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.     

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.     

Novel classification rule of two-phase test sample sparse representation

In this paper, a novel classifier based on two-phase test sample sparse representation (TPTSSR) classifier and coarse k nearest neighbor (C-kNN) classifier, called novel classification rule of two-phase test sample sparse representation (NCR-TPTSSR) classifier, is proposed for image recognition. Being similar to TPTSSR classifier and C-kNN classifier, NCR-TPTSSR classifier also uses the two phases to classify the test sample. However, the classification rule of NCR-TPTSSR classifier is different to the decision rule of TPTSSR classifier and C-kNN classifier. A large number of experiments on FERET face database, AR face database, JAFFE face database and PolyU FKP database are used to evaluate the proposed algorithm. The experimental results demonstrate that the proposed method achieves better recognition rate than TPTSSR classifier, C-kNN classifier, nearest feature center (NFC) classifier, nearest feature line (NFL) classifier, nearest neighbor (NN) and so on.     

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.     

Pairwise Elastic Net Representation-Based Classification for Hyperspectral Image Classification

The representation-based algorithm has raised a great interest in hyperspectral image (HSI) classification. l1-minimization-based sparse representation (SR) attempts to select a few atoms and cannot fully reflect within-class information, while l2-minimization-based collaborative representation (CR) tries to use all of the atoms leading to mixed-class information. Considering the above problems, we propose the pairwise elastic net representation-based classification (PENRC) method. PENRC combines the l1-norm and l2-norm penalties and introduces a new penalty term, including a similar matrix between dictionary atoms. This similar matrix enables the automatic grouping selection of highly correlated data to estimate more robust weight coefficients for better classification performance. To reduce computation cost and further improve classification accuracy, we use part of the atoms as a local adaptive dictionary rather than the entire training atoms. Furthermore, we consider the neighbor information of each pixel and propose a joint pairwise elastic net representation-based classification (J-PENRC) method. Experimental results on chosen hyperspectral data sets confirm that our proposed algorithms outperform the other state-of-the-art algorithms.     

An algorithm for computed tomography image reconstruction from limited-view projections

<正>With the development of the compressive sensing theory,the image reconstruction from the projections viewed in limited angles is one of the hot problems in the research of computed tomography technology.This paper develops an iterative algorithm for image reconstruction,which can fit most cases.This method gives an image reconstruction flow with the difference image vector,which is based on the concept that the difference image vector between the reconstructed and the reference image is sparse enough.Then the l_2-norm minimization method is used to reconstruct the difference vector to recover the image for flat subjects in limited angles.The algorithm has been tested with a thin planar phantom and a real object in limited-view projection data.Moreover,all the studies showed the satisfactory results in accuracy at a rather high reconstruction speed.     

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.     

Weighted Schatten p-Norm Low Rank Error Constraint for Image Denoising

Traditional image denoising algorithms obtain prior information from noisy images that are directly based on low rank matrix restoration, which pays little attention to the nonlocal self-similarity errors between clear images and noisy images. This paper proposes a new image denoising algorithm based on low rank matrix restoration in order to solve this problem. The proposed algorithm introduces the non-local self-similarity error between the clear image and noisy image into the weighted Schatten p-norm minimization model using the non-local self-similarity of the image. In addition, the low rank error is constrained by using Schatten p-norm to obtain a better low rank matrix in order to improve the performance of the image denoising algorithm. The results demonstrate that, on the classic data set, when comparing with block matching 3D filtering (BM3D), weighted nuclear norm minimization (WNNM), weighted Schatten p-norm minimization (WSNM), and FFDNet, the proposed algorithm achieves a higher peak signal-to-noise ratio, better denoising effect, and visual effects with improved robustness and generalization.     

Split Bregman iteration solution for sparse optimization in image restoration

It is always a challenging task to develop effective and accurate models for robust image restoration. In this paper, the family of sparse and redundant representation frameworks is considered as an alternative for the above problem. The principle of the family is expatiated on the development and research progress. Two well-known denoising methods are presented and analyzed on their properties. The K-SVD algorithm is an effective method for sparse representation. The iteratively approximate algorithms are always used for the solution of sparse coding operations. Here, a convexification of the l₀ norm to the l₁ norm is adopted in the implementation of K-SVD method. Then a split Bregman iteration solution is proposed for l₁ regularization problems in the performance of the sparse representation of the K-SVD algorithm. The split Bregman iterative method is well studied and fused into the famous K-SVD method. The PSNR (Peak Signal to Noise Ratio) and MSSIM (Mean Structural Similarity) are used to evaluate the performance of those methods. Experimental results on different types of images indicate that our proposed method not only achieve comparable results with the state of art methods, but also make the original method more efficient. Besides, it also provides a valuable and promising reference for image restoration techniques.     

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.     

Exploiting rank deficiency and transform domain sparsity for MR image reconstruction

The reconstruction of magnetic resonance (MR) images from the partial samples of their k-space data using compressed sensing (CS)-based methods has generated a lot of interest in recent years. To reconstruct the MR images, these techniques exploit the sparsity of the image in a transform domain (wavelets, total variation, etc.). In a recent work, it has been shown that it is also possible to reconstruct MR images by exploiting their rank deficiency. In this work, it will be shown that, instead of exploiting the sparsity of the image or rank deficiency alone, better reconstruction results can be achieved by combining transform domain sparsity with rank deficiency.To reconstruct an MR image using its transform domain sparsity and its rank deficiency, this work proposes a combined l₁-norm (of the transform coefficients) and nuclear norm (of the MR image matrix) minimization problem. Since such an optimization problem has not been encountered before, this work proposes and derives a first-order algorithm to solve it.The reconstruction results show that the proposed approach yields significant improvements, in terms of both visual quality as well as the signal to noise ratio, over previous works that reconstruct MR images either by exploiting rank deficiency or by the standard CS-based technique popularly known as the ‘Sparse MRI.’     

结合稀疏编码和空间约束的红外图像聚类分割研究

提出了结合稀疏编码和空间约束的红外图像聚类分割新算法, 在稀疏编码的基础上融合聚类算法, 扩展了传统的基于K-means聚类的图像分割方法. 结合稀疏编码的聚类分割算法能有效融合图像的局部信息, 便于利用像素之间的内在相关性, 但是对于分割会出现过分割和像素难以归类的问题.为此, 在字典的学习过程中, 将原子的聚类算法引入其中, 有助于缩减字典中原子所属类别的数目, 防止出现过分割; 考虑到像素及其邻域像素具有类别属性一致性的特点, 引入了空间类别属性约束信息, 并给出了一种交替优化算法. 联合学习字典、稀疏系数、聚类中心和隶属度, 将稀疏编码系数同原子对聚类中心的隶属程度相结合, 构造像素归属度来判断像素所属的类别. 实验结果表明, 该方法能够有效提高红外图像重要区域的分割效果, 具有较好的鲁棒性. 关键词： 图像分割 稀疏编码 聚类 空间约束     

Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography

In this paper, an efficient l₁-regularized reconstruction method named the primal-dual interior-point (PDIP) method is presented for three-dimensional bioluminescence tomography (BLT) based on the adaptive finite element framework. Taking into account the sparse characteristic of the bioluminescent source, the BLT inverse problem is considered to be a linear programming problem and is represented by its primal and dual form. The source localization and quantification are obtained by solving the primal-dual Newton equation system. The comparisons between PDIP and the classical conjugate gradient least square (CGLS) algorithm are implemented to validate our method. Results from numerical simulation and an in vivo mouse experiment demonstrate the credibility and the potential of the proposed method in practical BLT reconstruction.     

Sparse representation-based classification algorithm for optical Tibetan character recognition

A sparse representation-based two-phase classification algorithm is proposed for off-line handwritten Tibetan character recognition. The first phase realizes coarse classification with the K-NN classifier by finding the K nearest neighbours of a test sample in the dictionary constructed by K-SVD with samples of all classes, and the classes of these neighbours are regarded as the candidate classes of the test sample. The second phase performs fine classification with the sparse representation classifier by sparsely representing the test sample with all elements of the dictionary constructed by K-SVD with samples of all candidate classes, and the test sample is finally classified into the candidate class with the maximal contribution in sparse representation. Experiments on the Tibetan off-line handwritten character database show that an optimal recognition rate of 97.17% has been reached and it is 2.12% higher than that of K-NN.     

Laplacian regularized kernel minimum squared error and its application to face recognition

Kernel minimum squared error (KMSE) has been receiving much attention in data mining and pattern recognition in recent years. Generally speaking, training a KMSE classifier, which is a kind of supervised learning, needs sufficient labeled examples. However, labeled examples are usually insufficient and unlabeled examples are abundant in real-world applications. In this paper, we introduce a semi-supervised KMSE algorithm, called Laplacian regularized KMSE (LapKMSE), which explicitly exploits the manifold structure. We construct a p nearest neighbor graph to model the manifold structure of labeled and unlabeled examples. Then, LapKMSE incorporates the structure information of labeled and unlabeled examples in the objective function of KMSE by adding a Laplacian regularization term. As a result, the labels of labeled and unlabeled examples vary smoothly along the geodesics on the manifold. Experimental results on several synthetic and real-world datasets illustrate the effectiveness of our algorithm. Finally our algorithm is applied to face recognition and achieves the comparable results compared to the other supervised and semi-supervised methods.     

An algorithm for sparse MRI reconstruction by Schatten p-norm minimization

In recent years, there has been a concerted effort to reduce the MR scan time. Signal processing research aims at reducing the scan time by acquiring less K-space data. The image is reconstructed from the subsampled K-space data by employing compressed sensing (CS)-based reconstruction techniques. In this article, we propose an alternative approach to CS-based reconstruction. The proposed approach exploits the rank deficiency of the MR images to reconstruct the image. This requires minimizing the rank of the image matrix subject to data constraints, which is unfortunately a nondeterministic polynomial time (NP) hard problem. Therefore we propose to replace the NP hard rank minimization problem by its nonconvex surrogate — Schatten p-norm minimization. The same approach can be used for denoising MR images as well.Since there is no algorithm to solve the Schatten p-norm minimization problem, we derive an efficient first-order algorithm. Experiments on MR brain scans show that the reconstruction and denoising accuracy from our method is at par with that of CS-based methods. Our proposed method is considerably faster than CS-based methods.     

基于稀疏表示模型和自回归模型的高光谱分类

针对高光谱分类中对光谱信息和空间信息利用不足的问题,提出了一种基于稀疏表示模型和自回归模型相结合的分类算法。该算法利用稀疏表示模型和自回归模型,设计联合字典:在光谱维上,利用稀疏表示模型将高光谱的每个光谱向量表示为字典中训练样本的稀疏线性组合;在空间维上,利用自回归模型对每个光谱向量的8邻域进行约束。针对不同样本分别构造一个字典,在减少计算量的同时减小重构误差,最后在最小重构误差和邻域相关性的约束下求解稀疏表示问题,以最小重构误差为准则实现高光谱数据的分类。仿真结果表明,该方法能够有效地提高高光谱数据的分类精度。     

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.     

Improved l1-SPIRiT using 3D walsh transform-based sparsity basis

l1-SPIRiT is a fast magnetic resonance imaging (MRI) method which combines parallel imaging (PI) with compressed sensing (CS) by performing a joint l1-norm and l2-norm optimization procedure. The original l1-SPIRiT method uses two-dimensional (2D) Wavelet transform to exploit the intra-coil data redundancies and a joint sparsity model to exploit the inter-coil data redundancies. In this work, we propose to stack all the coil images into a three-dimensional (3D) matrix, and then a novel 3D Walsh transform-based sparsity basis is applied to simultaneously reduce the intra-coil and inter-coil data redundancies. Both the 2D Wavelet transform-based and the proposed 3D Walsh transform-based sparsity bases were investigated in the l1-SPIRiT method. The experimental results show that the proposed 3D Walsh transform-based l1-SPIRiT method outperformed the original l1-SPIRiT in terms of image quality and computational efficiency.