Split Bregman Method Research Articles

Learning-Based Split Unfolding Framework for 3-D mmW Radar Sparse Imaging

The application of the compressed sensing (CS) method in the radar field enables the radar imaging system to satisfy both low data cost and high reconstruction quality, however, it is accompanied by enormous iterative operations and difficult adjustments of parameters. In this paper, we propose a learning-based split unfolding framework, dubbed as split iterative sparse reconstruction network (SISR-Net), for near-field 3-D millimeter-wave (mmW) radar sparse imaging. Firstly, a sparse reconstruction algorithm, i.e., SISRA, is proposed to theoretically guide the structure of the imaging framework. Subsequently, by combining the model-based CS method and data-driven deep learning method, SISR-Net is constructed by SISRA to produce 3-D mmW radar images efficiently with excellent explainability and generalization ability. Joint the radar-imaging kernel, echo-generation kernel, and the split Bregman method, the efficiency and stability of SISR-Net are guaranteed, all parameters are layer-varied and learned steadily by end-to-end training to improve the convergence and robustness of the imaging network. Simulated data and the echo from a high-resolution mmW radar dataset 3DRIED, are used to train and test the SISR-Net based on the Adam optimizer. For both simulation and extensive 3-D mmW radar measured experiments, the proposed SISR-Net outperforms other state-of-the-art imaging methods in terms of imaging accuracy and generalization ability.

An improved numerical method for variational image registration model with intensity correction

In this paper, we propose an efficient numerical method for registration and intensity correction of images. We first introduced a variational image registration model in our previous work. The variational minimization of the functional in the model is solved by using an alternating minimization method. In this work, we improve the numerical solution of the minimization problem by applying the Split Bregman method together with the alternating minimization method. As expected, our numerical experiments show that the proposed method converges rapidly and is significantly more accurate than the method used in our previous work.

Combined total variation of first and fractional orders for Poisson noise removal in digital images

Introduction: Many methods have been proposed to handle the image restoration problem with Poisson noise. A popular approach to Poissonian image reconstruction is the one based on Total Variation. This method can provide significantly sharp edges and visually fine images, but it results in piecewise-constant regions in the resulting images. Purpose: Developing an adaptive total variation-based model for the reconstruction of images contaminated by Poisson noise, and an algorithm for solving the optimization problem. Results: We proposed an effective way to restore images degraded by Poisson noise. Using the Bayesian framework, we proposed an adaptive model based on a combination of first-order total variation and fractional order total variation. The first-order total variation model is efficient for suppressing the noise and preserving the keen edges simultaneously. However, the first-order total variation method usually causes artifact problems in the obtained results. To avoid this drawback, we can use high-order total variation models, one of which is the fractional-order total variation-based model for image restoration. In the fractional-order total variation model, the derivatives have an order greater than or equal to one. It leads to the convenience of computation with a compact discrete form. However, methods based on the fractional-order total variation may cause image blurring. Thus, the proposed model incorporates the advantages of two total variation regularization models, having a significant effect on the edge-preserving image restoration. In order to solve the considered optimization problem, the Split Bregman method is used. Experimental results are provided, demonstrating the effectiveness of the proposed method. Practical relevance: The proposed method allows you to restore Poissonian images preserving their edges. The presented numerical simulation demonstrates the competitive performance of the model proposed for image reconstruction. Discussion: From the experimental results, we can see that the proposed algorithm is effective in suppressing noise and preserving the image edges. However, the weighted parameters in the proposed model were not automatically selected at each iteration of the proposed algorithm. This requires additional research.

A fractional-order regularization with sparsity constraint for blind restoration of images

Blind image deconvolution is one of the most challenging problems in image restoration. Inspired by the work on sparsity constraint and deblurring of blind motion, we propose a model with fractional-order regularization and sparsity constraint for blind restoration and construct split Bregman combining an iterative thresholding algorithm. Fractional-order penalty term in Besov space is expanded by wavelet basis and computed using iterative thresholding algorithm. The regularized terms of blur kernel under tight wavelet frame systems are solved by the split Bregman method. Numerical experiments show that our algorithm can effectively remove different kinds of blur without requiring any prior information of the blur kernels and obtain higher signal-to-noise ratios and lower relative errors. In addition, fractional-order derivative in Besov space can preserve both edges and smoothness better than the integer-order derivative.

Strip-Map SAR Image Formulation Based on the Modified Alternating Split Bregman Method

Conventional compressive sensing (CS)-based imaging methods allow images to be reconstructed from a small amount of data, while they suffer from high computational burden even for a moderate scene. To address this problem, this paper presents a novel two-dimensional (2D) CS imaging algorithm for strip-map synthetic aperture radars (SARs) with zero squint angle. By introducing a 2D separable formulation to model the physical procedure of the SAR imaging, we separate the large measurement matrix into two small ones, and then the induced algorithm can deal with 2D signal directly instead of converting it into 1D vector. As a result, the computational load can be reduced significantly. Furthermore, thanks to its superior performance in maintaining contour information, the gradient space of the SAR image is exploited and the total variation (TV) constraint is incorporated to improve resolution performance. Due to the non-differentiable property of the TV regularizer, it is difficult to directly solve the induced TV regularization problem. To overcome this problem, an improved split Bregman method is presented by formulating the TV minimization problem into a sequence of unconstrained optimization problem and Bregman updates. It yields an accurate and simple solution. Finally, the synthesis and real experiment results demonstrate that the proposed algorithm remains competitive in terms of high resolution and high computational efficiency.

Multidirectional Anisotropic Total Variation and Its Application in the Tomography of the Surrounding Rock of Coal Mining Faces

The computed tomography (CT) reconstruction algorithm is one of the crucial components of the CT system. To date, total variation (TV) has been widely used in CT reconstruction algorithms. Although TV utilizes the a priori information of the longitudinal and lateral gradient sparsity of an image, it introduces some staircase artifacts. To overcome the current limitations of TV and improve imaging quality, we propose a multidirectional anisotropic total variation (MATV) that uses multidirectional gradient information. The surrounding rock of coal mining faces uses principles of tomography similar to those of medical X-rays. The velocity distribution for the surrounding rock can be obtained by the first-arrival traveltime tomography of the transmitted waves in the coal mining face. Combined with the geological data, we can interpret the geological hazards in the coal mining face. To perform traveltime tomography, we first established the objective function of the first-arrival traveltime tomography of the transmitted waves based on the MATV regularization and then used the split Bregman method to solve the objective function. The simulated data and real data show that the MATV regularization method proposed in this paper can better maintain the boundaries of geological anomalies and reduce the artifacts compared with the isotropic total variation regularization method and the anisotropic total variation regularization method. Furthermore, this approach describes the distribution of geological anomalies more accurately and effectively and improves imaging accuracy.

A neighbor level set framework minimized with the split Bregman method for medical image segmentation

Medical image segmentation has a huge challenge due to intensity inhomogeneity and the similarity of the background and the object. To meet this challenge, we propose an improved active contour model, in which we combine the level set method and the split Bregman method, and provide the two-phase formulation, the multi-phase formulation and 3D formulation. In this paper, the proposed model is presented in a level set framework by including the neighbor region information for segmenting medical images in which the energy functional contains the data fitting term and the length term. The neighbor region and the local intensity variances in the data fitting term are designed to optimize the minimization process. To minimize the energy functional then we apply the split Bregman method which contributes to get faster convergence. Besides, we extend our model to the multi-phase segmentation model and the 3D segmentation model for cardiac MR images, which have all achieved good results. Experimental results show that the new model not only has strong robustness to other cardiac tissue effects and image intensity inhomogeneity, but it also can much better conduce to the extraction of effective tissues. As we expected, our model has higher segmentation accuracy and efficiency for medical image segmentation.

Accurate and efficient image segmentation and bias correction model based on entropy function and level sets

With the development of society, image segmentation plays a pivotal role in real-world life. However, the images we obtain are often distorted or contaminated by noise and shade, causing low construct and weak boundaries. In this paper, we propose a new model suitable for segmenting and correcting images with inhomogeneous intensity simultaneously. According to the characteristics of the entropy function, we use it as the coefficient of the global and local terms of the energy functional, related to the intensity of the image. Most importantly, the compression function is added to control the global item and the local item in the same range, reducing parameters to be tuned. In addition, our model can be extended to the multi-objective model and the colour model to deal with a more difficult situation. Furthermore, by utilizing the split Bregman method to solve the energy functional, we reduce computational cost and maintain the convergence of the algorithm. A large number of experimental results demonstrate the superiority of our model over previous models. Our model has promising performance for the segmentation and correction of inhomogeneous medical images and colour images.

Nonlocal total variation regularization with Shape Adaptive Patches for image denoising via Split Bregman method

Non-Local or patch-based approaches are used in most of the state-of-the-art methods for image denoising. Denoising with regular square patches may cause noise halos around edges, particularly high contrasted edges. In order to overcome this drawback, this work presents an extension of the Non-Local TV framework that effectively exploits the potential local geometric features of the image by replacing the simple square patches with different oriented shapes. We first solve the denoised models of ROF-NLTV with different oriented patch shapes by Split Bregman algorithm. Then, use exponentially weighted aggregation method based on Stein’s unbiased risk estimate to combine the estimators obtained in the first step. The numerical results show our method outperforms some previous ones. Moreover, common noise halos around edges usually observed by denoising with Non-Local TV method are reduced thanks our approach.

Level set framework with transcendental constraint for robust and fast image segmentation

Though image segmentation models are plentiful and have many applications nowadays, it can be difficult to segment images with complex boundaries and serious intensity inhomogeneity. To some extent, the region-scalable fitting energy model can segment images suffering from intensity inhomogeneity since it considers image intensity as a function, but it relies on initial conditions dramatically. Nowadays, prior knowledge has been widely applied in image segmentation models, which can integrate automatic method and experts experience in one robust and fast segmentation model. In this paper we present a new model that can segment various images accurately by taking the advantages of the region-scalable fitting energy model and the advanced transcendental constraint from artificial experience. The proposed energy functional consists of a smooth length term, a target image data term and a transcendental constraint term. The transcendental constraint term plays a key role in the proposed model, which not only gives the accurate segmentation results but also provides us the chance to carry out the parallel computation. In the proposed-parallel model, the efficiency is improved a lot and the results become more precise compared with other methods. The split Bregman method is applied to minimize the energy functional. Furthermore, we present the convergence analysis and the time complexity analysis of our algorithm. Multiple experimental results and comparisons including parameters sensitivity discussion are shown to demonstrate the superiority of the proposed model such as high accuracy, robustness and efficiency.

Blocking sparse method for image denoising

In recent years, compressive sensing has been one promising technique for denoising images. This paper presents a new denoising model based on blocking sparsity. First, an image is blocked. Second, the split-Bregman method is used to solve for each block image. Finally, all denoised block images are combined into one image. Compared with the latest HTV, GHNS, FastATV, CSR and BM3D models, experimental results demonstrate that the proposed method is efficient, and has better denoising capability.

AFISTA: Accelerated FISTA for sparse signal recovery and compressive sensing

This paper presents a new fast iterative shrinkage-thresholding algorithm, termed AFISTA. The essential idea is to improve the convergence rate of FISTA using a new continuation strategy leading to a less number of iterations compared to FISTA. The convergence theorem of the AFISTA is proposed. In order to further accelerate the AFISTA method, it is equipped with the Barzilai-Borwein (BB) method. Also, for applications with orthogonal sensing matrix A, we proposed a specialized version of the AFISTA method. AFISTA is tailored for solving the basis pursuit problem which can be applied successfully on a variety of problems arising in signal and image processing issues such as sparse signal recovery, signal and image denoising, image restoration, and compressive sensing. To show the efficiency of the method, we compare our results with generalizations of linearized Bregman and fixed - point continuation (FPC) methods in sparse signal recovery applications, with split Bregman method in compressive sensing for sparse MRI and with Gradient projection for sparse reconstruction (GPSR) method in image deconvolution. Numerical results demonstrate that AFISTA overcomes all of the compared methods in convergence rate and some of them in both convergence rate and quality of reconstructed results.

Image denoising by a novel variable‐order total fractional variation model

The total variation model performs very well for removing noise while preserving edges. However, it gives a piecewise constant solution which often leads to the staircase effect, consequently small details such as textures are filtered out in the denoising process. Fractional‐order total variation method is one of the major approaches to overcome such drawbacks. Unlike their good quality of fractional order, all these methods use a fixed fractional order for the whole of the image. In this paper, a novel variable‐order total fractional variation model is proposed for image denoising, in which the order of fractional derivative will be allocated automatically for each pixel based on the context of the image. This kind of selection is able to capture the edges and texture of the image simultaneously. In this regard, we prove the existence and uniqueness of the presented model. The split Bregman method is adapted to solve the model. Finally, the results illustrate the efficiency of the proposed model that yielded good visual effects and a better signal‐to‐noise ratio.

Seismic ground roll attenuation in DWT domain by split Bregman iteration algorithm

Noise reduction of seismic data is one of the most substantial steps in seismic signal processing. Goudarzi and Riahi presented wavelet domain ground roll analysis (WDGA), a technique based on the data. The results of this method showed a good improvement of the signal-to-noise ratio, but the WDGA did not work well in the data with a high level of random noise because this method uses a comparison-based search algorithm. This problem requires additional research to find a way to overcome this limitation. The wavelet domain split Bregman (WDSB) method presented in this paper is based on the use of the split Bregman algorithm in a discrete wavelet domain. The goal is to simultaneously use the capabilities of the F-K filter and the split Bregman iteration algorithm in the discrete wavelet domain, to overcome the problem of high levels of random noise in ground roll attenuation. The results of using WDSB for synthetic and real seismic data show improvement in ground roll attenuation while protecting seismic signals. The new algorithm is studied in time and wavelet domains.

An infrared and visible image fusion method based on multi-scale transformation and norm optimization

In this paper, we propose a new infrared and visible image fusion method based on multi-scale transformation and norm optimization. In this method, a new loss function is designed with contrast fidelity (L2 norm) and sparse constraint (L1 norm), and the split Bregman method is used to optimize the loss function to obtain pre-fusion images. The final fused base layer is obtained by using a multi-level decomposition latent low-rank representation (MDLatLRR) method to decompose the pre-fusion images. Then, using the pre-fusion image as the reference, image structure similarity (SSIM) is introduced to evaluate the validity of detail information from the visible image, and the SSIM is then transformed into a weight map which is applied to the optimization method based on L2 norm to generate the final detail fusion layer. Our proposed method is evaluated and compared with 18 state-of-the-art image fusion methods, both qualitatively and quantitatively on four public datasets (i.e., CVC14 driving dataset, TNO dataset with natural scenarios, RoadScene dataset, and whole brain atlas dataset). The results show that our proposed method is generally better than the compared methods in terms of highlighting targets and retaining effective detail information.

A region fusion based split Bregman method for TV Denoising algorithm

A novel robust image denoising method based Total Variation (TV) model called Region Fusion based Split Bregman (RFSB) method is proposed in this paper. First, the structural characteristics of the edge region and that of the smooth region of the noisy image are analyzed and then separated. Second, split Bregman method is used for calculating two baseline TV-based model, the TV model and TV-penalized least squares functional model, and results of the two models are fused by proposed Region Fusion method. Third, for the fusion denoising result contains a wealth of repetitive redundant information, fast non-local means (FNLM) is introduced for post-processing the denoising result. Our proposed method has three contributions: (i) the use of split Bregman to calculate two baseline TV-based model because of its highly minimization speed; (ii) the proposal of a region fusion method to fuse two baseline models; (iii) the application of the FNLM method as the post-processing. Gray images and color images with different intensities of noise are used to compare our proposed method with other existing classical TV-based method. Experimental results show that Our proposed method has outperformed other TV-based methods in denoising visual effect and objective evaluation for different styles of noised images. The time complexity of our proposed method is about 2% and 50% lower than that of TV and Zhang’s Total Variation (ZTV). Our proposed method shows lower time complexity and does not distort the image information compared with two deep learning algorithm.

Adaptive segmentation model for liver CT images based on neural network and level set method

Accurate segmentation is difficult for liver computed tomography (CT) images, since the liver CT images do not always have obvious and smooth boundaries. The location of the tumor is not specified and the image intensity is similar to that of the liver. Although manual and automatic segmentation methods, traditional and deep learning models currently exist, none can be specifically and effectively applied to segment liver CT images. In this paper, we propose a new model based on a level set framework for liver CT images in which the energy functional contains three terms including the data fitting term, the length term and the bound term. Then we apply the split Bregman method to minimize the energy functional that leads the energy functional to converge faster. The proposed model is robust to initial contours and can segment liver CT images with intensity inhomogeneity and unclear boundaries. In the bound term, we use the U-Net to get constraint information which has a considerable influence on effective and accurate segmentation. We improve a multi-phase level set of our model to get contours of tumor and liver at the same time. Finally, a parallel algorithm is proposed to improve segmentation efficiency. Results and comparisons of experiments are shown to demonstrate the merits of the proposed model including robustness, accuracy, efficiency and intelligence.

Spectral-Image Decomposition With Energy-Fusion Sensing for Spectral CT Reconstruction

Spectral-computed tomography (CT) has been demonstrating its great advantages in lesion detection, tissue characterization, and material decomposition. However, the quality of images is often significantly corrupted with various noises, which brings a great challenge for its applications. Because the channel-wise images from different energy interval share similar structure and physical message, the spatial sparsity, global correlation across the spectrum (GCS), and nonlocal self-similarity (NSS) as three important characteristics are employed to spectral CT reconstruction. In this study, we propose a spectral-image decomposition with energy-fusion sensing (SIDES) reconstruction method, which encourages to obtain better quality spectral images and material decomposition results by establishing a unified tensor decomposition model. First, considering the noise distribution in channel-wise and the difference of linear attenuation coefficients within channel-cross, an adaptive weighted full-spectrum prior image as additional supervised information is incorporated to formulate a new weighted prior image-based tensor. Cooperating with original image tensor, they fully explore the spatial sparsity, GCS, and NSS properties. Then, we formulate nonlocal similar patch-based tensor groups to encode the NSS property from image-domain and residual-image-domain (which is expanded by prior-image and image-self). Next, low-rank regularized Tucker tensor decomposition is employed to fully explore the intrinsic knowledge with the help of prior-image supervision. Finally, the relaxed convex optimization model is optimized by dividing reconstruction model into several subproblem using split-Bregman method. Numerical simulations and real experiments are designed to validate and evaluate the SIDES method and the results demonstrate that the SIDES reconstruction outperforms the state-of-the-art.

Tensor Gradient L₀-Norm Minimization-Based Low-Dose CT and Its Application to COVID-19

Methods to recover high-quality computed tomography (CT) images in low-dose cases will be of great benefit. To reach this goal, sparse-data subsampling is one of the common strategies to reduce radiation dose, which is attracting interest among the researchers in the CT community. Since analytic image reconstruction algorithms may lead to severe image artifacts, the iterative algorithms have been developed for reconstructing images from sparsely sampled projection data. In this study, we first develop a tensor gradient L0-norm minimization (TGLM) for low-dose CT imaging. Then, the TGLM model is optimized by using the split-Bregman method. The Coronavirus Disease 2019 (COVID-19) has been sweeping the globe, and CT imaging has been deployed for detection and assessing the severity of the disease. Finally, we first apply our proposed TGLM method for COVID-19 to achieve low-dose scan by incorporating the 3-D spatial information. Two COVID-19 patients (64 years old female and 56 years old man) were scanned by the n}{}{mu }nCT 528 system, and the acquired projections were retrieved to validate and evaluate the performance of the TGLM.

Split Bregman Approach to Linear Prediction Based Dereverberation With Enforced Speech Sparsity

The recordings of speech in enclosures are corrupted by reverberation caused by multipath wave propagation from the speaker to the distant microphones. In this letter, we address the problem of reducing the late part of room reverberation. The presented blind dereverberation method consists in multichannel linear prediction (MCLP) and enforces sparsity of the dereverberated speech by adopting the split Bregman approach. The proposed algorithm alternately solves two optimization problems, where the former cost function is derived by assuming that speech is modelled using a sparse prior distribution, while the latter optimization emphasizes speech sparsity by an additional incorporation of a weighted $\ell _1$ -norm of the output signal to the standard linear prediction based cost function. The results of experiments performed using simulated and measured room impulse responses for various reverberation time values indicate superior performance of the proposed sparse split Bregman (SSB) method over state-of-the-art non-sparse and sparse MCLP-based dereverberation methods in terms of standard evaluation measures and as pre-processsing to the automatic speech recognition.