Smoothly Clipped Absolute Deviation (SCAD) regularization for compressed sensing MRI Using an augmented Lagrangian scheme

PurposeCompressed sensing (CS) provides a promising framework for MR image reconstruction from highly undersampled data, thus reducing data acquisition time. In this context, sparsity-promoting regularization techniques exploit the prior knowledge that MR images are sparse or compressible in a given transform domain. In this work, a new regularization technique was introduced by iterative linearization of the non-convex smoothly clipped absolute deviation (SCAD) norm with the aim of reducing the sampling rate even lower than it is required by the conventional l₁ norm while approaching an l₀ norm.Materials and MethodsThe CS-MR image reconstruction was formulated as an equality-constrained optimization problem using a variable splitting technique and solved using an augmented Lagrangian (AL) method developed to accelerate the optimization of constrained problems. The performance of the resulting SCAD-based algorithm was evaluated for discrete gradients and wavelet sparsifying transforms and compared with its l₁-based counterpart using phantom and clinical studies. The k-spaces of the datasets were retrospectively undersampled using different sampling trajectories. In the AL framework, the CS-MRI problem was decomposed into two simpler sub-problems, wherein the linearization of the SCAD norm resulted in an adaptively weighted soft thresholding rule with a sparsity enhancing effect.ResultsIt was demonstrated that the proposed regularization technique adaptively assigns lower weights on the thresholding of gradient fields and wavelet coefficients, and as such, is more efficient in reducing aliasing artifacts arising from k-space undersampling, when compared to its l₁-based counterpart.ConclusionThe SCAD regularization improves the performance of l₁-based regularization technique, especially at reduced sampling rates, and thus might be a good candidate for some applications in CS-MRI.     

Face image recognition via collaborative representation on selected training samples

In this paper, we present a collaborative representation-based classification on selected training samples (CRC_STS) for face image recognition. The CRC_STS uses a two stage scheme: The first stage is to select some most significant training samples from the original training set by using a multiple round of refining process. The second stage is to use collaborative representation classifier to perform classification on the selected training samples. Our method can be regarded as a sparse representation approach but without imposing l₁-norm constraint on representation coefficients. The experimental results on three well known face databases show that our method works very well.     

An Efficient Proximity Point Algorithm for Total-Variation-Based Image Restoration

In this paper, we propose a fast proximity point algorithm and apply it to total variation (TV) based image restoration. The novel method is derived from the idea of establishing a general proximity point operator framework based on which new first-order schemes for total variation (TV) based image restoration have been proposed. Many current algorithms for TV-based image restoration, such as Chambolle's projection algorithm, the split Bregman algorithm, the Bermúdez-Moreno algorithm, the Jia-Zhao denoising algorithm, and the fixed point algorithm, can be viewed as special cases of the new first-order schemes. Moreover, the convergence of the new algorithm has been analyzed at length. Finally, we make comparisons with the split Bregman algorithm which is one of the best algorithms for solving TV-based image restoration at present. Numerical experiments illustrate the efficiency of the proposed algorithms.     

Sparse hyperspectral unmixing using an approximate L0 norm

Sparse unmixing aims at finding an optimal subset of spectral signatures in a large spectral library to effectively model each pixel in the hyperspectral image and compute their fractional abundances. In most previous work concerned with the sparse unmixing, L₂ norm is used to measure the error tolerance and the L₁ norm is added as the sparsity regularization. However, in some applications, using L₁ norm to measure the error tolerance has significant robustness advantages over the L₂ norm. Besides, in some cases, using a smooth function to approximate the L₀ norm can obtain more accurate results than the L₁ norm in the field of sparse regression. Thus, in this paper, we consider the two alternative choices for sparse unmixing. A reweighted iteration algorithm is also proposed so that the unconvex regularizer (smoothed L₀ norm) can be efficiently solved through transforming it into a series of weighted L₁ regularizer problems. Experimental results on both synthetic and real hyperspectral data demonstrate the efficacy of the new models.     

Improved dynamic MRI reconstruction by exploiting sparsity and rank-deficiency

In this paper we address the problem of dynamic MRI reconstruction from partially sampled K-space data. Our work is motivated by previous studies in this area that proposed exploiting the spatiotemporal correlation of the dynamic MRI sequence by posing the reconstruction problem as a least squares minimization regularized by sparsity and low-rank penalties. Ideally the sparsity and low-rank penalties should be represented by the l₀-norm and the rank of a matrix; however both are NP hard penalties. The previous studies used the convex l₁-norm as a surrogate for the l₀-norm and the non-convex Schatten-q norm (0 < q ≤ 1) as a surrogate for the rank of matrix. Following past research in sparse recovery, we know that non-convex l_p-norm (0 < p ≤ 1) is a better substitute for the NP hard l₀-norm than the convex l₁-norm. Motivated by these studies, we propose improvements over the previous studies by replacing the l₁-norm sparsity penalty by the l_p-norm. Thus, we reconstruct the dynamic MRI sequence by solving a least squares minimization problem regularized by l_p-norm as the sparsity penalty and Schatten-q norm as the low-rank penalty. There are no efficient algorithms to solve the said problems. In this paper, we derive efficient algorithms to solve them. The experiments have been carried out on Dynamic Contrast Enhanced (DCE) MRI datasets. Both quantitative and qualitative analysis indicates the superiority of our proposed improvement over the existing methods.     

Non-convex algorithm for sparse and low-rank recovery: Application to dynamic MRI reconstruction

In this work we exploit two assumed properties of dynamic MRI in order to reconstruct the images from under-sampled K-space samples. The first property assumes the signal is sparse in the x-f space and the second property assumes the signal is rank-deficient in the x-t space. These assumptions lead to an optimization problem that requires minimizing a combined l_p-norm and Schatten-p norm. We propose a novel FOCUSS based approach to solve the optimization problem. Our proposed method is compared with state-of-the-art techniques in dynamic MRI reconstruction. Experimental evaluation carried out on three real datasets shows that for all these datasets, our method yields better reconstruction both in quantitative and qualitative evaluation.     

Single-image super-resolution with joint-optimization of TV regularization and sparse representation

A super-resolution (SR) reconstruction framework is proposed using regularization restoration combined with learning-based resolution enhancement via sparse representation. With the viewpoint of conventional learning methods, the original image can be split into low frequency (LF) and high frequency (HF) components. The reconstruction mainly focuses on the process of HF part, while the LF one is founded simply by typical interpolation function. For the severely blurred single-image, we first use regularization restoration technology to recover it. Then the regularized output remarkably betters the quality of LF used in traditional learning-based methods. Last, image resolution enhancement with characteristic of edge preserving can implement based on the acquired relatively sharp intermediate image and the pre-constructed over-complete dictionary for sparse representation. Specifically, the regularization can favorably weaken the dependence of atoms on the course of degradation. With both techniques, we can noticeably eliminate the blur and the edge artifacts in the enlarged image simultaneously. Various experimental results demonstrate that the proposed approach can produce visually pleasing resolution for severely blurred image.     

Wavelet frame based scene reconstruction from range data

How to reconstruct the scene (a visible surface) from a set of scattered, noisy and possibly sparse range data is a challenging problem in robotic navigation and computer graphics. As most real scenes can be modeled by piecewise smooth surfaces, traditional surface fitting techniques (e.g. smoothing spline) generally can not preserve sharp discontinuities of surfaces. Based on sparse approximation of piecewise smooth functions in frame domain, we propose a new tight frame based formulation for reconstructing a piecewise smooth surface from a sparse range data set, which is robust to both additive noise and outliers. Furthermore, the resulting minimization problem from our formulation can be efficiently solved by the split Bregman method [1], [2]. The numerical experiments show that the proposed approach is capable of reconstructing a piecewise smooth surface with sharp edges from sparse range data corrupted with noise and outliers.     

Dynamic MRI reconstruction from highly undersampled (k,t)-space data using weighted Schatten p-norm regularizer of tensor

Combination of the low-rankness and sparsity has been successfully used to reconstruct desired dynamic magnetic resonance image (MRI) from highly-undersampled (k, t)-space data. However, nuclear norm, as a convex relaxation of the rank function, can cause the solution deviating from the original solution of low-rank problem. Moreover, equally treating different rank component is not flexible to deal with real applications. In this paper, an efficient reconstruction model is proposed to efficiently reconstruct dynamic MRI. First, we treat dynamic MRI as a 3rd-order tensor, and formulate the low-rankness via non-convex Schatten p-norm of matrices unfolded from the tensor. Secondly, we assign different weight for each rank component in Schatten p-norm. Furthermore, we combine the proposed weighted Schatten p-norm of a tensor as low-rank regularizer, and spatiotemporal total variation as sparse regularizer to formulate the reconstruction model for dynamic MRI. Thirdly, to efficiently solve the formulated reconstruction model, we derive an algorithm based on Bregman iterations with alternating direction multiplier. Over two public data sets of dynamic MRI, experiments demonstrate that the proposed method achieves much better quality.     

Image restoration with dual-prior constraint models based on Split Bregman

In order to utilizing the local and non-local information in the image, we proposed a novel sparse scheme for image restoration in this paper. The new scheme includes two important contributions. The first one is that we extended the image prior model in conventional total variation to the dual-prior models for combining the local smoothness and non-local sparsity under regularization framework. The second one is we developed a modified iterative Split Bregman majorization method to solve the objective function with dual-prior models. The experimental results show that the proposed scheme achieved the state-of-the-art performance compared to the current restoration algorithms.     

基于稀疏先验的空间目标图像盲反演方法

针对图像盲反演算法未考虑空间目标图像自身特性,致使对空间目标图像细节信息恢复不理想、重构图像中易产生边界伪像等不足之处,提出了一种基于稀疏表示的联合稀疏先验约束盲反演算法.首先,结合空间目标图像梯度的稀疏特性,采用图像梯度的L 0范数提取有利于模糊核估计的图像显著边缘信息;其次,采用L p范数和L 0范数对图像的梯度分布和空间域进行稀疏约束,以保证反演图像的像素点间具有显著的对比度,同时保证图像中包含边缘和纹理等细节信息;最后,采用拉普拉斯分布先验对模糊核进行约束,以保证模糊核的稀疏特性.采取交替迭代策略对所提出的模型进行优化求解,从而得到模糊核和空间目标图像的估计值.实验结果表明,相比于几种具有代表性的盲反演算法,提出的方法能估计出更准确的模糊核,对图像边缘和纹理等细节信息具有更好的恢复能力,在主观评价和客观评价方面均取得了较好的反演性能.     

Multi-frame blind deconvolution using sparse priors

In this paper, we propose a method for multi-frame blind deconvolution. Two sparse priors, i.e., the natural image gradient prior and an l₁-norm based prior are used to regularize the latent image and point spread functions (PSFs) respectively. An alternating minimization approach is adopted to solve the resulted optimization problem. We use both gray scale blurred frames from a data set and some colored ones which are captured by a digital camera to verify the robustness of our approach. Experimental results show that the proposed method can accurately reconstruct PSFs with complex structures and the restored images are of high quality.     

A sparse representation method based on kernel and virtual samples for face recognition

To improve the classification accuracy of face recognition, a sparse representation method based on kernel and virtual samples is proposed in this paper. The proposed method has the following basic idea: first, it extends the training samples by copying the left side of the original training samples to the right side to form virtual training samples. Then the virtual training samples and the original training samples make up a new training set and we use a kernel-induced distance to determine M nearest neighbors of the test sample from the new training set. Second, it expresses the test sample as a linear combination of the selected M nearest training samples and finally exploits the determined linear combination to perform classification of the test sample. A large number of face recognition experiments on different face databases illustrate that the error ratios obtained by our method are always lower more or less than face recognition methods including the method mentioned in Xu and Zhu [21], the method proposed in Xu and Zhu [39], sparse representation method based on virtual samples (SRMVS), collaborative representation based classification with regularized least square (CRC_RLS), two-phase test sample sparse representation (TPTSSR), and the feature space-based representation method.     

Symmetric and traceless tensors on Minkowski space

The symmetric traceless projection of a tensor of rank 2l on Minkowski space is determined. These tensors form an invariant subspace under transformations by the 2l-fold product of an element of the Lorentz group SO₀(1, 3). As is well known, this representation is irreducible and equivalent to the representation (l₁,l₂) of the group $\text{SL(}\text{2,C}\text{)}$ for l₁ = l₂ = l. We derive the explicit form of the equivalence operator and its main properties. With these investigations we extend results obtained by S. Weinberg.     

The nonparaxial propagation of the TEM0l doughnut beam with the orbital angular momentum in the far field

The analytical expression for nonparaxial electric field amplitude of the TEM_0l^? doughnut beam with the orbital angular momentum quantum number l is derived in the far field by means of the angular spectrum representation and the stationary phase method. It is shown that the divergent angle of the far field of TEM_0l^? doughnut beam will be smaller with the decreasing of the orbital angular momentum quantum number l or the increasing of the beam waist width w of the initial beam. And the maximal radial intensity of the beam is decreased with its propagation at different rates for different l and w.     

融合自适应稀疏表示和相关系数的高光谱伪装分类方法

目前比较成熟的高光谱成像手段有卫星遥感和航空成像技术,这两种成像方式侦察时间大致相同,入射光方向基本一致,因而地物的光谱曲线比较固定;在陆基条件下,地物的光谱曲线受成像环境的影响凸显,因此应该对适用于陆基条件下的高光谱图像分类方法进行研究。在陆基高光谱图像中,对每个地物进行类型以及种类的判别有利于后续对目标的识别和处理,不同于传统遥感图像分类,陆基条件下的高光谱图像目标分类训练样本不仅较难获得,并且在陆基条件下的高光谱图像中,训练样本之间的相关性随着目标类型、探测器参数以及成像环境等因素时刻发生变化。基于稀疏性表示的分类方法已经被广泛应用于处理图像问题以及各种机器视觉问题。对于陆基高光谱图像来说,基于固定范数约束的稀疏编码策略无法适应陆基条件下高光谱成像多变的环境,而自适应稀疏表示可以根据样本相关性自适应的调节范数约束,相关系数可以提高图像中的破坏因素（阴影、噪声点等）的识别精度。通过引入正则化参数,融合了自适应稀疏表示和相关系数,提出了一种新的高光谱图像分类方法。为了验证所提方法的有效性,分别在绿色植被背景和荒漠背景中设置伪装物,通过不同的分类方法对图像进行分类,实验结果表明,不管是分类精度还是分类一致性,该方法都有明显的优势,可以应用于陆基条件下的高光谱图像分类,为目标分类提供了理论基础。     

Split bregman method-based background extraction for blob-mura defect detection in thin film transistor-liquid crystal display image

In this paper, we present the results of using a split Bregman method for blob-Mura defect detection in a thin-film transistor-liquid crystal display (TFT-LCD) image. A reference image obtained by the split Bregman method was simply subtracted from a test image to detect blob-Mura defects. For a test image, two resulting images containing black or white bob-Mura defects were obtained separately. Through simulation it was verified that the proposed method has a superior capability for detecting blob-Mura defects.     

基于K-SVD和残差比的低信噪比图像稀疏表示去噪算法

针对低信噪比图像去噪问题,提出了一种基于K-SVD(Singular Value Decomposition)和残差比(Residual Ratio Iteration Termination)的正交匹配追踪(Orthogonal Matching Pursuit,OMP)图像稀疏分解去噪算法。该算法利用K-SVD算法将离散余弦变换(Discrete cosine transform,DCT)框架产生的冗余字典训练成能够有效反映图像结构特征的超完备字典,以实现图像的有效表示。然后以残差比作为OMP算法迭代的终止条件来实现图像的去噪。实验表明,该算法相对于传统基于Symlets小波图像去噪、基于Contourlet变换的图像去噪,以及基于DCT冗余字典的稀疏表示图像去噪,能够更加有效地滤除低信噪比图像中的高斯白噪声,保留原图像的有用信息。     

Joint L1/Lp-regularized minimization in video recovery of remote sensing based on compressed sensing

L₁ regularization and L_p regularization are proposed for processing recovered images based on compressed sensing (CS). L₁ regularization can be solved as a convex optimization problem but is less sparse than L_p (0 < p < 1). L_p regularization is sparser than L₁ regularization but is more difficult to solve. This paper proposes joint L₁/L_p (0 < p < 1) regularization, which combines L_p regularization and L₁ regularization. This joint regularization is applied to recover video of remote sensing based on CS. Joint regularization is sparser than L₁ regularization but is as easy to solve as L₁ regularization. A linearized Bregman reweighted iteration algorithm is proposed to solve the joint L₁/L_p regularization problem. The performance and capabilities of the linearized Bregman algorithm and linearized Bregman reweighted algorithm for solving the joint L₁/L_p regularization model are analyzed and compared through numerical simulations.