissimilarity are both integrated. Extensive experiments on several benchmark datasets, including Handwritten Numerals and Coil20, demonstrate that our algorithms significantly outperform state-of-the-art multiple kernel k-means methods.

Chebyshev spectral methods are fundamental in numerical analysis and scientific computing. However, existing studies primarily focus on asymptotic error estimates for polynomial approximations of certain singular functions and lack sharp pointwise error estimates for functions with interior or endpoint singularities in fractional spaces. In this work, we introduce a novel analytical framework to rigorously derive explicit and sharp pointwise error estimates for Chebyshev polynomial approximations of functions exhibiting interior or endpoint singularities within fractional spaces. Our analysis provides three key contributions: (i) new definitions of fractional space that yield more precise theoretical results, (ii) explicit and sharp pointwise error estimates for Chebyshev approximations of functions with interior or endpoint singularities, and (iii) an extension to Chebyshev spectral differentiation, resulting in explicit and improved upper error bounds. Numerical experiments are presented to support the theoretical results. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Plug-and-Play (PnP) image restoration provides a flexible framework that integrates model-based optimization with powerful deep denoising priors. However, PnP methods based on deep neural networks are highly sensitive to perturbations. In this work, we propose a robust PnP framework embedding a frequency-domain robust correction block and an adaptive noise-level scheduling block for adaptive image restoration. The frequency-domain block selectively suppresses corrupted high-frequency components to align adversarial noise with AWGN assumptions, serving as a general-purpose preprocessing strategy that can be applied beyond PnP. The adaptive noise-level scheduling block based on patch-wise PCA estimation dynamically adjusts the denoiser noise level throughout the iterative process. Experiments on deblurring and superresolution demonstrate that our method achieves stronger robustness and restoration performance than existing PnP methods under adversarial attacks. Ablation studies validate the complementary contributions of the two proposed blocks.

Legendre spectral differentiation is a basic tool in the numerical solution of differential equations. More precise information on spectral differentiation errors is important for deriving reliable error estimates for related numerical algorithms. However, existing studies primarily focus on asymptotic error estimates for polynomial approximations of specific singular functions and lack sharp pointwise error estimates for functions with interior or endpoint singularities in fractional spaces. In this work, we present explicit and sharp pointwise error estimates for Legendre spectral differentiation of functions with limited regularity in fractional spaces. We start by specifying a fractional-space setting suited to the pointwise error analysis in order to deal with the pointwise error estimates. We then derive explicit upper error bounds for Legendre spectral differentiation of functions with interior or endpoint singularities. Numerical experiments are provided to demonstrate the sharpness of our results.

Deep neural networks (DNNs) have shown superior performance comparing to traditional image denoising algorithms. However, DNNs are inevitably vulnerable while facing adversarial attacks. In this paper, we propose an adversarial attack method named denoising-PGD which can successfully attack all the current deep denoising models while keep the noise distribution almost unchanged. We surprisingly find that the current mainstream non-blind denoising models (DnCNN, FFDNet, ECNDNet, BRDNet), blind denoising models (DnCNN-B, Noise2Noise, RDDCNN-B, FAN), plug-and-play (DPIR, CurvPnP) and unfolding denoising models (DeamNet) almost share the same adversarial sample set on both grayscale and color images, respectively. Shared adversarial sample set indicates that all these models are similar in term of local behaviors at the neighborhood of all the test samples. Thus, we further propose an indicator to measure the local similarity of models, called robustness similitude. Non-blind denoising models are found to have high robustness similitude across each other, while hybrid-driven models are also found to have high robustness similitude with pure data-driven non-blind denoising models. According to our robustness assessment, data-driven non-blind denoising models are the most robust. We use adversarial training to complement the vulnerability to adversarial attacks. Moreover, the model-driven image denoising BM3D shows resistance on adversarial attacks.

In this article, a spatial concept of direct sum space based on the discontinuous reproducing kernel method (RKM for short) is proposed for impulsive differential equations, and a reproducing kernel numerical solution method is constructed. Based on the piecewise smoothness of the solution, a discontinuous reproducing kernel is constructed, and a direct sum space is constructed in vector form to represent the structure of the equation system. Furthermore, the simplified RKM is used to solve the operator equation, and an approximate solution in series form is obtained. Finally, the regularity analysis and uniform convergence analysis are carried out, and numerical experiments verify the second-order convergence and stability of the algorithm.

Fast Magnetic Resonance Imaging (MRI) reconstruction is a fundamental task in medical imaging, where balancing computational efficiency and high-fidelity image restoration remains a critical challenge. Recent advances in State Space Models (SSMs), particularly Mamba, have demonstrated strong capabilities in modeling long-range dependencies while maintaining efficiency. However, directly applying Mamba to MRI reconstruction poses limitations, including local pixel forgetting and an inability to capture fine-grained structural details. To address these challenges, we propose Recurrent Mamba (RCM), a novel framework that integrates state-space modeling with recurrent structures for efficient and high-fidelity MRI reconstruction. At its core, RCM introduces the Recurrent Mamba Block (RCMB), a cyclic state-space module that employs a spatially aware 2D selective state-space mechanism (2D_SSM) to mitigate pixel forgetting and enhance local feature representation. Additionally, an Adaptive Convolution Block (ACB) is incorporated to optimize feature extraction while maintaining computational efficiency. Expanding on RCMB, we develop the Unfolded Recurrent Mamba Module (URCM), which hierarchically refines multi-scale features through iterative learning with Recurrent Learning Units (RLUs) and a Refinement Module (RM). Experimental results show that RCM achieves considerable performance improvements on four MRI datasets with different sequences, while maintaining excellent parameter efficiency, with only 1.03 M trainable parameters.

In this paper, we present pointwise error bounds for Legendre approximations of singular functions in fractional spaces. We begin by recalling the relevant fractional space definitions needed for the pointwise estimates. We then derive explicit and pointwise error bounds for Legendre approximations of functions with interior or endpoint singularities. New analytical techniques are developed to handle the case of endpoint singularities, and all coefficients in our estimates are given by explicit and computable formulas, allowing for a precise quantification of the singularity’s influence and clearly demonstrating the sharpness of the results. Numerical experiments are presented to support the theoretical results.

In this paper, we develop a novel bound-preserving flux limiting scheme within the discontinuous Galerkin framework for one-dimensional scalar nonlinear conservation laws and convection-diffusion equations. The proposed scheme integrates the global monolithic convex (GMC) limiter with the generalized local Lax–Friedrichs flux, which effectively eliminates maximum principle violations around shocks. Compared with the classical flux-corrected transport technique, the GMC limiter introduces fewer constraints and enjoys natural compatibility with spatial semi-discretizations. In addition, the generalized local Lax–Friedrichs flux contains two tunable parameters, which facilitate accurate shock capturing and high-resolution simulation of smooth solutions. The established framework is further extended to handle one-dimensional convection-diffusion problems. In addition, we couple the GMC strategy with several typical high-order Runge–Kutta methods and analyze the bound-preserving property of the corresponding fully discrete scheme. A series of numerical examples involving one- and two-dimensional problems are presented. Numerical results demonstrate that the developed scheme preserves physical bounds near shocks while maintaining high-order accuracy in smooth regions.

The removal of multiplicative Gamma noise is a critical research area in the application of synthetic aperture radar (SAR) imaging, where neural networks serve as a potent tool. However, real-world data often diverges from theoretical models, exhibiting various disturbances, which makes the neural network less effective. Adversarial attacks can be used as a criterion for judging the adaptability of neural networks to real data, since they can find the most extreme perturbations that make neural networks ineffective. In this work, we propose a tunable, regularized neural network framework that unrolls a shallow neural denoising block and a diffusion regularization block into a single network for end-to-end training. The linear heat equation, known for its inherent smoothness and low-pass filtering properties, is adopted as the diffusion regularization block. The smoothness of our outputs is controlled by a single time step hyperparameter that can be adjusted dynamically. The stability and convergence of our model are theoretically proven. Experimental results demonstrate that the proposed model effectively eliminates high-frequency oscillations induced by adversarial attacks. Finally, the proposed model is benchmarked against several state-of-the-art denoising methods on simulated images, adversarial samples, and real SAR images, achieving superior performance in both quantitative and visual evaluations.

Blind image deblurring remains challenging in computational imaging due to the unknown blur kernel, often relying on complex priors or heuristic edge selection. This study presents a novel gradient sparsity framework guided by curvature for robust blind image deblurring. By extracting curvature information from image gradients, we design an efficient L1 regularization term to enhance edge retention and image sharpness while minimizing computational overhead. A spatially adaptive edge-weighting function is introduced to dynamically adjust regularization intensity according to local image characteristics, ensuring robust performance across various regions. The optimization problem is decomposed into two convex sub-problems, which are efficiently solved in closed form via the half-quadratic splitting algorithm. Comprehensive experiments on benchmark datasets demonstrate that our approach outperforms cutting-edge methods in both peak signal-to-noise ratio and structural similarity, producing sharper images with reduced artifacts. This framework provides a computationally efficient and robust solution for blind deblurring, especially in resource-constrained environments.

3D Gaussian Splatting (3DGS) has revolutionized novel view synthesis with real-time rendering and high-quality reconstruction. However, its performance degrades under varying scales, causing aliasing and dilation artifacts. To address these challenges, we introduce Msa-Splatting, an enhanced adaptive Gaussian splatting method designed for high-fidelity multi-scale novel view synthesis. Our approach reimagines the Gaussian adaptive control process with a novel multi-fold splitting and cloning mechanism, which optimizes Gaussian properties and reduces spurious artifacts by halving the densification frequency. We further mitigate aliasing in scaled-down renderings with a weighted alpha blending sampling technique, ensuring accurate representation of high-frequency details. To eliminate scale-mismatch artifacts, we introduce 2D variable dilation Gaussians, which dynamically adjust dilation based on the rendering scale. Msa-Splatting significantly enhances the fidelity and adaptability of Gaussian-based rendering, achieving state-of-the-art performance in multi-scale novel view synthesis. Extensive quantitative and qualitative evaluations demonstrate the efficacy of our approach, showcasing its ability to produce perceptually superior renderings with reduced artifacts across varying scales, thus establishing a robust framework for future advancements in scalable rendering technologies.

This paper deals with a novel nonlinear coupled nonlocal reaction-diffusion system proposed for image restoration, characterized by the advantages of preserving low gray level features and textures.The gray level indicator in the proposed model is regularized using a new method based on porous media type equations, which is suitable for recovering noisy blurred images. The well-posedness, regularity, and other properties of the model are investigated, addressing the lack of theoretical analysis in those existing similar types of models. Numerical experiments conducted on texture and satellite images demonstrate the effectiveness of the proposed model in denoising and deblurring tasks.

Existing color image denoising methods often fail to adequately capture correlations among RGB channels, leading to structural blurring and the loss of fine details. To overcome this limitation, we propose QMSANet, a Quaternion Multi-Scale Attention Network designed to explicitly model inter-channel correlations (i.e., correlations among RGB channels) throughout the denoising process, thereby enabling stronger noise suppression and more faithful detail reconstruction. Our network is built around three complementary modules: the Quaternion Multi-Scale Sparse Block (QMSB), the Quaternion Stacked Enhancement Block (QSEB), and the Lightweight Quaternion Attention Block (LQAB). These modules form a cohesive processing pipeline. Specifically, the QMSB first extracts sparse multi-scale features, allowing the model to capture contextual information at different granularities. These features are then refined by the QSEB, which enhances deep inter-channel interactions and stabilizes feature propagation to improve representational quality. Finally, the LQAB adapts the refined features through a lightweight attention strategy that selectively highlights the most informative responses with minimal computational overhead. Together, these modules operate sequentially to address key denoising challenges, improving efficiency while reducing incomplete noise removal, detail loss, and edge artifacts. Extensive experiments on standard color image denoising benchmarks show that QMSANet consistently outperforms state-of-the-art models under both synthetic and real-world noise. Moreover, although blind denoisers typically underperform their non-blind counterparts, our blind variant (i.e., QMSANet-B) still surpasses most representative methods.

In this paper, we investigate the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear Korteweg-de Vries type equations.The numerical flux for the nonlinear convection term is chosen as the generalized Lax-Friedrichs flux, and the generalized alternating flux and upwind-biased flux are used for the dispersion term.The generalized Lax-Friedrichs flux with anti-dissipation property will compensate the numerical dissipation of the dispersion term, resulting in a nearly energy conservative scheme that is useful in resolving waves and is beneficial for long time simulations.To deal with the nonlinearity and different numerical flux weights, a suitable numerical initial condition is constructed, for which a modified global projection is designed.By establishing relationships between the prime variable and auxiliary variables in combination with sharp bounds for jump terms, optimal error estimates are obtained.Numerical experiments are shown to confirm the validity of theoretical results.

This study investigates Dirichlet boundary condition related to a class of nonlinear parabolic problem with nonnegative L^1-data, which has a variable-order fractional p-Laplacian operator. The existence and uniqueness of renormalized solutions and entropy solutions to the equation is proved. To address the significant challenges encountered during this process, we use approximation and energy methods. In the process of proving, the well-posedness of weak solutions to the problem has been established initially, while also establishing a comparative result of solutions.

In this paper, we propose a novel variational model for color-texture image segmentation by embedding the molecular beam epitaxy (MBE) equation into a multi-cue segmentation (MCS) framework. The MBE equation incorporates a fourth-order diffusion term to smooth high-frequency noise while preserving curvature variations, along with a non-equilibrium term to ensure mass conservation and suppress oscillations, thereby eliminating the need for frequent re-initialization. Inspired by the physical principles of crystal film growth, this approach regulates the level set evolution by controlling thin-film growth dynamics, improving both stability and accuracy. We derive the gradient flow equation of the proposed model and prove the existence of a weak solution using the Galerkin approximation method. To solve the model efficiently, we design an implicit-explicit (IMEX) scheme, and employ an additive operator splitting (AOS) method to obtain the diffusion tensor. Extensive experiments demonstrate that the MBE-MCS model achieves more stable level set evolutions, better preserves fine structural details, and delivers superior segmentation accuracy, even for images with noise, sharp corners, and complex backgrounds.