Variable Splitting Research Articles

A Variable-Iterative Fully Convolutional Neural Network for Sparse Unmixing

Neural networks have greatly promoted the development of hyperspectral unmixing (HU). Most data-driven deep networks extract features of hyperspectral images (HSIs) by stacking convolutional layers to achieve endmember extraction and abundance estimation. Some model-driven networks have strong interpretability but fail to mine the deep feature. We propose a variable-iterative fully convolutional neural network (VIFCNN) for sparse unmixing, combining the characteristics of these two networks. Under the model-driven iterative framework guided by sparse unmixing by variable splitting and augmented lagrangian (SUnSAL), a data-driven spatial-spectral feature learning module and a spatial information updating module are introduced to enhance the learning of data information. Experimental results on synthetic and real datasets show that VIFCNN significantly outperforms several traditional unmixing methods and two deep learning???based methods. On real datasets, our method improves signal-to-reconstruction error by 17.38%, reduces abundance root-mean-square error by 25.24%, and reduces abundance spectral angle distance by 31.40% compared with U-ADMM-ßUNet.

Variable Splitting and Fusing for Image Phase Retrieval.

Phase Retrieval is defined as the recovery of a signal when only the intensity of its Fourier Transform is known. It is a non-linear and non-convex optimization problem with a multitude of applications including X-ray crystallography, microscopy and blind deconvolution. In this study, we address the problem of Phase Retrieval from the perspective of variable splitting and alternating minimization for real signals and seek to develop algorithms with improved convergence properties. An exploration of the underlying geometric relations led to the conceptualization of an algorithmic step aiming to refine the estimate at each iteration via recombination of the separated variables. Following this, a theoretical analysis to study the convergence properties of the proposed method and justify the inclusion of the recombination step was developed. Our experiments showed that the proposed method converges substantially faster compared to other state-of-the-art analytical methods while demonstrating equivalent or superior performance in terms of quality of reconstruction and ability to converge under various setups.

Energy-stable auxiliary variable viscosity splitting (AVVS) method for the incompressible Navier–Stokes equations and turbidity current system

In this work, we develop a novel energy-stable linear approach, which we name as auxiliary variable viscosity splitting (AVVS) method, to efficiently solve the incompressible fluid flows. Different from the projection-type methods with pressure correction, the AVVS method adopts the viscosity splitting strategy to split the original momentum equation into an intermediate momentum equation without divergence-free constraint and an advection-free momentum equation. A time-dependent auxiliary variable which has exact value 1 is introduced to construct a supplementary equation. The new model not only inherits the same dynamics of original incompressible Navier–Stokes equations, but also facilitates us to design linearly decoupled and energy-stable time-marching scheme. Comparing with the conventional projection-type schemes, the present method leads to an energy dissipation law with respect to kinetic energy instead of a modified energy including velocity and pressure gradient. In each time step, only two parabolic equations with constant coefficients and one Poisson equation need to be solved. Therefore, the numerical implementation is highly efficient. Moreover, the proposed AVVS method can be directly extended to construct linear, decoupled, and energy-stable scheme for the turbidity current system with slight modifications on the right-hand side of supplementary equation. Extensive numerical experiments are implemented to validate the accuracy, energy stability, and capability in complex fluid simulations.

Reinforcement learning for distributed hybrid flowshop scheduling problem with variable task splitting towards mass personalized manufacturing

Mass personalization manufacturing (MPM), an emerging production pattern, aims to improve enterprise profit in modern industries. However, the processing of heterogeneous orders from the consumers complicates such production scheduling problem. In addition, different scale tasks should adopt different splitting strategies in practical manufacturing, which makes the task splitting method more worthy of investigation. Towards MPM, this paper presents a distributed hybrid flowshop scheduling problem with variable task splitting (DHFSP-VTS) to minimize the makespan and total energy consumption simultaneously. Meanwhile, the VTS allows the tasks to be split into different sublots so they can save setup and transfer time. To solve these problems, we present an order modularization processing method that can categorize multiple types of orders into specific generation tasks, and a highly effective reinforcement learning-multiple objective evolutionary algorithm based on decomposition (RL-MOEA/D) is designed. In RL-MOEA/D, there are three features: (1) three initial rules are used for initialization based on the current splitting scheme that can increase the diversity of solutions; (2) the reinforcement learning agent uses the Q-learning mechanism to dynamically select the scheme of task splitting as action; (3) a neighborhood search strategy improves the exploitation ability and expand the solution space. To verify the effectiveness of RL-MOEA/D, the MOEA/Ds based on four splitting schemes and four RL combined meta-heuristics are compared on 18 instances. The results show that RL-MOEA/D can obtain the best optimization and stability of all the other comparison algorithms. Therefore, it’s a new technique to solve DHFSP with large-scale tasks, especially for the problem of MPM.

Disjoint sections of positive semidefinite matrices and their applications in linear statistical models

Abstract Given matrices A A and B B of the same order, A A is called a section of B B if R ( A ) ∩ R ( B − A ) = { 0 } {\mathscr{R}}\left(A)\cap {\mathscr{R}}\left(B-A)=\left\{0\right\} and R ( A T ) ∩ R ( ( B − A ) T ) = { 0 } {\mathscr{R}}\left({A}^{T})\cap {\mathscr{R}}\left({\left(B-A)}^{T})=\left\{0\right\} , where R ( . ) {\mathscr{R}}\left(.) denotes the range (column space) of a matrix argument and the superscript T T stands for the transpose of a matrix. A matrix G G is called positive semidefinite (p.s.d.) if x T G x {x}^{T}Gx is nonnegative for all real vector x x . However, we refer to a symmetric p.s.d. matrix simply as a p.s.d. matrix as the applications discussed in this article are concerned with only symmetric p.s.d. matrices. An n × n n\times n p.s.d. matrix G G admits of a symmetric section G X {G}_{X} of G G with regard to an n × k n\times k matrix X X such that R ( G X ) = R ( G ) ∩ R ( X ) {\mathscr{R}}\left({G}_{X})={\mathscr{R}}\left(G)\cap {\mathscr{R}}\left(X) . In this article, sections of the type G X {G}_{X} are used in minimization of quadratic functions under linear constraints and splitting of vector random variables into uncorrelated vector random variables. In the general Gauss-Markoff model, y = X β + ε y=X\beta +\varepsilon , with design matrix X X and a singular covariance matrix σ 2 G {\sigma }^{2}G of ε \varepsilon , y y is decomposed into four uncorrelated vector random variables as y = M 1 y + M 2 y + M 3 y + M 4 y y={M}_{1}y+{M}_{2}y+{M}_{3}y+{M}_{4}y , where M i , i = 1 , 2 , 3 {M}_{i},i=1,2,3 consist of sections of G G and X X T X{X}^{T} , and M 4 {M}_{4} is a matrix whose row space is the null space of G + X X T G+X{X}^{T} .

C-reactive protein as robust laboratory value associated with prognosis in patients with stage III non-small cell lung cancer (NSCLC) treated with definitive radiochemotherapy

To evaluate the prognostic value of biomarkers from peripheral blood obtained as routine laboratory assessment for overall survival in a cohort of stage III non-small cell lung cancer (NSCLC) patients treated with definitive radiochemotherapy at a high-volume cancer center. Seven blood biomarkers from 160 patients treated with definitive radiochemotherapy for stage III NSCLC were analyzed throughout the course treatment. Parameters were preselected using univariable and multivariable proportional hazards analysis and were assessed for internal validity using leave-one-out cross validation. Cross validated classifiers including biomarkers in addition to important clinical parameters were compared with classifiers containing the clinical parameters alone. An increased C-reactive protein (CRP) value in the final week of radiotherapy was found as a prognostic factor for overall survival, both as a continuous (HR 1.099 (1.038–1.164), p < 0.0012) as well as categorical variable splitting data at the median value of 1.2 mg/dl (HR 2.214 (1.388–3.531), p < 0.0008). In the multivariable analysis, the CRP value-maintained significance with an HR of 1.105 (1.040–1.173) and p-value of 0.0012. The cross validated classifier using CRP at the end of radiotherapy in addition to clinical parameters separated equally sized high and low risk groups more distinctly than a classifier containing the clinical parameters alone (HR = 2.786 (95% CI 1.686–4.605) vs. HR = 2.287 (95% CI 1.407–3.718)). Thus, the CRP value at the end of radiation therapy has successfully passed the crucial cross-validation test. The presented data on CRP levels suggests that inflammatory markers may become increasingly important during definitive radiochemotherapy, particularly with the growing utilization of immunotherapy as a consolidation therapy for stage III NSCLC.

Image reconstruction based on TV-L1 model for planar array capacitance defect detection

Planar array capacitive imaging is an emerging nondestructive testing (NDT) technology with broad application prospects. The severe ill-posed inverse problem of planar array capacitive imaging reduces quality of reconstructed images. In this paper, a total variation regularization model with an L1-norm (TV-L1 model) of 3 × 4 planar array capacitance imaging is established for imaging the defects detected in the inner layers of multi-layer composite components. Considering the solution instability caused by the non-differentiability of the TV-L1 model, a variable splitting self-adaptive augmented Lagrange alternating direction algorithm (VS-SALAD) is proposed. An augmented Lagrange method enhances model splitability, and a series of subproblems are obtained by variable splitting operation. The subproblems are solved iteratively using alternating minimization method with self-adaptive penalty parameter. Experimental results demonstrate that TV-L1 model effectively alleviates the ill-posed problem, and the proposed algorithm significantly improves both the quality of reconstructed images and solving speed of the TV-L1 model.

3D Poissonian image deblurring via patch-based tensor logarithmic Schatten-p minimization

In medical and biological image processing, multi-dimensional images are often corrupted by blur and Poisson noise. In this paper, we first propose a new tensor logarithmic Schatten-p (t-log-Sp ) low-rank measure and a tensor iteratively reweighted Schatten-p minimization algorithm for minimizing such measure. Furthermore, we adopt this low-rank measure to regularize the non-local tensors formed by similar 3D image patches and develop a patch-based non-local low-rank model. The data fidelity term of the model characterizes the Poisson noise distribution and blur operator. The optimization model is further solved by an alternating minimization technique combined with variable splitting. Experimental results tested on 3D fluorescence microscope images show that the proposed patch-based tensor logarithmic Schatten-p minimization method outperforms state-of-the-art methods in terms of image evaluation metrics and visual quality.

Model-based iterative reconstruction for direct imaging with point spread function encoded echo planar MRI

BackgroundEcho planar imaging (EPI) is a fast measurement technique commonly used in magnetic resonance imaging (MRI), but is highly sensitive to measurement non-idealities in reconstruction. Point spread function (PSF)-encoded EPI is a multi-shot strategy which alleviates distortion, but acquisition of encodings suitable for direct distortion-free imaging prolongs scan time. In this work, a model-based iterative reconstruction (MBIR) framework is introduced for direct imaging with PSF-EPI to improve image quality and acceleration potential. MethodsAn MBIR platform was developed for accelerated PSF-EPI. The reconstruction utilizes a subspace representation, is regularized to promote local low-rankedness (LLR), and uses variable splitting for efficient iteration. Comparisons were made against standard reconstructions from prospectively accelerated PSF-EPI data and with retrospective subsampling. Exploring aggressive partial Fourier acceleration of the PSF-encoding dimension, additional comparisons were made against an extension of Homodyne to direct PSF-EPI in numerical experiments. A neuroradiologists' assessment was completed comparing images reconstructed with MBIR from retrospectively truncated data directly against images obtained with standard reconstructions from non-truncated datasets. ResultsImage quality results were consistently superior for MBIR relative to standard and Homodyne reconstructions. As the MBIR signal model and reconstruction allow for arbitrary sampling of the PSF space, random sampling of the PSF-encoding dimension was also demonstrated, with quantitative assessments indicating best performance achieved through nonuniform PSF sampling combined with partial Fourier. With retrospective subsampling, MBIR reconstructs high-quality images from sub-minute scan datasets. MBIR was shown to be superior in a neuroradiologists' assessment with respect to three of five performance criteria, with equivalence for the remaining two. ConclusionsA novel image reconstruction framework is introduced for direct imaging with PSF-EPI, enabling arbitrary PSF space sampling and reconstruction of diagnostic-quality images from highly accelerated PSF-encoded EPI data.

Analysis of the Flow Capacity of Variable Cycle Split Fans at the Middle Speed

The next-generation variable cycle engine imposes stricter requirements on a fan’s flow capacity at the middle speed. To tackle this challenge, the implementation of split fans presents as a potential solution. In the present study, we conducted numerical simulations using the commercial software NUMECA to investigate the aerodynamic performance variation with bypass ratios for variable cycle split fans in “1 + 2” and “2 + 1” configurations at 80% rpm. The results indicate that the flow capacity of the split fans exhibits an increasing trend with a rise in the bypass ratio at 80% rpm and subsequently stabilizes upon reaching a certain bypass ratio. Specifically, the flow capacity of the “2 + 1” split fans is particularly stronger at the small bypass ratios, whereas the “1 + 2” split fans exhibit superior maximum flow capacity at the high bypass ratios. Additionally, there is a significantly faster increase in the flow capacity of the “1 + 2” split fans compared to that of the “2 + 1” split fans. Furthermore, when the flow capacity of the split fans reaches its maximum, both the efficiency and stall margin achieve their optimal values, indicating that the corresponding bypass ratio is optimal.

Blind multi-Poissonian image deconvolution with sparse log-step gradient prior.

Blind image deconvolution plays a very important role in the fields such as astronomical observation and fluorescence microscopy imaging, in which the noise follows Poisson distribution. However, due to the ill-posedness, it is a very challenging task to reach a satisfactory result from a single blurred image especially when the power of the Poisson noise is at a high level. Therefore, in this paper, we try to achieve high-quality restoration results with multi-blurred images which are contaminated by Poisson noise. Firstly, we design a novel sparse log-step gradient prior which adopts a mixture of logarithm and step functions to regularize the image gradients and combine it with the Poisson distribution to formulate the blind multi-image deconvolution problem. Secondly, we incorporate the methods of variable splitting and Lagrange multiplier to convert the original problem into sub-problems, then we alternately solve them to achieve the estimation of all the blur kernels of corresponding blurred images. Besides, we also design a non-blind multi-image deconvolution algorithm which is based on the log-step gradient prior to reach the final restored image. Experimental results on both synthetic and real-world blurred images show that the proposed prior is very capable of suppressing negative artifacts caused by ill-posedness. The algorithm can achieve restored image of very high quality which is competitive with some state-of-the-art methods.

Dehazing, enhancing the boundaries and corners in hazed images using optimal adaptive technique

ABSTRACT Objects or images captured from the remote area will not have good quality and will not be visible because of mist, smoke, moisture and fog. Here, we rub/remove the unwanted hazes from the image using the optimal adaptive technique. An examination of the built-in boundary constraint of the transmission function is extremely valuable to our method. This constraint is handled as an optimisation problem to predict the unknown scene transmission. In addition, transmission refining techniques are used to remedy the issue based on variable splitting methodology. With only a few broad assumptions, the proposed method may recover haze-free, high-quality images having original colours and better image information. Experiment findings are done with plenty of foggy images indicating the efficiency and effectiveness of the proposed technique by using image quality parameters.

Mixed Gaussian-impulse noise removal using non-convex high-order TV penalty

To restore images with clear edge details and no staircase artifacts from degraded versions, this paper incorporates the ℓ2 plus ℓ0 data fidelity and non-convex high-order total variation regularizer to establish an optimization model for eliminating mixed Gaussian-impulse noise. Among them, the ℓ2 fidelity is adopted to suppress Gaussian noise, while the ℓ0-norm is more suitable for detecting and removing impulse noise. In addition, the non-convex regularization displays excellent performance in overcoming the staircase effect and preserving edge details. Computationally, by using the iteratively reweighted ℓ1 algorithm and variable splitting method, this work designs a modified alternating minimization method to solve the optimization problem we construct. In theory, the convergence proof of our resulting algorithm is also presented. Finally, in the experimental section, we conduct extensive numerical experiments on degraded images and compare with other existing techniques. From the intuitive effects and restoration accuracy, it follows that our newly proposed method is effective and competitive for image deblurring and mixed noise removal.

An initially robust minimum simplex volume-based method for linear hyperspectral unmixing

ABSTRACT Initialization plays an important role in the accuracy of endmember extraction algorithms (EEAs) in linear hyperspectral unmixing (LHU). Random initialization can lead to varying endmembers generated by EEAs. To address this challenge, an initialization strategy has been introduced, encompassing vertex component analysis (VCA), automatic target generation process (ATGP), among others. These techniques significantly contribute to enhancing the accuracy of EEAs. However, complex initialization is sometimes less preferable, prompting the unexplored question of whether there exists an EEA robust to initialization. This paper focuses on analyzing this issue within the context of minimum simplex volume-based (MV) methods, which have received considerable attention in the past two decades due to their robustness against the absence of pure pixels. MV methods typically formulate LHU as an optimization problem, most of which includes a non-convex volume term. Additionally, many MV methods use VCA as an initialization strategy. Firstly, this paper demonstrates that the variable splitting augmented Lagrangian approach (SISAL), as a representative non-convex MV method, heavily depends on initialization. To our knowledge, the impact of initialization for MV methods has not been thoroughly analyzed before. Furthermore, this paper proposes an initially robust MV method by introducing a new convex MV term. Numerical experiments conducted on simulated and real datasets demonstrate its outstanding performance in accuracy and robustness to initialization. Throughout the experiments the proposed method proves to be the most stable, which is crucial in real scene where the ground truth is unknown beforehand.

DeepRED Based Sparse SAR Imaging

The integration of deep neural networks into sparse synthetic aperture radar (SAR) imaging is explored to enhance SAR imaging performance and reduce the system’s sampling rate. However, the scarcity of training samples and mismatches between the training data and the SAR system pose significant challenges to the method’s further development. In this paper, we propose a novel SAR imaging approach based on deep image prior powered by RED (DeepRED), enabling unsupervised SAR imaging without the need for additional training data. Initially, DeepRED is introduced as the regularization technique within the sparse SAR imaging model. Subsequently, variable splitting and the alternating direction method of multipliers (ADMM) are employed to solve the imaging model, alternately updating the magnitude and phase of the SAR image. Additionally, the SAR echo simulation operator is utilized as an observation model to enhance computational efficiency. Through simulations and real data experiments, we demonstrate that our method maintains imaging quality and system downsampling rate on par with deep-neural-network-based sparse SAR imaging but without the requirement for training data.

The Design and Characterization of an Ultra-Compact Asymmetrical Multimode Interference Splitter on Lithium Niobate Thin Film

We propose and demonstrate a high-performance asymmetrical multimode interference splitter on X-cut lithium niobate on insulator (LNOI) with an ultra-compact size of 5.8 μm × (26.4–35.6) μm. A rectangle with a small region is removed from the upper left corner of the multimode interference (MMI) coupler to achieve a variable splitting ratio. Here, we design and characterize MMIs in six different distribution ratios ranging from 50:50 to 95:5 on a 600 nm thick LNOI. Based on the cascade structure, the linear fitting method accurately shows the device loss (~0.1–0.9 dB). Our fabricated devices demonstrate robustness across a 30 nm optical bandwidth (1535–1565 nm). In addition, we numerically simulate the Z-cut LNOI, showing that the structure corresponding to the TM mode can also achieve a good variable splitting ratio.

A combined first and fractional order regularization method for mixed Poisson-white spike noisy image restoration

In this paper we study the problem of restoring images affected by mixed Poisson-Square Cauchy noise. A multi-convex variational model is derived by integrating infimal convolution likelihood into a process of joint maximum a posteriori estimation, where a modified four-directional fractional-order total variation is united with the first order total variation to characterize the image prior. To solve the proposed model numerically, a block coefficient descent based algorithm is derived, in which variable splitting and alternating direction minimization of multipliers are utilized along with anisotropic diffusion and additive operator splitting to gain efficiency and quality. The obtained numerical results are compared with results obtained from two other total variation regularized models with different fidelities. The numerical discuss confirms the flexibility and validity of the proposed model.

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.

Convolutional attention with roll padding: Classifying PM2.5 concentration levels in the city of Beijing

A precise and timely classification of particulate matter 2.5 concentration levels is important for the design of air quality regulatory measures in a contemporaneous context characterized by the transition to a low-carbon economy. This study uses a well-known air quality dataset retrieved from the University of California at Irvine repository, which consists of a multivariate time series covering particulate matter 2.5 concentration levels in the city of Beijing for a period of 5 years. We train, test, and validate several deep learning architectures for a multinomial classification of the target variable in the period of 24 h ahead from the contemporaneous moment of action relying on historical information about the last 168 h and considering a sliding window of 24 h to construct examples. Results indicate that the internationally patented Variable Split Convolutional Attention model exhibits the best accuracy. The main novelty of this model consists of introducing bidimensional convolutional operations inside the attention block to capture the relative attention weight given to patterns of contiguous segments within different time-steps for each input variable. Therefore, a valuable deep learning architecture is presented to properly classify particulate matter 2.5 concentration levels in the atmosphere.

Assessment of the possibility of imitating experts' aesthetic judgments about the impact of knots on the attractiveness of furniture fronts made of pine wood

Our research aims to reconstruct expert preferences regarding the visual attractiveness of furniture fronts made of pine wood using machine learning algorithms. A numerical experiment was performed using five machine learning algorithms of various paradigms. To find the answer to the question of what determines the expert's decision, we determined the importance of variables for some machine learning models. For random forest and classification trees, it involves the overall reduction in node impurities resulting from variable splitting, while for neural networks it uses the Garson algorithm. Based on the numerical experiments we can conclude that the best results of expert decision reconstruction are provided by a neural network model. The expert's decision is better reconstructed for more beautiful images. The decision for nice images is made based on the best 4 or 5 variables, while for ugly images many more features are important. Prettier images and those for which the expert's decision is better reconstructed have fewer knots.