非精确交替方向总变分最小化重建算法

CT (computed tomography)系统实际应用当中, 经常会出现扫描数据不满足数据完备性条件的情况. 针对不完全角度重建问题的研究, 是目前迭代型算法研究中的一个热点. 一系列基于带有约束的总变分最小化的重建算法近年来在不完全角度重建中取得了较好的效果, 这其中基于交替方向法 (alternating direction method, ADM) 的重建算法表现出更好的性能. 然而, ADM方法在求解过程中对矩阵求逆的处理效率不高, 导致极大的计算开销. 本文针对该问题, 使用非精确ADM方法, 利用线性近似的方式替换掉计算开销较大的项, 使得矩阵求逆问题可以通过快速傅里叶变换加速实现. 实验结果表明, 本文提出的非精确交替方向总变分最小化重建算法与精确ADM重建算法相比, 没有明显的精度损失, 计算时间缩减30%左右. 关键词： 不完全角度重建 总变分最小化 非精确交替方向法     

结合透射CT的康普顿背散射图像重建

在康普顿散射成像(CST)技术中可以结合透射成像重建出衰减系数来消除散射重建的非线性,但这样得到的投影矩阵带有误差。而CST重建问题的不适定性对噪声和投影矩阵的误差非常敏感,重建结果会有较大误差。针对此问题,基于压缩感知理论提出了一种新的CST重建算法。新方法将图像重建问题归结为一个图像的全变分(TV)最小化问题,并使用收敛速度较快的基于交替方向法的Split-Bregman方法进行求解。在仿真实验中,通过与代数重建技术(ART)进行比较,在测量数据充足和测量数据不足两种情况下,本文算法都具有更好的重建质量,证明了所提算法在重建精度和抗噪性能方面的优势。     

总变差约束的数据分离最小图像重建模型及其Chambolle-Pock求解算法

基于优化的迭代法,可以结合压缩感知和低秩矩阵等稀疏优化技术高精度地重建图像.其中,总变差最小（total variation minimization,TV）模型是一种简单有效的优化模型.传统的约束TV模型,使用数据保真项为约束项,TV正则项为目标函数.本文研究TV约束的、数据分离最小（TV constrained,data divergence minimization,TVcDM）新型TV模型及其求解算法.详细推导了TVcDM模型的Chambolle-Pock（CP）算法,验证了模型及算法的正确性;分析了算法的收敛行为;评估了模型的稀疏重建能力;分析了模型参数的选择对重建的影响及算法参数对收敛速率的影响.研究表明,TVcDM模型有高精度稀疏重建能力;TVcDM-CP算法确保收敛,但迭代过程中有振荡现象;TV限对重建有重要影响,参数值过大会引入噪声而过小会模糊图像细节;算法参数的不同选取会导致不同的收敛速率.     

Reconstruction of band-limited signals from multichannel and periodic nonuniform samples in the linear canonical transform domain

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. This paper addresses the problem of signal reconstruction from multichannel and periodic nonuniform samples in the LCT domain. Firstly, the multichannel sampling theorem (MST) for band-limited signals with the LCT is proposed based on multichannel system equations, which is the generalization of the well-known sampling theorem for the LCT. We consider the problem of reconstructing the signal from its samples which are acquired using a multichannel sampling scheme. For this purpose, we propose two alternatives. The first scheme is based on the conventional Fourier series and inverse LCT operation. The second is based on the conventional Fourier series and inverse Fourier transform (FT) operation. Moreover, the classical Papoulis MST in FT domain is shown to be special case of the achieved results. Since the periodic nonuniformly sampled signal in the LCT has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived MST and the specific space-shifting property of the LCT. Last, the potential applications of the MST are presented to show the advantage of the theory.     

A modified equally sloped algorithm based on the total variation algorithm in computed tomography for insufficient data

Computed tomography (CT) has become an important technique for analyzing the inner structures of material, biological and energy fields. However, there are often challenges in the practical application of CT due to insufficient data. For example, the maximum rotation angle of the sample stage is limited by sample space or image reconstruction from the limited number of views required to reduce the X‐ray dose delivered to the sample. Therefore, it is difficult to acquire CT images with complete data. In this work, an iterative reconstruction algorithm based on the minimization of the image total variation (TV) has been utilized to develop equally sloped tomography (EST), and the reconstruction was carried out from limited‐angle, few‐view and noisy data. A synchrotron CT experiment on hydroxyapatite was also carried out to demonstrate the ability of the TV‐EST algorithm. The results indicated that the new TV‐EST algorithm was capable of achieving high‐quality reconstructions from projections with insufficient data.     

基于全变分最小化和交替方向法的康普顿散射成像重建算法

康普顿散射成像技术利用射线与物质作用后的散射光子信息对物质的电子密度进行成像.与传统的透射成像方式相比,康普顿散射成像具有系统结构灵活、成像对比度高、辐射剂量低等优势,在无损检测、医疗诊断、安全检查等领域有着广阔的应用前景.但其重建问题是一个非线性的逆问题,通常是不适定的,其解对噪声和测量误差非常敏感.为解决此问题,本文结合全变分最小化正则化方法和交替方向法提出了一种新的康普顿散射成像重建算法.该算法首先将问题对应的TV模型转化为与之等价的带约束的优化问题,然后利用增广拉格朗日乘子法将优化问题分解为两个具有解析解的子问题,并通过交替求解子问题使增广拉格朗日函数达到最小,进而得到重建的图像.在仿真实验中,通过与主流的ASD-POCS方法进行对比,证明了该算法在重建精度和重建效率方面的优势.     

Adaptive k-space sampling design for edge-enhanced DCE-MRI using compressed sensing

The critical challenge in dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is the trade-off between spatial and temporal resolution due to the limited availability of acquisition time. To address this, it is imperative to under-sample k-space and to develop specific reconstruction techniques. Our proposed method reconstructs high-quality images from under-sampled dynamic k-space data by proposing two main improvements; i) design of an adaptive k-space sampling lattice and ii) edge-enhanced reconstruction technique. A high-resolution data set obtained before the start of the dynamic phase is utilized. The sampling pattern is designed to adapt to the nature of k-space energy distribution obtained from the static high-resolution data. For image reconstruction, the well-known compressed sensing-based total variation (TV) minimization constrained reconstruction scheme is utilized by incorporating the gradient information obtained from the static high-resolution data. The proposed method is tested on seven real dynamic time series consisting of 2 breast data sets and 5 abdomen data sets spanning 1196 images in all. For data availability of only 10%, performance improvement is seen across various quality metrics. Average improvements in Universal Image Quality Index and Structural Similarity Index Metric of up to 28% and 24% on breast data and about 17% and 9% on abdomen data, respectively, are obtained for the proposed method as against the baseline TV reconstruction with variable density random sampling pattern.     

Sparsity-constrained SENSE reconstruction: An efficient implementation using a fast composite splitting algorithm

Parallel imaging and compressed sensing have been arguably the most successful and widely used techniques for fast magnetic resonance imaging (MRI). Recent studies have shown that the combination of these two techniques is useful for solving the inverse problem of recovering the image from highly under-sampled k-space data. In sparsity-enforced sensitivity encoding (SENSE) reconstruction, the optimization problem involves data fidelity (L2-norm) constraint and a number of L1-norm regularization terms (i.e. total variation or TV, and L1 norm). This makes the optimization problem difficult to solve due to the non-smooth nature of the regularization terms. In this paper, to effectively solve the sparsity-regularized SENSE reconstruction, we utilize a new optimization method, called fast composite splitting algorithm (FCSA), which was developed for compressed sensing MRI. By using a combination of variable splitting and operator splitting techniques, the FCSA algorithm decouples the large optimization problem into TV and L1 sub-problems, which are then, solved efficiently using existing fast methods. The operator splitting separates the smooth terms from the non-smooth terms, so that both terms are treated in an efficient manner. The final solution to the SENSE reconstruction is obtained by weighted solutions to the sub-problems through an iterative optimization procedure. The FCSA-based parallel MRI technique is tested on MR brain image reconstructions at various acceleration rates and with different sampling trajectories. The results indicate that, for sparsity-regularized SENSE reconstruction, the FCSA-based method is capable of achieving significant improvements in reconstruction accuracy when compared with the state-of-the-art reconstruction method.     

High total variation-based method for sparse-view photoacoustic reconstruction

We propose a novel method by combining the total variation（TV） with the high-degree TV（HDTV） to improve the reconstruction quality of sparse-view sampling photoacoustic imaging（PAI）. A weighing function is adaptively updated in an iterative way to combine the solutions of the TV and HDTV minimizations. The fast iterative shrinkage/thresholding algorithm is implemented to solve both the TV and the HDTV minimizations with better convergence rate. Numerical results demonstrate the superiority and efficiency of the proposed method on sparse-view PAI. In vitro experiments also illustrate that the method can be used in practical sparse-view PAI.     

Compressive sampling photoacoustic tomography based on edge expander codes and TV regularization

A new photoacoustic （PA） signal sampling and image reconstruction method, called compressive sampling PA tomography （CSPAT）, is recently proposed to make low sampling rate and high-resolution PA tomogra- phy possible. A key problem within the CSPAT framework is the design of optic masks. We propose to use edge expander codes-based masks instead of the conventional random distribution masks, and efficient total variation （TV） regularization-based model to formulate the associated problem. The edge expander codesbased masks, corresponding to non-uniform sampling schemes, are validated by both theoretical analysis and results from computer simulations. The proposed method is expected to enhance the capability of CSPAT for reducing the number of measurements and fast data acquisition.     

Total variation based gradient descent algorithm for sparse-view photoacoustic image reconstruction

In photoacoustic imaging (PAI), reconstruction from sparse-view sampling data is a remaining challenge in the cases of fast or real-time imaging. In this paper, we present our study on a total variation based gradient descent (TV-GD) algorithm for sparse-view PAI reconstruction. This algorithm involves the total variation (TV) method in compressed sensing (CS) theory. The objective function of the algorithm is modified by adding the TV value of the reconstructed image. With this modification, the reconstructed image could be closer to the real optical energy distribution map. Additionally in the proposed algorithm, the photoacoustic data is processed and the image is updated individually at each detection point. In this way, the calculation with large matrix can be avoided and a more frequent image update can be obtained. Through the numerical simulations, the proposed algorithm is verified and compared with other reconstruction algorithms which have been widely used in PAI. The peak signal-to-noise ratio (PSNR) of the image reconstructed by this algorithm is higher than those by the other algorithms. Additionally, the convergence of the algorithm, the robustness to noise and the tunable parameter are further discussed. The TV-based algorithm is also implemented in the in vitro experiment. The better performance of the proposed method is revealed in the experiments results. From the results, it is seen that the TV-GD algorithm may be a practical and efficient algorithm for sparse-view PAI reconstruction.     

Multichannel sampling expansion in the linear canonical transform domain and its application to superresolution

The linear canonical transform (LCT) describes the effect of first-order quadratic phase optical system on a wave field. In this paper, we address the problem of signal reconstruction from multichannel samples in the LCT domain based on a new convolution theorem. Firstly, a new convolution structure is proposed for the LCT, which states that a modified ordinary convolution in the time domain is equivalent to a simple multiplication operation for LCT and Fourier transform (FT). Moreover, it is expressible by a one dimensional integral and easy to implement in the designing of filters. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, a practical multichannel sampling expansion for band limited signal with the LCT is introduced. This sampling expansion which is constructed by the new convolution structure can reduce the effect of spectral leakage and is easy to implement. Last, the potential application of the multichannel sampling is presented to show the advantage of the theory. Especially, the application of multichannel sampling in the context of the image superresolution is also discussed. The simulation results of superresolution are also presented.     

Anisotropic total variation minimization approach in in-line phase-contrast tomography and its application to correction of ring artifacts

In-line phase-contrast computed tomography(IL-PC-CT) imaging is a new physical and biochemical imaging method.IL-PC-CT has advantages compared to absorption CT when imaging soft tissues. In practical applications, ring artifacts which will reduce the image quality are commonly encountered in IL-PC-CT, and numerous correction methods exist to either pre-process the sinogram or post-process the reconstructed image. In this study, we develop an IL-PC-CT reconstruction method based on anisotropic total variation(TV) minimization. Using this method, the ring artifacts are corrected during the reconstruction process. This method is compared with two methods: a sinogram preprocessing correction technique based on wavelet-FFT filter and a reconstruction method based on isotropic TV. The correction results show that the proposed method can reduce visible ring artifacts while preserving the liver section details for real liver section synchrotron data.     

一种光学层析图像的多准则重建方法

光学层析成像是一个病态重建问题,为克服重建过程的病态性,提出将多准则优化理论引入到图像重建中。利用了三个用于光学层析图像重建的准则:平方误差函数、图像熵函数和局部平滑函数。采用向量优化方法将多准则优化问题转化为单准则优化问题求解。为了确定各个目标函数间的权重系数,提出一种动态权重系数求解方法。重建过程目标函数关于光学参量的梯度计算是关键,因此提出一种基于梯度树的计算方法。实验过程中对多准则重建结果和基于平方误差函数的单准则重建结果做了比较,证明该方法能够克服传统的偏重单一目标的单准则重建的不足,有效地重建光学层析图像,提高图像重建质量。     

Fast multi-contrast MRI reconstruction

Multi-contrast magnetic resonance imaging (MRI) is a useful technique to aid clinical diagnosis. This paper proposes an efficient algorithm to jointly reconstruct multiple T1/T2-weighted images of the same anatomical cross section from partially sampled k-space data. The joint reconstruction problem is formulated as minimizing a linear combination of three terms, corresponding to a least squares data fitting, joint total variation (TV) and group wavelet-sparsity regularization. It is rooted in two observations: 1) the variance of image gradients should be similar for the same spatial position across multiple contrasts; 2) the wavelet coefficients of all images from the same anatomical cross section should have similar sparse modes. To efficiently solve this problem, we decompose it into joint TV regularization and group sparsity subproblems, respectively. Finally, the reconstructed image is obtained from the weighted average of solutions from the two subproblems, in an iterative framework. Experiments demonstrate the efficiency and effectiveness of the proposed method compared to existing multi-contrast MRI methods.     

Novel Fourier-based iterative reconstruction for sparse fan projection using alternating direction total variation minimization

Sparse-view x-ray computed tomography(CT) imaging is an interesting topic in CT field and can efficiently decrease radiation dose. Compared with spatial reconstruction, a Fourier-based algorithm has advantages in reconstruction speed and memory usage. A novel Fourier-based iterative reconstruction technique that utilizes non-uniform fast Fourier transform(NUFFT) is presented in this work along with advanced total variation(TV) regularization for a fan sparse-view CT. The proposition of a selective matrix contributes to improve reconstruction quality. The new method employs the NUFFT and its adjoin to iterate back and forth between the Fourier and image space. The performance of the proposed algorithm is demonstrated through a series of digital simulations and experimental phantom studies. Results of the proposed algorithm are compared with those of existing TV-regularized techniques based on compressed sensing method, as well as basic algebraic reconstruction technique. Compared with the existing TV-regularized techniques, the proposed Fourier-based technique significantly improves convergence rate and reduces memory allocation, respectively.     

Few-Projection Tomography. I. Radio-Astronomical Approach for the Problem and the 2-CLEAN DSA Method

We develop the radio-astronomical approach for solving the few-projection tomography problem. It is shown that the 2-CLEAN DSA method proposed for determination of the permissible solution area is an efficient way for solving this problem. The method is based on solving the deconvolution problem with allowance for the synthesized beam or the synthesized Green's function. The distortions due to the sidelobe responses of the synthesized transfer function are eliminated by using two realizations of the well-known iterative radio-astronomical CLEAN algorithm with nonlinear constraints. The proposed 2-CLEAN DSA method allows one to decrease the number of projections required for two-dimensional reconstruction by a factor of about 10 as compared with the conventional tomography approach, provided that a wide spatial-frequency spectrum limited from above is reconstructed. The method can easily be adapted to introducing additional constraints. Examples of astrotomography reconstruction are presented. We show that the proposed method is promising for a large number of remote sensing applications and compare it with other well-known reconstruction techniques. The papers by radio astronomers, who contributed significantly to the development of components of the method, are pointed out.     

基于加权TV正则化的X射线CT系统能谱估计方法

在CT硬化伪影校正、双能CT图像重建以及CT辐射剂量计算等实际应用中,X射线的能谱信息具有重要的作用。然而,由于透射测量方程组系数矩阵的病态性、X光子的统计涨落和噪声的干扰,使得EM等能谱估计方法难以获得较精细的能谱刻画。针对该问题,提出一种基于加权TV正则化的X射线CT系统能谱估计方法。首先采用能谱能量范围内带有不同K-edge的材料作为体模以降低投影测量方程之间的相关性。然后,利用CT成像系统的几何参数来获取投影测量数据所对应的准确透射长度信息来减小投影方程的测量误差。最后,综合利用透射衰减测量数据的保真性、轫致辐射能谱部分的连续性、特征辐射能谱部分的离散性、能谱的非负性和归一性以及平均有效衰减系数等信息,使用加权TV正则化方法建立目标函数,利用正则化理论中的L曲线准则通过黄金分割参数搜索策略求得最优正则化参数及对应的能谱估计结果。分别通过不同统计波动模型下的仿真能谱和实际测量数据对算法性能进行了验证,结果表明该方法与EM等方法相比,不受初始能谱的影响,有效提高了能谱估计结果的稳定性和准确性。     

基于广义变分正则化的闪光照相图像重建算法

 针对闪光照相图像信噪比低的特点,提出了一种基于广义变分正则化的图像重建算法,该方法采用p-范数取代目前广泛采用的全变分范数作为正则项,构造了用于图像重建的展平泛函,将图像重建问题转化为目标泛函最优化问题,采用固定点迭代法求解图像重建的最优解。数值计算结果表明,该算法在重建过程中能够有效抑制图像噪声,并加大对图像边缘的保持能力,从而提高了图像重建质量,是一种有效且性能优良的闪光照相图像重建算法。     

基于径向基函数神经网络的线性衰减系数的重建算法

层析γ扫描(TGS)技术是非破坏性分析(NDA)中的一项重要技术.在TGS透射测量中,线性衰减系数值的图像重建问题是TGS的难点和核心问题.在文[1]的基础上,提出了将神经网络方法应用于TGS重建线性衰减系数图像的算法.计算机上的仿真模拟结果表明,在一定范围内,径向基函数(RBF)神经网络方法重建的线性衰减系数值与实际值的相对误差小于4%,且具有快速、高精度等优点,表明了此方法的有效性.