svd-based deblurring

This work addresses the challenge of image restoration under spatially varying degradation, particularly non-uniform motion blur, by proposing a singular value decomposition (SVD)-based restoration framework. The method models the degradation process using position-dependent point spread functions and introduces a novel cumulative singular value energy ratio criterion to systematically determine the number of small singular values retained. This strategy effectively balances noise suppression and detail preservation. Experimental results demonstrate that the proposed approach successfully mitigates the ill-posedness of inverse problems across various motion blur models—including bidirectional linear, Gaussian, and harmonic motions—yielding significantly improved recovery of fine image details and reduced blur-related artifacts.

Real-world image restoration faces two fundamental challenges: inaccurate modeling of image priors and difficulty in precisely characterizing degradation processes—especially in practical scenarios where explicit degradation modeling is often constrained by restrictive parametric assumptions. To address this, we propose Noise Density-Guided Restoration (NDGR), a novel zero-shot, training-free, and degradation-agnostic paradigm. NDGR leverages a pre-trained diffusion model to perform deterministic mapping in the latent space, optimizing standard Gaussian noise density via gradient-guided inversion—steering input noise toward high-probability density regions without explicit degradation modeling. This is the first method to unify fully blind (unknown degradation type and parameters) and partially blind (known degradation type but unknown parameters) restoration under a single, degradation-independent framework. Extensive experiments demonstrate state-of-the-art performance across diverse real-world degradations—including blur, noise, and compression—with strong generalization capability, requiring neither fine-tuning nor dedicated degradation estimation modules.

This paper addresses the problem of estimating unknown probability density functions without prior reference or ground-truth knowledge. It systematically compares two maximum-likelihood-based deconvolution approaches: Richardson–Lucy (RL) deconvolution and a novel MISE-optimized data unfolding method. Crucially, the study innovatively adopts the mean integrated squared error (MISE) and the condition number of the response matrix as unified, internal quality metrics—eliminating reliance on external truth. Numerical experiments demonstrate that the MISE-optimized method consistently outperforms RL in reconstruction accuracy (lower MISE), numerical stability (smaller condition number), and robustness across diverse scenarios, indicating superior generalization capability. This work establishes a verifiable, self-consistent performance evaluation paradigm for density estimation in uncalibrated settings, advancing the theoretical and practical foundations of reference-free statistical inference.

This study addresses the numerical instability in extreme learning machine (ELM) training caused by ill-conditioned hidden-layer matrices during pseudoinverse computation. From a spectral perspective, it reveals how perturbations in output weights are amplified by the smallest singular value and quantifies instability via the condition number. The work establishes, for the first time, a systematic theoretical link between ELM numerical stability and the singular value structure of the hidden-layer matrix, proposing a spectral-based stability criterion. Leveraging singular value decomposition (SVD) and iterative hyperpower methods to compute the pseudoinverse, combined with random feature theory, it analyzes how network width influences the condition number. Experiments demonstrate that SVD is the most robust under ill-conditioned scenarios, whereas iterative methods exhibit greater sensitivity to spectral properties, confirming that stability is predominantly governed by the singular value spectrum.

This work investigates the impact of matrix concatenation operations on singular value spectra, aiming to bridge a theoretical gap in the structural stability of SVDs under concatenation. Addressing the central question—“How are the singular values of a concatenated matrix determined by those of its constituent submatrices?”—we extend Weyl’s inequality to block-wise concatenation for the first time, establishing a quantitative analytical framework grounded in matrix perturbation theory and norm inequalities. We derive computable upper bounds on singular value deviations and rigorously prove that dominant singular values remain stable when the operator norms of the submatrices are comparable. This result provides theoretical guarantees and principled design guidance for low-rank approximation, robust matrix clustering, and compression algorithms.

This work addresses the instability of reconstruction in ill-posed inverse problems caused by noise, as well as limitations of conventional regularization methods—such as reliance on manual hyperparameter tuning—and the lack of interpretability and cross-resolution generalization in deep learning approaches. To this end, we propose the Spectral Correction Network (SC-Net), which learns a signal-to-noise-ratio-adaptive, pointwise filtering function in the spectral domain of the forward operator to reweight spectral coefficients, yielding stable and interpretable solutions. Our method uniquely integrates interpretable adaptive spectral filtering with operator learning, and we theoretically prove that it can approximate continuous inverse operators, possesses discretization invariance, and achieves minimax optimal convergence rates. Experiments on 1D integral equations demonstrate a convergence rate of $O(\delta^{0.5})$, matching theoretical optimality; the learned filter outperforms Oracle Tikhonov regularization and generalizes zero-shot from $N=256$ to $N=2048$ with reconstruction error maintained at approximately 0.23.

This work addresses the significant challenges in ultra-high-definition (UHD) image restoration, which arise from high resolution, content complexity, and intricate structural details. The authors propose a progressive spectral decoupling paradigm that decomposes the restoration process into three sequential stages: zero-frequency enhancement, low-frequency recovery, and high-frequency refinement, integrated within a unified ERR framework. Key innovations include a novel stage-wise spectral processing mechanism, a frequency-windowed Kolmogorov–Arnold Network (FW-KAN), and LSUHDIR—the first large-scale, high-quality benchmark dataset for UHD image restoration. Extensive experiments demonstrate state-of-the-art performance across multiple UHD restoration tasks, while ablation studies confirm the effectiveness of each component, substantially advancing the field.

This work addresses the challenge of deploying large-scale pretrained models under high computational and memory costs. Existing low-rank compression methods, such as randomized SVD, often fail to preserve approximation accuracy when the singular value spectrum of weight matrices decays slowly, leading to degraded predictive performance. By analyzing the perturbation in softmax outputs induced by low-rank compression, this study establishes a theoretical connection between approximation error and classification performance. To mitigate this issue, the authors propose employing randomized subspace iteration with multiple power iterations to enhance spectral gap separation and improve approximation quality. The method consistently outperforms conventional randomized SVD across both convolutional and Transformer architectures, achieving near-optimal low-rank approximations under aggressive compression while maintaining higher prediction accuracy.

This work addresses the challenges of high-dimensional Bayesian inference and computational scalability in semi-blind image deconvolution, particularly for large-scale applications such as seismic imaging. The authors propose a Bayesian conjugate hierarchical model based on cyclic lattice embedding. By reformulating the Gibbs sampler in the Fourier domain and introducing a Hamiltonian Monte Carlo (HMC) strategy for blur kernel updates that analytically marginalizes over image variables, they present the first scalable Fourier-domain HMC sampling method applicable to general semi-blind deconvolution problems. Evaluated on a 300×50 seismic imaging task involving approximately 80,000 parameters, the proposed approach significantly outperforms conventional Gibbs samplers, achieving notable improvements in posterior exploration efficiency and mixing performance.