Half-quadratic Splitting Research Articles

ISRSL0: compressed sensing MRI with image smoothness regularized-smoothed \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\:{\\ell}_{0}$$\\end{document} norm

The reconstruction of MR images has always been a challenging inverse problem in medical imaging. Acceleration of MR scanning is of great importance for clinical research and cutting-edge applications. One of the primary efforts to achieve this is using compressed sensing (CS) theory. The CS aims to reconstruct MR images using a small number of sampled data in k-space. The CS-MRI techniques face challenges, including the potential loss of fine structure and increased computational complexity. We introduce a novel framework based on a regularized sparse recovery problem and a sharpening step to improve the CS-MRI approaches regarding fine structure loss under high acceleration factors. This problem is solved via the Half Quadratic Splitting (HQS) approach. The inverse problem for reconstructing MR images is converted into two distinct sub-problems, each of which can be solved separately. One key feature of the proposed approach is the replacement of one sub-problem with a denoiser. This regularization assists the optimization of the Smoothed \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\:{\\ell}_{0}$$\\end{document} (SL0) norm in escaping local minimums and enhances its precision. The proposed method consists of smoothing, feature modification, and Smoothed \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\:{\\ell}_{0}$$\\end{document} cost function optimization. The proposed approach improves the SL0 algorithm for MRI reconstruction without complicating it. The convergence of the proposed approach is illustrated analytically. The experimental results show an acceptable performance of the proposed method compared to the network-based approaches.

Blind Deblurring Method for CASEarth Multispectral Images Based on Inter-Band Gradient Similarity Prior.

Multispectral remote sensing images contain abundant information about the distribution and reflectance of ground objects, playing a crucial role in target detection, environmental monitoring, and resource exploration. However, due to the complexity of the imaging process in multispectral remote sensing, image blur is inevitable, and the blur kernel is typically unknown. In recent years, many researchers have focused on blind image deblurring, but most of these methods are based on single-band images. When applied to CASEarth satellite multispectral images, the spectral correlation is unutilized. To address this limitation, this paper proposes a novel approach that leverages the characteristics of multispectral data more effectively. We introduce an inter-band gradient similarity prior and incorporate it into the patch-wise minimal pixel (PMP)-based deblurring model. This approach aims to utilize the spectral correlation across bands to improve deblurring performance. A solution algorithm is established by combining the half-quadratic splitting method with alternating minimization. Subjectively, the final experiments on CASEarth multispectral images demonstrate that the proposed method offers good visual effects while enhancing edge sharpness. Objectively, our method leads to an average improvement in point sharpness by a factor of 1.6, an increase in edge strength level by a factor of 1.17, and an enhancement in RMS contrast by a factor of 1.11.

Blind image deblurring with a difference of the mixed anisotropic and mixed isotropic total variation regularization

This paper proposes a simple model for image deblurring with a new total variation regularization. Classically, the L1-21 regularizer represents a difference of anisotropic (i.e. L1) and isotropic (i.e. L21) total variation, so we define a new regularization as Le-2e, which is the weighted difference of the mixed anisotropic (i.e. L0 + L1 = Le) and mixed isotropic (i.e. L0 + L21 = L2e), and it is characterized by sparsity-promotingand robustness in image deblurring. Then, we merge the L0-gradient into the model for edge-preserving and detail-removing. The union of the Le-2e regularization and L0-gradient improves the performance of image deblurring and yields high-quality blur kernel estimates. Finally, we design a new solution format that alternately iterates the difference of convex algorithm, the split Bregman method, and the approach of half-quadratic splitting to optimize the proposed model. Experimental results on quantitative datasets and real-world images show that the proposed method can obtain results comparable to state-of-the-art works.

An unrolled neural network for accelerated dynamic MRI based on second-order half-quadratic splitting model

The reconstruction of dynamic magnetic resonance images from incomplete k-space data has sparked significant research interest due to its potential to reduce scan time. However, traditional iterative optimization algorithms fail to faithfully reconstruct images at higher acceleration factors and incur long reconstruction time. Furthermore, end-to-end deep learning-based reconstruction algorithms suffer from large model parameters and lack robustness in the reconstruction results. Recently, unrolled deep learning models, have shown immense potential in algorithm stability and applicability flexibility. In this paper, we propose an unrolled deep learning network based on a second-order Half-Quadratic Splitting(HQS) algorithm, where the forward propagation process of this framework strictly follows the computational flow of the HQS algorithm. In particular, we propose a degradation-sense module by associating random sampling patterns with intermediate variables to guide the iterative process. We introduce the Information Fusion Transformer(IFT) to extract both local and non-local prior information from image sequences, thereby removing aliasing artifacts resulting from random undersampling. Finally, we impose low-rank constraints within the HQS algorithm to further enhance the reconstruction results. The experiments demonstrate that each component module of our proposed model contributes to the improvement of the reconstruction task. Our proposed method achieves comparably satisfying performance to the state-of-the-art methods and it exhibits excellent generalization capabilities across different sampling masks. At the low acceleration factor, there is a 0.7% enhancement in the PSNR. Furthermore, when the acceleration factor reached 8 and 12, the PSNR achieves an improvement of 3.4% and 5.8% respectively.

Blind deblurring text images via Beltrami regularization

This article proposes a blind image deblurring model based on Beltrami regularization. The existence and uniqueness of the Beltrami model are proved, and we perform a theoretical analysis of the convergence and error estimation of the algorithm for solving the Beltrami regularization model. Meanwhile, we apply a half-quadratic splitting algorithm to solve it and obtain the blur kernel of the blurred image. Furthermore, we get a clean image through some non-blind deblurring algorithms. Finally, we compare this with the state-of-art methods, and our algorithm has specific performance improvements and advantages in the text images.© 2024 Elsevier Ltd. All rights reserved.

Robust Blind Text Image Deblurring via Maximum Consensus Framework

The blind text image deblurring problem presents a formidable challenge, requiring the recovery of a clean and sharp text image from a blurry version with an unknown blur kernel. Sparsity-based strategies have demonstrated their efficacy by emphasizing the sparse priors of the latent image and kernel. However, these existing strategies have largely neglected the influence of additional noise, imposing limitations on their performance. To overcome this limitation, we propose a novel framework designed to effectively mitigate the impact of extensive noise prevalent in blurred images. Our approach centers around a robust Maximum Consensus Framework, wherein we optimize the quantity of interest from the noisy blurry image based on the maximum consensus criterion. Furthermore, we propose the integration of the Alternating Direction Method of Multipliers (ADMM) and the Half-Quadratic Splitting (HQS) method to address the computationally intractable L0 norm problem. This innovative strategy enables improvements in the deblurring performance of blurry text images with the additional synthetic noise. Experimental evaluations conducted on various noisy blurry text images demonstrate the superiority of the proposed approach over existing methods.

Deep Unfolded Network with Intrinsic Supervision for Pan-Sharpening

Existing deep pan-sharpening methods lack the learning of complementary information between PAN and MS modalities in the intermediate layers, and exhibit low interpretability due to their black-box designs. To this end, an interpretable deep unfolded network with intrinsic supervision for pan-sharpening is proposed. Building upon the observation degradation process, it formulates the pan-sharpening task as a variational model minimization with spatial consistency prior and spectral projection prior. The former prior requires a joint component decomposition of PAN and MS images to extract intrinsic features. By being supervised in the intermediate layers, it can selectively provide high-frequency information for spatial enhancement. The latter prior constrains the intensity correlation between MS and PAN images derived from physical observations, so as to improve spectral fidelity. To further enhance the transparency of network design, we develop an iterative solution algorithm following the half-quadratic splitting to unfold the deep model. It rigorously adheres to the variational model, significantly enhancing the interpretability behind network design and efficiently alternating the optimization of the network. Extensive experiments demonstrate the advantages of our method compared to state-of-the-arts, showcasing its remarkable generalization capability to real-world scenes. Our code is publicly available at https://github.com/Baixuzx7/DISPNet.

Hierarchical Similarity Learning for Aliasing Suppression Image Super-Resolution.

As a highly ill-posed issue, single-image super-resolution (SISR) has been widely investigated in recent years. The main task of SISR is to recover the information loss caused by the degradation procedure. According to the Nyquist sampling theory, the degradation leads to the aliasing effect and makes it hard to restore the correct textures from low-resolution (LR) images. In practice, there are correlations and self-similarities among the adjacent patches in the natural images. This article considers the self-similarity and proposes a hierarchical image super-resolution network (HSRNet) to suppress the influence of aliasing. We consider the SISR issue in the optimization perspective and propose an iterative solution pattern based on the half-quadratic splitting (HQS) method. To explore the texture with local image prior, we design a hierarchical exploration block (HEB) and progressive increase the receptive field. Furthermore, multilevel spatial attention (MSA) is devised to obtain the relations of adjacent feature and enhance the high-frequency information, which acts as a crucial role for visual experience. The experimental result shows that HSRNet achieves better quantitative and visual performance than other works and remits the aliasing more effectively.

Super-resolution image reconstruction of a neutron thick pinhole imaging system using image denoiser prior based on half quadratic splitting method

Thick pinhole imaging system is a powerful tool to diagnose intense pulsed radiation sources. However, high spatial resolution of a thick pinhole imaging system cannot be achieved due to the limitation of field of view (FOV) and detection efficiency. In this paper, we propose a novel super-resolution image reconstruction algorithm to improve the spatial resolution of a practical thick pinhole imaging system designed for 14.1 MeV neutrons with 100 mm FOV. Point spread function (PSF) of the imaging system is obtained through complete simulation methods, indicating that the spatial resolution of the imaging system is about 500 μ m. A 4 × super-resolution image reconstruction algorithm using image denoiser as implicit image prior, named image denoiser prior, based on half quadratic splitting (HQS) method is firstly applied to restore more details of neutron sources. The results show that the proposed algorithm performs better than the Richardson-Lucy (RL) algorithm with fewer artifacts and sharper boundaries in reconstructed images. Notably, at a low noise level, the proposed algorithm can surpass the Nyquist sampling limit of 200 μ m, which is determined by the equivalent camera pixel size, to achieve a spatial resolution of 150 μ m.

Combining Low-Rank and Deep Plug-and-Play Priors for Snapshot Compressive Imaging.

Snapshot compressive imaging (SCI) is a promising technique that captures a 3-D hyperspectral image (HSI) by a 2-D detector in a compressed manner. The ill-posed inverse process of reconstructing the HSI from their corresponding 2-D measurements is challenging. However, current approaches either neglect the underlying characteristics, such as high spectral correlation, or demand abundant training datasets, resulting in an inadequate balance among performance, generalizability, and interpretability. To address these challenges, in this article, we propose a novel approach called LR2DP that integrates the model-driven low-rank prior and data-driven deep priors for SCI reconstruction. This approach not only captures the spectral correlation and deep spatial features of HSI but also takes advantage of both model-based and learning-based methods without requiring any extra training datasets. Specifically, to preserve the strong spectral correlation of the HSI effectively, we propose that the HSI lies in a low-rank subspace, thereby transforming the problem of reconstructing the HSI into estimating the spectral basis and spatial representation coefficient. Inspired by the mutual promotion of unsupervised deep image prior (DIP) and trained deep denoising prior (DDP), we integrate the unsupervised network and pre-trained deep denoiser into the plug-and-play (PnP) regime to estimate the representation coefficient together, aiming to explore the internal target image prior (learned by DIP) and the external training image prior (depicted by pre-trained DDP) of the HSI. An effective half-quadratic splitting (HQS) technique is employed to optimize the proposed HSI reconstruction model. Extensive experiments on both simulated and real datasets demonstrate the superiority of the proposed method over the state-of-the-art approaches.

Plug-and-Play Split Gibbs Sampler: embedding deep generative priors in Bayesian inference.

This paper introduces a stochastic plug-and-play (PnP) sampling algorithm that leverages variable splitting to efficiently sample from a posterior distribution. The algorithm based on split Gibbs sampling (SGS) draws inspiration from the half quadratic splitting method (HQS) and the alternating direction method of multipliers (ADMM). It divides the challenging task of posterior sampling into two simpler sampling problems. The first problem depends on the likelihood function, while the second is interpreted as a Bayesian denoising problem that can be readily carried out by a deep generative model. Specifically, for an illustrative purpose, the proposed method is implemented in this paper using state-of-the-art diffusion-based generative models. Akin to its deterministic PnP-based counterparts, the proposed method exhibits the great advantage of not requiring an explicit choice of the prior distribution, which is rather encoded into a pre-trained generative model. However, unlike optimization methods (e.g., PnP-ADMM and PnP-HQS) which generally provide only point estimates, the proposed approach allows conventional Bayesian estimators to be accompanied by confidence intervals at a reasonable additional computational cost. Experiments on commonly studied image processing problems illustrate the efficiency of the proposed sampling strategy. Its performance is compared to recent state-of-the-art optimization and sampling methods.

Boosting with fine-tuning for deep image denoising

While deep learning is ruling the image denoising domain in recent years, earlier works primarily focused on design of network architecture or training strategy. In this paper, we raise two questions: how to combine the advantages of traditional iterative methods and deep learning-based approaches, and how to avoid performance degradation caused by inaccurate modeling and estimation of noise. To answer the questions, we integrate recursive strategies and fine-tuning schemes to boost existing deep denoisers in a plug-and-play fashion. Specifically, based on the framework of plug-and-play priors, the image denoising problem is solved with the half quadratic splitting (HQS) algorithm to achieve iterative denoising. Different from the standard solving process, we develop a joint optimization scheme with regard to image restoration and network fine-tuning, realizing the matching between network and noise, thereby enabling better adaptation to the images contaminated by complex non-Gaussian noise. As such, two types of adaptive denoising boosters with convergence guarantee based on the fixed-point strategy and steepest-descent method are obtained. It is demonstrated in the experiments that the proposed schemes provide promising performance on additive white Gaussian noise (AWGN) and real-noise denoising for both supervised and self-supervised deep learning-based image denoising algorithms.

An Interpretable Unsupervised Unrolling Network for Hyperspectral Pansharpening.

Existing deep convolutional neural networks (CNNs) have recently achieved great success in pansharpening. However, most deep CNN-based pansharpening models are based on "black-box" architecture and require supervision, making these methods rely heavily on the ground-truth data and lose their interpretability for specific problems during network training. This study proposes a novel interpretable unsupervised end-to-end pansharpening network, called as IU2PNet, which explicitly encodes the well-studied pansharpening observation model into an unsupervised unrolling iterative adversarial network. Specifically, we first design a pansharpening model, whose iterative process can be computed by the half-quadratic splitting algorithm. Then, the iterative steps are unfolded into a deep interpretable iterative generative dual adversarial network (iGDANet). Generator in iGDANet is interwoven by multiple deep feature pyramid denoising modules and deep interpretable convolutional reconstruction modules. In each iteration, the generator establishes an adversarial game with the spatial and spectral discriminators to update both spectral and spatial information without ground-truth images. Extensive experiments show that, compared with the state-of-the-art methods, our proposed IU2PNet exhibits very competitive performance in terms of quantitative evaluation metrics and qualitative visual effects.

Why projection over convex sets works and how to make it better

Projection over convex sets (POCS) is one of the most widely used algorithms to interpolate seismic data sets. A formal understanding of the underlying objective function and the associated optimization process is, however, lacking to date in the literature. Here, POCS is shown to be equivalent to the application of the half-quadratic splitting (HQS) method to the [Formula: see text] norm of an orthonormal projection of the sought after data, constrained on the available traces. Similarly, the apparently heuristic strategy of using a decaying threshold in POCS is revealed to be the result of the continuation strategy that HQS must use to converge to a solution of the minimizer. In light of this theoretical understanding, another methods able to solve this convex optimization problem, namely the Chambolle-Pock primal-dual algorithm, is shown to lead to a new POCS-like method with superior interpolation capabilities at nearly the same computational cost of the industry-standard POCS method.

MGDUN: An interpretable network for multi-contrast MRI image super-resolution reconstruction

Magnetic resonance imaging (MRI) Super-Resolution (SR) aims to obtain high resolution (HR) images with more detailed information for precise diagnosis and quantitative image analysis. Deep unfolding networks outperform general MRI SR reconstruction methods by providing better performance and improved interpretability, which enhance the trustworthiness required in clinical practice. Additionally, current SR reconstruction techniques often rely on a single contrast or a simple multi-contrast fusion mechanism, ignoring the complex relationships between different contrasts. To address these issues, in this paper, we propose a Model-Guided multi-contrast interpretable Deep Unfolding Network (MGDUN) for medical image SR reconstruction, which explicitly incorporates the well-studied multi-contrast MRI observation model into an unfolding iterative network. Specifically, we manually design an objective function for MGDUN that can be iteratively computed by the half-quadratic splitting algorithm. The iterative MGDUN algorithm is unfolded into a special model-guided deep unfolding network that explicitly takes into account both the multi-contrast relationship matrix and the MRI observation matrix during the end-to-end optimization process. Extensive experimental results on the multi-contrast IXI dataset and the BraTs 2019 dataset demonstrate the superiority of our proposed model, with PSNR reaching 37.3366 and 35.9690 respectively. Our proposed MGDUN provides a promising solution for multi-contrast MR image super-resolution reconstruction. Code is available at https://github.com/yggame/MGDUN.

Flexible and High-Gain DOFS Deconvolution Based on Data-Driven Denoising Prior

Spatial resolution is an essential parameter for distributed optical fiber sensing (DOFS). Deconvolution is a promising solution to improve spatial resolution because of its good universality and ability to restore infinitely high spatial resolution in theory. However, conventional iterative deconvolution algorithms require manually designed priors, which makes it difficult to restore the signal accurately. Although end-to-end methods based on artificial neural networks provide more accurate results based on data-driven priors, a neural network can only be applied to data with similar characteristics to its training data. For systems with different parameters, the neural network needs to be retrained or fine-tuned, which is time and computational resource consuming. Here, we employ a new deconvolution method for DOFS based on half-quadratic splitting and denoising neural networks that exploits the advantages of both iterative and machine learning methods. Besides, we propose a method to calculate the deconvolution gain to evaluate the deconvolution performance quantitatively. In an experiment with 10 deconvolution ratios, the deconvolution gain of the employed method is 15.8 dB, while that of a conventional iterative method and an end-to-end machine learning method are 7.9 dB and 15 dB, respectively. Moreover, this new deconvolution method can be adapted to data with arbitrary system parameters, making it more flexible than end-to-end neural networks in practical applications.

Optimization-inspired Cumulative Transmission Network for image compressive sensing

Compressive Sensing (CS) techniques enable accurate signal reconstruction with few measurements. Deep Unfolding Networks (DUNs) have recently been shown to increase the efficiency of CS by emulating iterative CS optimization procedures by neural networks. However, most of these DUNs suffer from redundant update procedures or complex matrix operations, which can impair their reconstruction performances. Here we propose the optimization-inspired Cumulative Transmission Network (CT-Net), a DUN approach for natural image CS. We formulate an optimization procedure introducing an auxiliary variable similar to Half Quadratic Splitting (HQS). Unfolding this procedure defines the basic structure of our neural architecture, which is then further refined. A CT-Net is composed of Reconstruction Fidelity Modules (RFMs) for minimizing the reconstruction error and Constraint Gradient Approximation (CGA) modules for approximating (the gradient of) sparsity constraints instead of relying on an analytic solutions such as soft-thresholding. Furthermore, a lightweight Cumulative Transmission (CT) between CGAs in each reconstruction stage is proposed to facilitate a better feature representation. Experiments on several widely used natural image benchmarks illustrate the effectiveness of CT-Net with significant performance improvements and fewer network parameters compared to existing state-of-the-art methods. The experiments also demonstrate the scene and noise robustness of the proposed method.

Efficient FMT reconstruction based on L1-αL2 regularization via half-quadratic splitting and a two-probe separation light source strategy.

Fluorescence molecular tomography (FMT) can achieve noninvasive, high-contrast, high-sensitivity three-dimensional imaging in vivo by relying on a variety of fluorescent molecular probes, and has excellent clinical transformation prospects in the detection of tumors in vivo. However, the limited surface fluorescence makes the FMT reconstruction have some ill-posedness, and it is difficult to obtain the ideal reconstruction effect. In this paper, two different emission fluorescent probes and L 1-L 2 regularization are combined to improve the temporal and spatial resolution of FMT visual reconstruction by introducing the weighting factor α and a half-quadratic splitting alternating optimization (HQSAO) iterative algorithm. By introducing an auxiliary variable, the HQSAO method breaks the sparse FMT reconstruction task into two subproblems that can be solved in turn: simple reconstruction and image denoising. The weight factor α (α>1) can increase the weight of nonconvex terms to further promote the sparsity of the algorithm. Importantly, this paper combines two different dominant fluorescent probes to achieve high-quality reconstruction of dual light sources. The performance of the proposed reconstruction strategy was evaluated by digital mouse and nude mouse single/dual light source models. The simulation results show that the HQSAO iterative algorithm can achieve more excellent positioning accuracy and morphology distribution in a shorter time. In vivo experiments also further prove that the HQSAO algorithm has advantages in light source information preservation and artifact suppression. In particular, the introduction of two main emission fluorescent probes makes it easy to separate and reconstruct the dual light sources. When it comes to localization and three-dimensional morphology, the results of the reconstruction are much better than those using a fluorescent probe, which further facilitates the clinical transformation of FMT.

Blind turbulent image deblurring through dual patch-wise pixels prior

Deblurring turbulent images is an active topic in image processing and low-level vision research. Existing methods usually use the parametric physical model for nonblind image restoration, which lacks adaptability to different turbulent scenes. To overcome this challenge, a dual patch-wise pixels (DPP) prior is proposed for effective blind deblurring of turbulent images. A DPP-based turbulent image deblurring model was established based on the fact that the value of the DPP decreases through the turbulent blurring process, which has been proven both mathematically and experimentally. To solve the nonlinear DPP in the model, a linear mapping operator was constructed. Additionally, half-quadratic splitting and threshold methods were used to solve the L0 regularization term. Experimental results showed that the proposed algorithm performs well on various types of turbulent scenes as well as real images and outperforms state-of-the-art algorithms in terms of computational efficiency and effectiveness.

Simultaneous denoising and resolution enhancement of seismic data based on elastic convolution dictionary learning

Enhancing seismic resolution is a key component in seismic data processing, which plays a valuable role in raising the prospecting accuracy of oil reservoirs. However, in noisy situations, existing resolution enhancement methods are difficult to yield satisfactory processing outcomes for reservoir characterization. To solve this problem, we develop a new approach for simultaneous denoising and resolution enhancement of seismic data based on convolution dictionary learning. First, an elastic convolution dictionary learning algorithm is presented to efficiently learn a convolution dictionary with stronger representation capability from the noisy data to be processed. Specifically, the algorithm introduces the elastic L1/2 norm as a sparsity constraint and employs a steepest gradient descent strategy to efficiently solve the frequency-domain linear system with substantial computational cost in a half-quadratic splitting framework. Then, based on the learned convolution dictionary, a weighted convolutional sparse representation paradigm is designed to encode the noisy data to acquire an optimal sparse approximation of the effective signal. Subsequently, a high-resolution dictionary with a broadband spectrum is constructed by the proposed parameter scaling strategy and matched filtering technique on the basis of atomic spectrum modeling. Finally, the optimal sparse approximation of the effective signal and the constructed high-resolution dictionary are used for data reconstruction to obtain the seismic signal with high resolution and high signal-to-noise ratio. Synthetic and field dataset examples are executed to check the effectiveness and reliability of the developed method. The results indicate that this method has a more competitive performance in seismic applications compared with the conventional deconvolution and spectral whitening methods.