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.’     

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.     

Adaptive fixed-point iterative shrinkage/thresholding algorithm for MR imaging reconstruction using compressed sensing

Recently compressed sensing (CS) has been applied to under-sampling MR image reconstruction for significantly reducing signal acquisition time. To guarantee the accuracy and efficiency of the CS-based MR image reconstruction, it necessitates determining several regularization and algorithm-introduced parameters properly in practical implementations. The regularization parameter is used to control the trade-off between the sparsity of MR image and the fidelity measures of k-space data, and thus has an important effect on the reconstructed image quality. The algorithm-introduced parameters determine the global convergence rate of the algorithm itself. These parameters make CS-based MR image reconstruction a more difficult scheme than traditional Fourier-based method while implemented on a clinical MR scanner. In this paper, we propose a new approach that reveals that the regularization parameter can be taken as a threshold in a fixed-point iterative shrinkage/thresholding algorithm (FPIST) and chosen by employing minimax threshold selection method. No extra parameter is introduced by FPIST. The simulation results on synthetic and real complex-valued MRI data show that the proposed method can adaptively choose the regularization parameter and effectively achieve high reconstruction quality. The proposed method should prove very useful for practical CS-based MRI applications.     

Non-Iterative Regularized reconstruction Algorithm for Non-CartesiAn MRI: NIRVANA

We introduce a novel noniterative algorithm for the fast and accurate reconstruction of nonuniformly sampled MRI data. The proposed scheme derives the reconstructed image as the nonuniform inverse Fourier transform of a compensated dataset. We derive each sample in the compensated dataset as a weighted linear combination of a few measured k-space samples. The specific k-space samples and the weights involved in the linear combination are derived such that the reconstruction error is minimized. The computational complexity of the proposed scheme is comparable to that of gridding. At the same time, it provides significantly improved accuracy and is considerably more robust to noise and undersampling. The advantages of the proposed scheme makes it ideally suited for the fast reconstruction of large multidimensional datasets, which routinely arise in applications such as f-MRI and MR spectroscopy. The comparisons with state-of-the-art algorithms on numerical phantoms and MRI data clearly demonstrate the performance improvement.     

Generalized analytic signal associated with linear canonical transform

Analytic signal is tightly associated with Hilbert transform and Fourier transform. The linear canonical transform is the generalization of many famous linear integral transforms, such as Fourier transform, fractional Fourier transform and Fresnel transform. Based on the parameter (a, b)-Hilbert transform and the linear canonical transform, in this paper, we develop some issues on generalized analytic signal. The generalized analytic signal can suppress the negative frequency components in the linear canonical transform domain. Furthermore, we prove that the kernel function of the inverse linear canonical transform satisfies the generalized analytic condition and get the generalized analytic pairs. We show the generalized Bedrosian theorem is valid in the linear canonical transform domain.     

基于多级微反射镜的时空联合调制傅里叶变换成像光谱仪:原理及数据处理

介绍了一种基于多级阶梯微反射镜的时空联合调制傅里叶变换成像光谱仪的原理及数据处理方法.仪器利用一块多级阶梯微反射镜取代传统迈克尔逊干涉仪中的动镜以实现静态干涉,通过摆镜扫描使目标物体成像在不同的子阶梯反射面上从而获得目标物体不同光程差的干涉信息.某一时刻,目标物体经摆镜与前置成像系统后在平面镜与多级阶梯微反射镜上形成两个一次像点,两个一次像点被平面镜和多级阶梯微反射镜反射之后经后置成像系统最终成像在探测器焦平面上.平面镜与多级阶梯微反射镜之间的高度差会使到达探测器的两束光的光程差不同,因此探测器焦平面上可以获得目标物体的二维空间信息及一维干涉信息.根据多级阶梯微反射镜参数及光学系统设计参数计算得到摆镜步进角度为0.095°.利用实验获得的三维数据立方体进行了图像拼接与光谱复原.针对子阶梯反射镜存在宽度差异的问题,提出了一种基于极坐标霍夫变换的图像分割方法.为缓解拼接全景图中的间断线效应,将图像变换到HSI颜色空间并插值拟合其亮度分量后再变换回原空间.对拼接后的干涉图像进行了降维、去直流、寻址、切趾、相位校正、傅里叶变换及光谱分辨率增强等处理,完成了光谱复原工作.复原光谱分辨率为194 cm-1,优于设计指标(250 cm-1).     

分数傅里叶变换计算全息

在计算全息和分数傅里叶变换的基础上提出了不对称分数傅里叶变换计算全息和双随机相位不对称分数傅里叶变换计算全息。在这种方法中,首先用一随机相位函数乘以输入图像信息,然后沿x方向实施α级次的一维分数傅里叶变换,再乘以第二个随机相位函数,最后,沿y方向实施β级次的一维分数傅里叶变换。采用迂回位相编码法对变换后的结果编码,绘出计算全息图。为了恢复原始图像,需要知道变换级次和随机相位函数。利用这种方法进行图像加密,使加密图像的密钥由原来两重增加到四重,从而提高了系统的保密性能。     

二维小波变换及在青藏高原地形重构中的应用

计算和检验了4种不同小波基底和不同阶小波基底对单脉冲和尖峰两个信号的分解及重构的性质,结果表明双正交小波的精度最高,Coifman小波的精度最低.通过设阈值对小波和快速Fourier变换进行截断后再重构,说明小波具有重构精度高、所用非零系数少以及误差被局域化等优点,这就削弱了"Gibbs现象",限制了它的影响范围.在此基础上,用二维快速Fourier和二维离散小波变换对实际青藏高原及其附近地形做了分解、重构及截断后重构等,说明二维小波变换能以较之FFT更少的非零系数和误差局域化等重构大地形,表明了二维小波变换在未来大气环流数值模式中的应用潜力.     

非高斯环境下的深度学习脉冲信号去噪与重构*

侧扫声呐进行沉底小目标探测时,底混响是主要背景干扰。底混响通常是一种非平稳、非高斯的带限噪声,它使得白噪声条件下的滤波器性能受到限制。在混响背景下常利用自回归模型对接收信号进预行白化处理,但对于实际侧扫声呐应用,白化后直接匹配滤波的处理效果不甚理想。针对此问题,在自回归模型预白化的基础上,提出采用一种次最佳检测与多分辨二分奇异值分解相结合的改进方法。该方法首先对接收信号进行分段处理,利用改进Burg算法估计每段数据自回归模型的系数及阶数;然后构造白化滤波器对分段数据预白化,并对白化后的数据进行多分辨二分奇异值分解;最后应用ostu方法对原始声图和处理后的声图进行目标检测。仿真与实验结果表明,该方法明显提高了信混比,改善了侧扫声呐沉底静态小目标的成图质量,有利于后期实现基于图像的目标自动检测。     

Finite-aperture effects for Fourier transform systems with convergent illumination. Part II: 3-D system analysis

One of the most important optical signal processing operations is the optical Fourier transform (OFT). Of the arrangements for implementation of the OFT, perhaps the most flexible is that for the scaled optical Fourier transform (SOFT), as it allows control over the scale of the output Fourier transform distribution. By means of an analysis in cylindrical coordinates, we examine some of the practical limits introduced by the use of a thin lens of finite aperture in the implementation of the SOFT. Using simple rules of thumb that are based on an examination of the phase and magnitude deviations from the ideal (infinite-lens) diameter case, we define a volume inside the geometric shadow, which we refer to as a sub-geometric shadow. We then show that inside this sub-geometric shadow errors introduced by diffraction can be quantified.     

A radial propagator for axisymmetric pressure fields

The concept of a propagator is central to the angular spectrum formulation of diffraction theory, which expresses the pressure field diffracted by a two-dimensional aperture as a superposition of a continuum of plane waves. In the conventional form, an exponential term, known as a propagator, is multiplied by the wavenumber spectrum obtained from a two-dimensional spatial Fourier transform of the aperture boundary condition, to obtain the wavenumber spectrum in a plane parallel to the boundary, offset by some distance specified in the propagator. By repeated use of this propagator and Fourier inversion, it is possible to completely construct the homogeneous part of the pressure field in the positive half-space beyond the planar boundary containing the aperture. Drawing upon preceding work relating the boundary condition to the axial pressure [Pees, J. Acoust. Soc. Am. 127(3), 1381-1390 (2010)], it is shown in this article that when the aperture is axially symmetric, an alternative type of propagator can be derived that propagates an axial wavenumber spectrum away from the axis of the aperture. Use of this radial propagator can be computationally advantageous since it allows for field construction using one-dimensional Fourier transforms instead of Hankel transforms or two-dimensional Fourier transforms.     

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.     

傅里叶计算全息图的数字再现及混叠的消除

提出了一种傅里叶计算全息图数字再现中混叠消除的方法。通过对傅里叶变换计算全息图再现过程的分析,在空域扩展待编码图像的范围,实现了数字图像的傅里叶Burch型计算全息图的制作。对编码的实值图像作逆傅里叶变换,可以准确地重建原图像。该编码方法解决了再现过程中图像混叠的问题,适用于对所有灰度图像的编码。实验结果表明,提出的编码方法可以得到清晰的再现图像。     

Effects of permeable boundaries on the diffusion-attenuated MR signal: insights from a one-dimensional model

The local magnetization distribution M(x,t) and the net MR signal S arising from a one-dimensional periodic structure with permeable barriers in a Tanner-Stejskal pulsed-field gradient experiment are considered. In the framework of the narrow pulse approximation, the general expressions for M(x,t) and S as functions of diffusion time and the bipolar field gradient strength are obtained and analyzed. In contrast to a system with impermeable boundaries, the signal S as a function of the b-value is modeled well as a bi-exponential decay not only in the short-time regime but also in the long-time regime. At short diffusion times, the local magnetization M(x,t) is strongly spatially inhomogeneous and the two exponential components describing S have a clear physical interpretation as two "population fractions" of the slow- and fast-diffusing quasi-compartments (pools). In the long-diffusion time regime, the two exponential components do not have clear physical meaning but rather serve to approximate a more complex functional signal form. The average diffusion propagator, obtained by means of standard q-space analysis procedures in the long-diffusion time regime is explored; its structure creates the deceiving appearance of a system with multiple compartments of different sizes, while in reality, it reflects the permeable nature of boundaries in a system with multiple compartments all of the same size.     

数字无透镜傅里叶变换全息术中非傍轴及离焦像差的校正

对数字无透镜傅里叶变换全息图直接采用逆傅里叶变换进行物场的数值重建时.需要满足两个条件:第一,全息图的记录过程必须满足傍轴近似条件,否则再现过程中会产生非傍轴像差;第二,记录全息图时物平面与参考点光源到全息图记录平面的距离必须相等,否则再现过程中会产生离焦像差.理论分析了非傍轴及离焦记录条件下数字无透镜傅里叶变换全息图的灰度分布特点,并提出了相应的非傍轴及离焦像差的数值校正方法.根据实际的非傍轴或离焦记录情况.分别给所记录的数字全息图灰度分布矩阵乘以适当的非傍轴或离焦校正因子,以消除灰度矩阵中非傍轴或离焦因素的影响.然后再对校正后的伞息图灰度矩阵做逆傅里叶变换处理.即可得到准确的数字再现像.实验结果表明.该数值重建方法能够有效地消除无透镜傅里叶变换全息术中数字再现像的非傍轴像差及离焦像差,提高再现像的质量.     

相干场成像全相位目标直接重构法

回波信号处理和目标重构算法是相干场成像中的核心数据处理技术, 直接影响系统的成像质量. 基于全相位谱分析理论, 提出一种新的系统融合处理算法, 对接收端回波信号直接提取全相位谱相位及幅值信息实现目标图像重构, 能够有效抑制各种因素带来的频率误差. 经室外实验系统验证, 成像能力大大优于传统重构算法, 重构目标分辨率接近理论极限值.     

Multi component LFM signal detection and parameter estimation based on EEMD–FRFT

This paper studied multi component LFM signal detection and parameter estimation under the noise circumstance of various signal-to-noise ratios. Based on the analysis of fractional Fourier transform detection and parameter estimation on simple component LFM signal, this paper proposed the method of multi component LFM signal detection and parameter estimation based on EEMD–FRFT (Ensemble Empirical Mode Decomposition–Fractional Fourier transform), and this method was that with the EEMD algorithm, from the frequency domain decompose the analyzable signal to narrow-bandwidth components, whose center frequency changed from high to low, then accurately estimate the parameter and detect the signal of each component out of the pseudo-component with FrFT. This method solved the problem of mode aliasing of signal decomposition; meanwhile, the problem of detecting the multi component LFM signal would be simplified as the problem of one-dimensional search in small scope, which could reduce the amount of operation and improved the detection accuracy. A simulation computation for multi component LFM signal of various SNR (signal-to-noise ratios) was made and the result showed that the error of parameter estimation was less than 5% in the case of SNR not less than −10 dB.     

A simple application of compressed sensing to further accelerate partially parallel imaging

Compressed sensing (CS) and partially parallel imaging (PPI) enable fast magnetic resonance (MR) imaging by reducing the amount of k-space data required for reconstruction. Past attempts to combine these two have been limited by the incoherent sampling requirement of CS since PPI routines typically sample on a regular (coherent) grid. Here, we developed a new method, “CS+GRAPPA,” to overcome this limitation. We decomposed sets of equidistant samples into multiple random subsets. Then, we reconstructed each subset using CS and averaged the results to get a final CS k-space reconstruction. We used both a standard CS and an edge- and joint-sparsity-guided CS reconstruction. We tested these intermediate results on both synthetic and real MR phantom data and performed a human observer experiment to determine the effectiveness of decomposition and to optimize the number of subsets. We then used these CS reconstructions to calibrate the generalized autocalibrating partially parallel acquisitions (GRAPPA) complex coil weights. In vivo parallel MR brain and heart data sets were used. An objective image quality evaluation metric, Case-PDM, was used to quantify image quality. Coherent aliasing and noise artifacts were significantly reduced using two decompositions. More decompositions further reduced coherent aliasing and noise artifacts but introduced blurring. However, the blurring was effectively minimized using our new edge- and joint-sparsity-guided CS using two decompositions. Numerical results on parallel data demonstrated that the combined method greatly improved image quality as compared to standard GRAPPA, on average halving Case-PDM scores across a range of sampling rates. The proposed technique allowed the same Case-PDM scores as standard GRAPPA using about half the number of samples. We conclude that the new method augments GRAPPA by combining it with CS, allowing CS to work even when the k-space sampling pattern is equidistant.     

Regularized maximum likelihood algorithm for PET image reconstruction using a detail and edges preserving anisotropic diffusion

The traditional iterative reconstruction algorithms of positron emission tomography cannot effectively suppress the noise in low SNR case. Recently anisotropic diffusion (AD) is introduced into tomography reconstruction, which can improve the reconstructed image. Although AD reconstruction algorithm can suppress noise, it does not perverse the detail edge information accurately, especially the thin edges. In order to solve the problem, we introduce a new anisotropic diffusion term, which can preserve the detail edges effectively, into the maximum likelihood algorithm, and combine with median filter, forming the regularized maximum likelihood algorithm in PET image reconstruction (PML_NewAD). Results of computer simulated demonstrate that compared with the other classical reconstruction algorithms, PML_NewAD not only availably suppress the noise and produce a higher quality image, but also preserve the structure of image's edge excellently.