Bregman Iteration Research Articles

Enhancing joint reconstruction and segmentation with non-convex Bregman iteration

All imaging modalities such as computed tomography, emission tomography and magnetic resonance imaging require a reconstruction approach to produce an image. A common image processing task for applications that utilise those modalities is image segmentation, typically performed posterior to the reconstruction. Recently, the idea of tackling both problems jointly has been proposed. We explore a new approach that combines reconstruction and segmentation in a unified framework. We derive a variational model that consists of a total variation regularised reconstruction from undersampled measurements and a Chan–Vese-based segmentation. We extend the variational regularisation scheme to a Bregman iteration framework to improve the reconstruction and therefore the segmentation. We develop a novel alternating minimisation scheme that solves the non-convex optimisation problem with provable convergence guarantees. Our results for synthetic and real data show that both reconstruction and segmentation are improved compared to the classical sequential approach.

A fast splitting method for efficient Split Bregman iterations

In this paper we propose a new fast splitting algorithm to solve the Weighted Split Bregman minimization problem in the backward step of an accelerated Forward–Backward algorithm. Beside proving the convergence of the method, numerical tests, carried out on different imaging applications, prove the accuracy and computational efficiency of the proposed algorithm.

Point cloud surface segmentation based on volumetric eigenfunctions of the Laplace-Beltrami operator

In the process of surface modeling from scanned point data, a segmentation that partitions a point cloud into meaningful regions or extracts important features from the 3D point data plays an important role in compressing the scanned data and fitting surface patches. In this paper, a new spectral point cloud surface segmentation method is proposed based on volumetric eigenfunctions of the Laplace-Beltrami operator. The proposed algorithm consists of two main steps. Firstly, the point cloud surface is modeled as the union of a bunch of level set surfaces, on which the eigenfunctions are computed from the level set form of the Laplace-Beltrami operator using the finite element method. Secondly, a new vectorial volumetric eigenfunction segmentation model is developed based on the classical Mumford-Shah model, in which we approximate volumetric eigenfunctions by piecewise-constant functions, and the point cloud surface is segmented via segmenting the volumetric eigenfunctions. Instead of solving the Euler-Lagrange equation by evolution implementation, the split Bregman iteration, which is shown to be a fast algorithm, is utilized. Experimental results demonstrate that our volumetric eigenfunction based technique yields superior segmentation results in terms of accuracy and robustness, compared with the surface eigenfunction based method.

Single image blind deblurring based on the fractional-order differential

Image deblurring is a fundamental problem in image processing. When the blur type is unknown, the problem becomes more challenging, which is called blind deblurring. In this paper, we propose a novel variational method for single image blind deblurring based on the fractional-order differential, which can overcome the staircase effect produced by the total variation regularization and alleviate the ringing artifact in deblurring. The fractional-order gradient fidelity term is added in the cost functional to improve the restoration. Moreover, we make use of the edge detect function with the fractional-order gradient to preserve sharp edges, and the Bregman iteration to reconstruct more structures. The primal–dual algorithm is developed to solve the proposed model. Numerical experiments show that our method is able to get the sharp image without the staircase effect and correctly estimate the unknown blur kernel.

An improved algorithm for basis pursuit problem and its applications

We propose an algorithm for solving the basis pursuit problem minu∈Cn{∥u∥1:Au=f}. Our starting motivation is the algorithm for compressed sensing, proposed by Qiao, Li and Wu, which is based on linearized Bregman iteration with generalized inverse. Qiao, Li and Wu defined new algorithm for solving the basis pursuit problem in compressive sensing using a linearized Bregman iteration and the iterative formula of linear convergence for computing the matrix generalized inverse. In our proposed approach, we combine a partial application of the Newton’s second order iterative scheme for computing the generalized inverse with the Bregman iteration. Our scheme takes lesser computational time and gives more accurate results in most cases. The effectiveness of the proposed scheme is illustrated in two applications: signal recovery from noisy data and image deblurring.

Learning Deep CNN Denoiser Priors for Depth Image Inpainting

Due to the rapid development of RGB-D sensors, increasing attention is being paid to depth image applications. Depth images play an important role in computer vision research. In this paper, we address the problem of inpainting for single depth images without corresponding color images as a guide. Within the framework of model-based optimization methods for depth image inpainting, the split Bregman iteration algorithm was used to transform depth image inpainting into the corresponding denoising subproblem. Then, we trained a set of efficient convolutional neural network (CNN) denoisers to solve this subproblem. Experimental results demonstrate the effectiveness of the proposed algorithm in comparison with three traditional methods in terms of visual quality and objective metrics.

Enhanced Non-Local Total Variation Model and Multi-Directional Feature Prediction Prior for Single Image Super Resolution.

It is widely acknowledged that single image super-resolution (SISR) methods play a critical role in recovering the missing high-frequencies in an input low-resolution image. As SISR is severely ill-conditioned, image priors are necessary to regularize the solution spaces and generate the corresponding high-resolution image. In this paper, we propose an effective SISR framework based on the enhanced non-local similarity modeling and learning-based multi-directional feature prediction (ENLTV-MDFP). Since both the modeled and learned priors are exploited, the proposed ENLTV-MDFP method benefits from the complementary properties of the reconstruction-based and learning-based SISR approaches. Specifically, for the non-local similarity-based modeled prior [enhanced non-local total variation, (ENLTV)], it is characterized via the decaying kernel and stable group similarity reliability schemes. For the learned prior [multi-directional feature prediction prior, (MDFP)], it is learned via the deep convolutional neural network. The modeled prior performs well in enhancing edges and suppressing visual artifacts, while the learned prior is effective in hallucinating details from external images. Combining these two complementary priors in the MAP framework, a combined SR cost function is proposed. Finally, the combined SR problem is solved via the split Bregman iteration algorithm. Based on the extensive experiments, the proposed ENLTV-MDFP method outperforms many state-of-the-art algorithms visually and quantitatively.

Reduced quaternion matrix‐based sparse representation and its application to colour image processing

The traditional colour image sparse models ignore the relationship among the three separate colour channels. The authors propose a novel colour image sparse model by employing reduced quaternion matrix, which can treat independent colour channels as a whole. In addition, reduced quaternion matrix singular value decomposition is employed to design the corresponding dictionary learning algorithm. To make the proposed model robust and tractable, a reduced quaternion split Bregman iteration is developed to solve the minimisation problem. The proposed model cannot only preserve inherent colour structures but also avoid hue bias issue efficiently. Extensive experiments on colour image de-noising, in-painting, and super-resolution manifest that the proposed sparse representation model outperforms the state-of-the-art schemes.

Adjusted Non-Local Regression and Directional Smoothness for Image Restoration

Image restoration (IR) problems are very important in many low-level vision tasks. Due to their ill-posed natures, image priors are widely used to regularize the solution spaces. Recently, patch-based non-local self-similarity has shown great potential in IR problems, leading to many effective non-local priors. Their performance largely depends on whether the non-local self-similarity of the underlying image can be fully exploited. However, most of these priors, including non-local regression (NLR), only utilize the center pixel of each patch to model the non-local feature, which is suboptimal. We propose an effective overlap-based non-local regression (ONLR) to fully exploit the non-local similar patches: first, the concept of overlap-based similar pixels group (OSPG) is introduced; second, for each pixel within an OSPG, the non-local weight is obtained via a novel similarity measurement method; third, based on the consistency assumption, the non-local fitting deviations (NLFDs) by using OSPGs are uniformly constrained. Because of the uniform constraints, the restoration may be poor in regions where OSPGs are not reliable. Consequently, a weighting scheme is proposed to measure the OSPG reliability, leading to a novel adjusted non-local regression (ANLR). In addition, the integral image technique (IIT) is adopted to speed up the similar patches search process. To further boost the ANLR, a local directional smoothness (DS) prior is proposed as a good complement of the non-local feature. Finally, a fast split Bregman iteration algorithm is designed to solve the ANLR-DS minimization problem. Extensive experiments on two typical IR problems, that is, image deblurring and super resolution, demonstrate the superiority of the proposed method compared to many state-of-the-art IR methods.

Split Bregman iteration based image reconstruction algorithm for electrical capacitance tomography

Practical applications of the electrical capacitance tomography (ECT) rely mainly on the effectiveness of reconstruction algorithms. In this paper the solution of the inverse problem with the focus on the ECT imaging is reformulated to be an optimization problem by introducing a new loss function with regularizes encoding multiple features of solution. An iterative scheme that decomposes a complex optimization problem into several simpler sub-problems is developed to solve the proposed loss function, in which the linearization approximation and the acceleration strategy are introduced to improve numerical performances. Numerical experiments validate the effectiveness of the proposed imaging method in tackling the ECT inverse problem.

Bregman inverse filter

An inverse problem in which the observed signal is produced by passing the original signal through a filter that is derived from a convex minimisation problem is dealt. The filter is available as a black box. It is shown that the Bregman iteration, which is a tool for convex optimisation, can be used to solve this inverse problem. This novel application of the Bregman iteration is demonstrated through successful restoration of the original image from images filtered by a low-pass filter. Theoretical analysis is conducted to explain the inversion process.

Sparse modified marginal fisher analysis for facial expression recognition

Marginal Fisher analysis (MFA) is an efficient method for dimension reduction, which can extract useful discriminant features for image recognition. Since sparse learning can achieve better generalization ability and lessen the amount of computations in recognition tasks, this paper introduces sparsity into MFA and proposes a novel sparse modified MFA (SMMFA) method for facial expression recognition. The goal of SMMFA is to extract discriminative features by using the resulted sparse projection matrix. First, a modified MFA is proposed to find the original projection matrix. Similar to MFA, the modified MFA also defines the intra-class graph and the inter-class graph to describe geometry structure in the same class and local discriminant structure between different classes, respectively. In addition, the modified MFA removes the null space of the total scatter matrix. The sparse solution of SMMFA can be gained by solving the l1 –minimization problem on the original projection matrix using the linearized Bregman iteration. Experimental results show that the proposed SMMFA can effectively extract intrinsic features and has better discriminant power than the state-of-the-art methods.

General regularization framework for DEER spectroscopy.

Tikhonov regularization is the standard processing technique for the inversion of double electron-electron resonance (DEER) data to distance distributions without assuming a parametrized model. In other fields it has been surpassed by modern regularization methods. We analyze such alternative regularization methods based on the Tikhonov, total variation (TV) and Huber penalties with and without the use of Bregman iterations. For this, we provide a general mathematical framework and its open-source software implementation. We extend an earlier approach by Edwards and Stoll for the selection of an optimal regularization parameter to all of these penalties and use their big test data set of noisy DEER traces with known ground truth for assessment. The results indicate that regularization methods based on Bregman iterations provide an improvement upon Tikhonov regularization in recognizing features and recovering distribution width at moderate signal-to-noise ratio, provided that noise variance is known. Bregman-iterative methods are robust with respect to the method used in the choice of regularization parameter.

Risk for antipsychotic-induced extrapyramidal symptoms: a focus on monoamine receptors on peripheral blood lymphocytes

Single image super-resolution (SR) aims to estimate a high-resolution (HR) image from only one observed low-resolution (LR) image. It is a severely ill-posed problem that needs image priors to ensure a reliable HR estimation. Since different priors usually emphasize different aspects of image characteristics, it’s a big challenge to impose a balanced-and-overall image property constraint on single image SR. In this paper, we cast single image SR into a maximum a posteriori optimization problem and combine two types of complementary priors to answer this challenge. One prior is a novel local gradient field prior derived from example-based gradient field estimation (EGFE) that focuses on recovering the sharpness of gradient profiles. It is good at enhancing the edge sharpness and restoring fine texture details. Whereas the other is a powerful non-local low rank prior implemented in a weighted adaptive p-norm model (WANM). By imposing lp penalties adaptive to regional saliency and weighted constraints, the WANM prior performs well in preserving edge smoothness and suppressing image noise and reconstruction artifacts. An improved Split Bregman Iteration method that adaptively attenuates the regularization strength is further developed to solve the proposed EGFE-WANM SR problem. Comprehensive experiments are conducted and the results show that the proposed EGFE-WANM SR method outperforms many state-of-the-art methods in both objective evaluations and subjective visual comparisons.

Image Compressive Sensing Reconstruction Based on z-Score Standardized Group Sparse Representation

Non-local similarity-based group sparse representation (GSR) has shown great potential in image restoration. Considering the universal existing non-stationarity of natural images and the statistic characteristic differences of different components in the sparse domain of image patch group, this paper proposes a new image compressive sensing reconstruction (ICSR) algorithm based on z-score standardized group sparse representation (ZSGSR). Specifically, the image is first partitioned into overlapping patches, and the similar patch groups are further generated to be decomposed by adaptive PCA dictionary; then, the resulting sparse coefficients are performed component-wise on z-score standardization; finally, the l 1 norm of the standardized sparse coefficients are used to regularize the ICSR. The reconstruction model is solved by splitting Bregman iteration (SBI) and soft threshold shrinking algorithms. The z-score standardization could enhance sparse representation ability, which reflects the importance of different sparse coefficients well; this is beneficial to effectively preserve the crucial small coefficients and to better recovery, the edges and texture details of images, thus improving the reconstructed image quality. Using objective and subjective quality evaluation, the extensive experiments show that the proposed method can obtain a better performance than the existing state-of-the-art algorithms.

On image restoration from random sampling noisy frequency data with regularization

ABSTRACTConsider the image restoration using random sampling noisy frequency data by total variation regularization. By exploring image sparsity property under wavelet expansion, we establish an optimization model with two regularizing terms specifying image sparsity and edge preservation on the restored image. The choice strategy for the regularizing parameters is rigorously set up together with corresponding error estimate on the restored image. The cost functional with data-fitting in the frequency domain is minimized using the Bregman iteration scheme. By deriving the gradient of the cost functional explicitly, the minimizer of the cost functional at each Bregman step is also generated by an inner iteration process with Tikhonov regularization, which is implemented stably and efficiently due to the special structure of the regularizing iterative matrix. Numerical tests are given to show the validity of the proposed scheme.

Adaptive $$l_1$$ l 1 -regularization for short-selling control in portfolio selection

We consider the $$l_1$$ -regularized Markowitz model, where a $$l_1$$ -penalty term is added to the objective function of the classical mean-variance one to stabilize the solution process, promoting sparsity in the solution. The $$l_1$$ -penalty term can also be interpreted in terms of short sales, on which several financial markets have posed restrictions. The choice of the regularization parameter plays a key role to obtain optimal portfolios that meet the financial requirements. We propose an updating rule for the regularization parameter in Bregman iteration to control both the sparsity and the number of short positions. We show that the modified scheme preserves the properties of the original one. Numerical tests are reported, which show the effectiveness of the approach.

Single-image super-resolution via joint statistic models-guided deep auto-encoder network

Recent researches on super-resolution (SR) with deep learning networks have achieved amazing results. However, most of the existing studies neglect the internal distinctiveness of an image and the output of most methods tends to be of blurring, smoothness and implausibility. In this paper, we proposed a unified model which combines the deep model with the image restoration model for single-image SR. This model can not only reconstruct the SR image, but also keep the distinct fine structures for the low-resolution image. Two statistic priors are used to guide the updating of the output of the deep neural network: One is the non-local similarity and the other is the local smoothness. The former is modeled as the non-local total variation regularization, and the latter as the steering kernel regression total variation regularization. For this unified model, a new optimization function is formulated under a regularization framework. To optimize the total variation problem, a novel algorithm based on split Bregman iteration is developed with the theoretical proof of convergence. The experimental results demonstrate that the proposed unified model improves the peak signal-to-noise ratio of the deep SR model. Quantitative and qualitative results on four benchmark datasets show that the proposed model achieves better performance than the deep SR model without regularization terms.

Split Bregman iteration based reconstruction algorithm for electrical capacitance tomography

The electrical capacitance tomography (ECT) technique uses the measured capacitance data to reconstruct the permittivity distribution in a specific measurement area, in which the performances of reconstruction algorithms play a crucial role in the reliability of measurement results. According to the Tikhonov regularization technique, a new cost function with the total least squares technique and the ℓ1-norm based regularizer is presented, in which measurement noises, model deviations and the influence of the outliers in the measurement data are simultaneously considered. The split Bregman technique and the fast-iterative shrinkage-thresholding method are combined into a new iterative scheme to solve the proposed cost function efficiently. Numerical experiment results show that the proposed algorithm achieves the boost in the precision of reconstruction, and under the noise-free condition the image errors for the imaging targets simulated in this paper, that is, 8.4%, 12.4%, 13.5% and 6.4%, are smaller than the linear backprojection (LBP) algorithm, the Tikhonov regularization (TR) algorithm, the truncated singular value decomposition (TSVD) algorithm, the Landweber algorithm and the algebraic reconstruction technique (ART).

Iterative reconstruction algorithm for the inverse problems in electrical capacitance tomography

The electrical capacitance tomography (ECT) technology is a promising tomography technology that can image the distribution information of permittivity in a measurement region. The imaging algorithms play crucial roles in practical ECT image reconstruction problems. Different from previous numerical methods, this study proposes a novel cost function to model the ECT inverse problem, in which the L1-norm is used as the data fidelity to weaken the influence of the outliers contained in the capacitance data, the L1 regularization is introduced to enhance the sparseness of reconstructed objects, and the second order total variation (STV) regularization is used to weaken the staircasing effects caused by the first order total variation (FTV) regularization. The split Bregman iteration (SBI) algorithm that splits a complicated ECT imaging problem into several simpler sub-problems is developed to solve the proposed cost function effectively. The numerical simulation results show that the imaging algorithm proposed in this paper can reconstruct satisfactory results.