基于奇异值分解和中位数绝对偏差的拉曼成像数据去噪方法

拉曼成像是一种无损伤、无需标记的光谱成像技术,它可以提供样品的不同组分的分子指纹信息以及空间分布特征,相比其他成像技术有着更重要的应用。但是拉曼散射的截面积小,灵敏度低,加上在很多实验中为了观察某些组分的动态分布而缩短扫描时间,导致最终得到的成像数据被噪声干扰,因此往往需要对信号进行去噪处理。常规的算法一般都是基于一个给定的数学模型对光谱进行处理,容易造成过滤波,使得信号失真;另外,在处理拉曼成像数据时,常规算法往往是对数据进行逐条光谱去噪,从而忽略了多条光谱之间的相互关系,导致最终的拉曼图像仍然受许多噪点干扰。因此,提出了一种基于奇异值分解和中位数绝对偏差的拉曼成像的信号处理方法,用于拉曼成像数据的去噪处理。该方法首先对拉曼成像数据进行奇异值分解,获得一个奇异值矩阵与两个正交矩阵;然后通过中位数绝对偏差法对奇异值矩阵中的各奇异值进行离群值检测,选取前k个被连续标记的离群值作为要保留的奇异值,并将其余的奇异值赋值为零,得到新的奇异值矩阵;最后用新的奇异值矩阵与两个正交矩阵重新求解得到去噪后的拉曼成像数据。实验中,首先验证了中位数绝对偏差法确定前k个奇异值的正确性,其次分别从处理后的图像质量和信号波形两方面对比了该算法与常规算法的去噪效果。结果证明,中位数绝对偏差法可以快速地确定出合理的k值大小,而且,依据该k值处理后的成像数据不仅在成像质量上消除了大量的噪点,使得组分的空间分布特征清晰可见,也在信号波形上较完美地保留了微小谱峰,并恢复光谱信号。该算法不同于常规算法,能同时对整个拉曼成像数据进行处理,并保留光谱之间的统计特征,是一种更加有效的拉曼成像数据的去噪方法。     

A Nyström method for weakly singular integral operators on surfaces

We describe a modified Nyström method for the discretization of the weakly singular boundary integral operators which arise from the formulation of linear elliptic boundary value problems as integral equations. Standard Nyström and collocation schemes proceed by representing functions via their values at a collection of quadrature nodes. Our method uses appropriately scaled function values in lieu of such representations. This results in a scheme which is mathematically equivalent to Galerkin discretization in that the resulting matrices are related to those obtained by Galerkin methods via conjugation with well-conditioned matrices, but which avoids the evaluation of double integrals. Moreover, we incorporate a new mechanism for approximating the singular integrals which arise from the discretization of weakly singular integral operators which is considerably more efficient than standard methods. We illustrate the performance of our method with numerical experiments.     

Invariant Measures for Passive Scalars in the Small Noise Inviscid Limit

We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel–Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.     

Large dimension forecasting models and random singular value spectra

We present a general method to detect and extract from a finite time sample statistically meaningful correlations between input and output variables of large dimensionality. Our central result is derived from the theory of free random matrices, and gives an explicit expression for the interval where singular values are expected in the absence of any true correlations between the variables under study. Our result can be seen as the natural generalization of the Marčenko-Pastur distribution for the case of rectangular correlation matrices. We illustrate the interest of our method on a set of macroeconomic time series.     

2-D time-harmonic BEM for solids of general anisotropy with application to eigenvalue problems

We present the direct formulation of the two-dimensional boundary element method (BEM) for time-harmonic dynamic problems in solids of general anisotropy. We split the fundamental solution, obtained by Radon transform, into static singular and dynamics regular parts. We evaluate the boundary integrals for the static singular part analytically and those for the dynamic regular part numerically over the unit circle.We apply the developed BEM to eigenvalue analysis. We determine eigenvalues of full non-symmetric complex-valued matrices, depending non-linearly on the frequency, by first reducing them to the generalized linear eigenvalue problem and then applying the QZ algorithm. We test the performance of the QZ algorithm thoroughly in comparison with the FEM solution. The proposed BEM is not only a strong candidate to replace the FEM for industrial eigenvalue problems, but it is also applicable to a wider class of two-dimensional time-harmonic problems.     

基于强影响度的近红外奇异样本识别算法研究

校正样本选择以及奇异样本剔除对于近红外光谱定量和定性建模非常重要。现有的识别奇异样本的方法一般都基于数据重心估计,需要一个经验的判断阈值,在很大程度上限制了其识别准确性和实用性。针对现有方法奇异样本识别准确率低的问题,改进了一种现有度量尺度-杠杆值,构造出一种新的基于强影响度的奇异样本识别算法。这种度量尺度在一定程度上减少了对数据重心的依赖,使正常样本更加聚集,拉开了奇异样本与正常样本的距离;同时,为了避免人工根据经验设定阈值的不合理性,引入统计学领域中跳跃度的概念,提出了一种自动阈值设定方法判别奇异样本。为了验证该算法的有效性,利用马氏距离、杠杆值-光谱残差法与该算法分别对200个代表性校正集样本中的异常样品进行剔除,然后通过偏最小二乘法(PLS)对剩余的校正集样本(以烟碱为指标)定量建模,并对60个代表性测试集样本进行预测,以交互验证均方根误差(RMSECV)、相关系数(r)和预测均方根误差(RMSEP)为评价指标比较各算法的优劣。实验对比结果表明,基于强影响度的奇异样本识别算法较现有方法明显提高了奇异样本识别的准确率,具有较低的RMSECV(0.104),RMSEP(0.112)以及较高的R(0.983),提高了模型的稳定性和预测能力。     

A method for multi-frequency calculation of boundary integral equation in acoustics based on series expansion

The paper presents a method to solve the problem of multi-frequency calculation of Helmholtz boundary integral equation in acoustics. Based on series expansion, system matrices are independent of wavenumber and become the matrix power series of wavenumber. As a result, all matrices in the matrix power series are only dependent on the structure geometry. In addition, an element transform method to calculate the singular integral and Cauchy singular integral is also discussed because the singular integral need to be solved using the method. The convergence of the series expansion method is also proved in this paper. The effectiveness of the method is confirmed by two numerical examples.     

基于迭代TSVD的NMR二维谱反演算法

针对现有基于TSVD的二维反演算法中截断位置判断不准确、容易产生虚假峰等问题,提出了一种改进的方法. 首先,通过逐步求精的方法对L曲线上的拐角位置进行定位,得到迭代的最大截断位置;然后,根据反演核奇异值的集中程度获取迭代的最小截断位置;最后,在给定的截断位置范围内从小到大进行TSVD,每次迭代都以上一次迭代的结果为依据. 对仿真数据和实验数据的反演结果表明,该算法都能够得到比较好的二维反演效果. 与现有基于TSVD的方法相比,该算法具有更高的鲁棒性,能够得到更清晰的二维谱,可满足实际应用需求.     

Optimized quantum singular value thresholding algorithm based on a hybrid quantum computer

Quantum singular value thresholding (QSVT) algorithm, as a core module of many mathematical models, seeks the singular values of a sparse and low rank matrix exceeding a threshold and their associated singular vectors. The existing all-qubit QSVT algorithm demands lots of ancillary qubits, remaining a huge challenge for realization on nearterm intermediate-scale quantum computers. In this paper, we propose a hybrid QSVT (HQSVT) algorithm utilizing both discrete variables (DVs) and continuous variables (CVs). In our algorithm, raw data vectors are encoded into a qubit system and the following data processing is fulfilled by hybrid quantum operations. Our algorithm requires O[log(MN)] qubits with O(1) qumodes and totally performs O(1) operations, which significantly reduces the space and runtime consumption.     

基于等几何分析的边界元法求解Helmholtz问题

将基于一类局部双变量B样条函数的等几何分析方法和Burton-Miller方法相结合,分析三维Helmholtz问题.对于某些从二维参数域映射到三维空间具有奇异点的参数曲面,该方法可以有效地避免奇异点处大量奇异与近奇异积分的计算.数值算例表明该方法具有较好的计算精度和计算效率.复杂问题的分析表明,该方法具有良好的工程应用前景.     

虚拟阵列Khatri-Rao积与子空间联合稀疏表示的DOA估计方法

利用目标信号在空域分布的稀疏性,该文提出了一种基于虚拟阵列Khatri-Rao(KR)积与信号子空间联合稀疏表示的单快拍DOA估计方法;该方法利用单次快拍的采样数据,构造出双向虚拟阵列数据,并对虚拟阵列数据的协方差矩阵进行KR积变换处理,然后对向量化后的数据进行顺序重构,利用重构矩阵的大奇异值对应的左奇异向量为估计信号子空间;最后,利用凸优化工具箱对稀疏模型进行二阶凸规划的优化求解,得到高精度的DOA估计值;仿真实验验证了算法的有效性,在低信噪比下比传统MUSIC和OMP算法具有更高的估计精度。     

Best tree wavelet packet transform based copyright protection scheme for digital images

In this paper, a dual watermarking scheme based on discrete wavelet transform (DWT), wavelet packet transform (WPT) with best tree, and singular value decomposition (SVD) is proposed. In our algorithm, the cover image is sub-sampled into four sub-images and then two sub-images, having the highest sum of singular values are selected. Two different gray scale images are embedded in the selected sub-images. For embedding first watermark, one of the selected sub-image is decomposed via WPT. The entropy based algorithm is adopted to find the best tree of WPT. Watermark is embedded in all frequency sub-bands of the best tree. For embedding second watermark, l-level discrete wavelet transform (DWT) is performed on the second selected sub-image. The watermark is embedded by modifying the singular values of the transformed image. To enhance the security of the scheme, Zig-Zag scan in applied on the second watermark before embedding. The robustness of the proposed scheme is demonstrated through a series of attack simulations. Experimental results demonstrate that the proposed scheme has good perceptual invisibility and is also robust against various image processing operations, geometric attacks and JPEG Compression.     

奇异值非线性修正的低对比度红外图像实时增强

由于场景中目标与背景的温差相对较小,红外图像会存在对比度低、视觉效果差的问题,针对这一问题,提出一种基于奇异值非线性修正的红外图像对比度实时增强方法。该方法首先对红外图像进行奇异值分解得到其原始奇异值,然后采用一个对数型非线性变换对图像奇异值进行优化,最后根据修正的奇异值重构出对比度增强的红外图像。利用对数型非线性变换修正图像奇异值不仅能够有效拉伸奇异值的动态范围,同时可优化奇异值的变化梯度,使图像的能量信息得到更充分地表达,改善红外图像不良的视觉效果。实验结果表明,该方法较几种对比方法在视觉效果和客观评价方面均具有更优的增强性能;同时体现出良好的实时性,为实现红外图像的实时增强提供了新途径。     

Community Partitioning over Feature-Rich Networks Using an Extended K-Means Method

This paper proposes a meaningful and effective extension of the celebrated K-means algorithm to detect communities in feature-rich networks, due to our assumption of non-summability mode. We least-squares approximate given matrices of inter-node links and feature values, leading to a straightforward extension of the conventional K-means clustering method as an alternating minimization strategy for the criterion. This works in a two-fold space, embracing both the network nodes and features. The metric used is a weighted sum of the squared Euclidean distances in the feature and network spaces. To tackle the so-called curse of dimensionality, we extend this to a version that uses the cosine distances between entities and centers. One more version of our method is based on the Manhattan distance metric. We conduct computational experiments to test our method and compare its performances with those by competing popular algorithms at synthetic and real-world datasets. The cosine-based version of the extended K-means typically wins at the high-dimension real-world datasets. In contrast, the Manhattan-based version wins at most synthetic datasets.     

字典奇异值分解加权压缩感知多径信号参数估计

为提高水声信道多径参数估计的分辨率,提出了一种基于字典奇异值分解的加权压缩感知算法。对于有源声呐,根据发射信号构造字典,对字典进行奇异值分解,利用大特征值对应的特征向量构造信号子空间,然后使用信号子空间对接收信号进行滤波。对滤波结果进行加权压缩感知参数估计,得出最终时延估计结果。仿真实验表明,所提方法能够对水声多径参数进行超分辨估计,适用于任何脉冲信号。湖试处理结果显示,混响背景下该方法也有较好的多径参数估计性能,能够降低接收数据的噪声成分,提高对水声信道的多径时延、个数和幅度的估计精度。      

Distributions of singular values for some random matrices

The singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data, such as complex space-time images obtained in both physical and biological systems. In this paper, we examine the distribution of singular values of low-rank matrices corrupted by additive noise. Past studies have been limited to uniform uncorrelated noise. Using diagrammatic and saddle point integration techniques, we extend these results to heterogeneous and correlated noise sources. We also provide perturbative estimates of error bars on the reconstructed low-rank matrix obtained by truncating a singular value decomposition.     

The minimum norm method for the determination of the charge density from elastic electron scattering data

Unphysical behavior in the QR algorithm based least squares determination of the expansion coefficients of the charge density obtained from limited information about the charge form factor occurs when the spread of the singular values in the matrix relating these quantities becomes too large. Setting the smallest singular values equal to zero in the singular value decomposition used in the minimum norm method yields a much more reasonable determination of the charge density. Increasing the size of the basis without increasing the range of the prior information about the charge form factor leads to ambiguities in the determination of the charge density. Numerical results in an analytic model are presented.     

The minimum norm method for the determination of the charge density from elastic electron scattering data

Unphysical behavior in the QR algorithm based least squares determination of the expansion coefficients of the charge density obtained from limited information about the charge form factor occurs when the spread of the singular values in the matrix relating these quantities becomes too large. Setting the smallest singular values equal to zero in the singular value decomposition used in the minimum norm method yields a much more reasonable determination of the charge density. Increasing the size of the basis without increasing the range of the prior information about the charge form factor leads to ambiguities in the determination of the charge density. Numerical results in an analytic model are presented.     

基于多尺度Radon变换的图像检索

根据Radon变换的统计特性构造了不变量,提出一种新的基于多尺度Radon变换的图像形状检索方法.对检索图像作小波变换,根据小波模极大原理得到边缘图像,对边缘图像构造Radon变换中心矩,在中心矩的基础上根据Radon的统计原理构造出尺度不变矩.由于矩阵的奇异值具有旋转不变性,因此针对不变矩向量矩阵求奇异值,该奇异值特征向量具有平移、尺度和旋转不变性.将该Radon变换的不变量作为形状特征,并进行高斯归一化,按照欧氏距离计算不同图像间的形状相似度.试验结果表明,该方法对高斯噪音具有较强的鲁棒性,与其它方法相比具有较好的检索效果.     

High speed ghost imaging based on a heuristic algorithm and deep learning

We report an overlapping sampling scheme to accelerate computational ghost imaging for imaging moving targets,based on reordering a set of Hadamard modulation matrices by means of a heuristic algorithm. The new condensed overlapped matrices are then designed to shorten and optimize encoding of the overlapped patterns, which are shown to be much superior to the random matrices. In addition, we apply deep learning to image the target, and use the signal acquired by the bucket detector and corresponding real image to train the neural network. Detailed comparisons show that our new method can improve the imaging speed by as much as an order of magnitude, and improve the image quality as well.