Split Bregman Framework Research Articles

Efficient Iterative Regularization Method for Total Variation-Based Image Restoration

Total variation (TV) regularization has received much attention in image restoration applications because of its advantages in denoising and preserving details. A common approach to address TV-based image restoration is to design a specific algorithm for solving typical cost function, which consists of conventional ℓ2 fidelity term and TV regularization. In this work, a novel objective function and an efficient algorithm are proposed. Firstly, a pseudoinverse transform-based fidelity term is imposed on TV regularization, and a closely-related optimization problem is established. Then, the split Bregman framework is used to decouple the complex inverse problem into subproblems to reduce computational complexity. Finally, numerical experiments show that the proposed method can obtain satisfactory restoration results with fewer iterations. Combined with the restoration effect and efficiency, this method is superior to the competitive algorithm. Significantly, the proposed method has the advantage of a simple solving structure, which can be easily extended to other image processing applications.

Deconvolution-based partial volume correction of PET images with parallel level set regularization

The partial volume effect (PVE), caused by the limited spatial resolution of positron emission tomography (PET), degrades images both qualitatively and quantitatively. Anatomical information provided by magnetic resonance (MR) images has the potential to play an important role in partial volume correction (PVC) methods. Post-reconstruction MR-guided PVC methods typically use segmented MR tissue maps, and further, assume that PET activity distribution is uniform in each region, imposing considerable constraints through anatomical guidance. In this work, we present a post-reconstruction PVC method based on deconvolution with parallel level set (PLS) regularization. We frame the problem as an iterative deconvolution task with PLS regularization that incorporates anatomical information without requiring MR segmentation or assuming uniformity of PET distributions within regions. An efficient algorithm for non-smooth optimization of the objective function (invoking split Bregman framework) is developed so that the proposed method can be feasibly applied to 3D images and produces sharper images compared to PLS method with smooth optimization. The proposed method was evaluated together with several other PVC methods using both realistic simulation experiments based on the BrainWeb phantom as well as in vivo human data. Our proposed method showed enhanced quantitative performance when realistic MR guidance was provided. Further, the proposed method is able to reduce image noise while preserving structure details on in vivo human data, and shows the potential to better differentiate amyloid positive and amyloid negative scans. Overall, our results demonstrate promise to provide superior performance in clinical imaging scenarios.

Numerical computations of split Bregman method for fourth order total variation flow

The split Bregman framework for Osher-Solé-Vese (OSV) model and fourth order total variation flow are studied. We discretize the problem by piecewise constant function and compute ∇(−Δav)−1 approximately and exactly. Furthermore, we provide a new shrinkage operator for Spohn's fourth order model. Numerical experiments are demonstrated for fourth order problems under periodic boundary condition.

Formula omitted] norm regularization based group sparse representation for image compressed sensing recovery

One important challenge in image compressed sensing (CS) recovery methods is to develop a sparsity inducing model which can reflect the image priors appropriately and hence yields high quality recovery results. Recent advances have suggested that group sparse representation based models, which exploit nonlocal self-similarity prior, lead to superior results in image CS recovery. In this paper, we propose a CS recovery method via group sparse representation with nonconvex LLp norm regularization (GSR-LLp). In the proposed method, nonconvex LLp norm with (0<p≤1) is introduced as a new sparsity metric to better promote the sparsity of the group coefficients, rather than the l0 norm. Furthermore, the principle component analysis (PCA) is utilized to learn an adaptive orthogonal dictionary for each group. To solve the GSR-driven LLp minimization problem, an efficient algorithm based on split Bregman framework and Majorization–Minimization (MM) algorithm is developed. Moreover, the proposed model is combined with a robust M-estimate to cope with the case where measurements are corrupted by impulsive noise. In this case, we substitute the l2 norm data fidelity with Welch m-estimate which has shown the advantage of robustness against heavy-tailed impulsive noise. We develop an efficient scheme based on split Bregman framework and half-quadratic (HQ) theory to solve the resulting optimization problem (called as RGSR-LLp). Extensive experimental results show effectiveness of the proposed methods compared with the state-of-the-arts methods in CS image recovery.

Compressive Hyperspectral Imaging With Spatial and Spectral Priors

This paper proposes a new compressive hyperspectral imaging method, including the design of a cost-effective distributed sampling (DS) scheme and an efficient reconstruction model. The new sampling scheme, named as distributed separate sampling (DSS), encodes different hyperspectral bands with mutually independent two-dimensional separate sensing operators. Compared with existing DS schemes, DSS reduces lots of resource overhead in the premise of generating measurements with low redundancy. Furthermore, in contrast to the existing DS schemes, DSS keeps the original structure of hyperspectral images (HSIs) during sampling procedure. The new joint reconstruction model, namely, joint nuclear/total variation/ $L_{1}$ norm minimization, exploits both spatial and spectral priors of HSIs. Unlike the other joint reconstruction models, the proposed model utilize an $L_{1}$ -based distance function to measure the similarity between adjacent bands, which improves the recovery quality of HSIs. Besides, we develop a new compressive inversion algorithm under the split Bregman framework, which is of low computational complexity, to solve our proposed reconstruction model. Comprehensive experimental results demonstrate the efficiency of our method.

Pulsed-high Intensity Focused Ultrasound Therapy of Rabbit Middle Cerebral Artery Thromboembolism Accompanied by Anticoagulant Nanoparticles Administration

One important challenge in image compressed sensing (CS) recovery methods is to develop a sparsity inducing model which can reflect the image priors appropriately and hence yields high quality recovery results. Recent advances have suggested that group sparse representation based models, which exploit nonlocal self-similarity prior, lead to superior results in image CS recovery. In this paper, we propose a CS recovery method via group sparse representation with nonconvex LLp norm regularization (GSR-LLp). In the proposed method, nonconvex LLp norm with (0<p≤1) is introduced as a new sparsity metric to better promote the sparsity of the group coefficients, rather than the l0 norm. Furthermore, the principle component analysis (PCA) is utilized to learn an adaptive orthogonal dictionary for each group. To solve the GSR-driven LLp minimization problem, an efficient algorithm based on split Bregman framework and Majorization–Minimization (MM) algorithm is developed. Moreover, the proposed model is combined with a robust M-estimate to cope with the case where measurements are corrupted by impulsive noise. In this case, we substitute the l2 norm data fidelity with Welch m-estimate which has shown the advantage of robustness against heavy-tailed impulsive noise. We develop an efficient scheme based on split Bregman framework and half-quadratic (HQ) theory to solve the resulting optimization problem (called as RGSR-LLp). Extensive experimental results show effectiveness of the proposed methods compared with the state-of-the-arts methods in CS image recovery.

Hybrid spectral CT reconstruction.

Current photon counting x-ray detector (PCD) technology faces limitations associated with spectral fidelity and photon starvation. One strategy for addressing these limitations is to supplement PCD data with high-resolution, low-noise data acquired with an energy-integrating detector (EID). In this work, we propose an iterative, hybrid reconstruction technique which combines the spectral properties of PCD data with the resolution and signal-to-noise characteristics of EID data. Our hybrid reconstruction technique is based on an algebraic model of data fidelity which substitutes the EID data into the data fidelity term associated with the PCD reconstruction, resulting in a joint reconstruction problem. Within the split Bregman framework, these data fidelity constraints are minimized subject to additional constraints on spectral rank and on joint intensity-gradient sparsity measured between the reconstructions of the EID and PCD data. Following a derivation of the proposed technique, we apply it to the reconstruction of a digital phantom which contains realistic concentrations of iodine, barium, and calcium encountered in small-animal micro-CT. The results of this experiment suggest reliable separation and detection of iodine at concentrations ≥ 5 mg/ml and barium at concentrations ≥ 10 mg/ml in 2-mm features for EID and PCD data reconstructed with inherent spatial resolutions of 176 μm and 254 μm, respectively (point spread function, FWHM). Furthermore, hybrid reconstruction is demonstrated to enhance spatial resolution within material decomposition results and to improve low-contrast detectability by as much as 2.6 times relative to reconstruction with PCD data only. The parameters of the simulation experiment are based on an in vivo micro-CT experiment conducted in a mouse model of soft-tissue sarcoma. Material decomposition results produced from this in vivo data demonstrate the feasibility of distinguishing two K-edge contrast agents with a spectral separation on the order of the energy resolution of the PCD hardware.

An Efficient Sparse Optimization Algorithm for Weighted $\ell _{0}$ Shearlet-Based Method for Image Deblurring

Sparsity is one of the key concepts that allows the signal recovery at a significantly lower subsample rate than required by the Nyquist-Shannon sampling theorem. By using a multistate transform, such as wavelets and shearlets system, the sparse representation of signals can be obtained. To further exploit the sparsity of the reconstructed signal, a generalized gradient regularizer is introduced to the proposed model. Motivated by the idea of iterative support detection (ISD), an optimization algorithm framework for image deblurring is given. The algorithm aims to solve a reweighted $\ell _{0}$ -minimization problem in split Bregman framework, and the weights used for the next iteration are decided by an ISD process. The advantage of this process is that it forms an iterative-feedback mechanism, which improves the effectiveness for solution searching. A series of experiments are presented to demonstrate the availability of the proposed framework. Experimental results show that this method yields significant improvement in peak signal-to-noise ratio when compared to other counterparts. However, the numerical experiments also show that more computing time is required due to the utilization of the redundant multiscale system.

Image/video compressive sensing recovery using joint adaptive sparsity measure

Compressive sensing (CS) is a recently emerging technique and an extensively studied problem in signal and image processing, which enables joint sampling and compression into a unified approach. Recently, local smoothness and nonlocal self-similarity have both led to superior sparsity priors for CS image restoration. In this paper, first, a new sparsity measure called joint adaptive sparsity measure (JASM) is introduced. The proposed JASM enforces both local sparsity and nonlocal 3D sparsity in transform domain, concurrently, providing a powerful mechanism for characterizing the structured sparsities of natural image. More precisely, the local sparsity depicts the local smoothness redundancies exploited by an adaptively learned sparsifying basis, and the nonlocal 3D sparsity corresponds to the nonlocal self-similarity constraint achieved by a new proposed nonlocal statistical sparse modeling. Then, two novel techniques for high-fidelity CS image and video recovery via JASM are proposed. The proposed methods are formulated in the form of minimization functional under regularization-based framework which is solved via an efficient alternating minimization algorithm based on split Bregman framework. Comprehensive experimental results are reported to manifest the effectiveness of the proposed methods compared with the current state-of-the-art methods in CS image/video restoration.

Pan-sharpening of multi-spectral images using a new variational model

In remote-sensing image processing, pan-sharpening is used to obtain a high-resolution multi-spectral image by combining a low-resolution multi-spectral image with a corresponding high-resolution panchromatic image. In this article, to preserve the geometry, spectrum, and correlation information of the original images, three hypotheses are presented, i.e. (1) the geometry information contained in the pan-sharpened image should also be contained in the panchromatic bands; (2) the upsampled multi-spectral image can be seen as a blurred form of the fused image with an unknown kernel; and (3) the fused bands should keep the correlation between each band of the upsampled multi-spectral image. A variational energy functional is then built based on the assumptions, of which the minimizer is the target fused image. The existence of a minimizer of the proposed energy is further analysed, and the numerical scheme based on the split Bregman framework is presented. To verify the validity, the new proposed method is compared with several state-of-the-art techniques using QuickBird data in subjective, objective, and efficiency aspects. The results show that the proposed approach performs better than some compared methods according to the performance metrics.

Image denoising by generalized total variation regularization and least squares fidelity

Inspired by the ability of $$\ell _p$$ l p -regularized algorithms and the close connection of total variation (TV) to the $$\ell _1$$ l 1 norm, a $$p$$ p th-power type TV denoted as TV $$_p$$ p is proposed for $$0\le p \le 1$$ 0 ≤ p ≤ 1 . The TV $$_p$$ p -regularized problem for image denoising is nonconvex thus difficult to tackle directly. Instead, we deal with the problem by proposing a weighted TV (WTV) minimization where the weights are updated iteratively to locally approximate the TV $$_p$$ p -regularized problem. The difficulty of WTV minimization is dealt with in a modified split Bregman framework. Numerical results are presented to demonstrate improved denoising performance of the new algorithm with $$p<1$$ p < 1 relative to that obtained by the standard TV minimization and several recent denoising methods from the literature on a variety of images.

A few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction

In recent years, the total variation (TV) minimization method has been widely used for compressed sensing (CS) based CT image reconstruction. In this paper, we propose a few-view reweighted sparsity hunting (FRESH) method for CT image reconstruction, and demonstrate the superior performance of this method. Specifically, the key of the purposed method is that a reweighted total variation (RwTV) measure is used to characterize image sparsity in the cost function, outperforming the conventional TV counterpart. To solve the RwTV minimization problem efficiently, the Split-Bregman method and other state-of-the-art L1 optimization methods are compared. Inspired by the fast iterative shrinkage/thresholding algorithm (FISTA), a predication step is incorporated for fast computation in the Split-Bregman framework. Extensive numerical experiments have shown that our FRESH approach performs significantly better than competing algorithms in terms of image quality and convergence speed for few-view CT. High-quality images were reconstructed by our FRESH method after 250 iterations using only 15 few-view projections of the Forbild head phantom while other competitors needed more than 800 iterations. Remarkable improvements in details in the experimental evaluation using actual sheep thorax data further indicate the potential real-world application of the FRESH method.