Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues

The influence of the multiplication order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex tissue-like turbid media exhibiting simultaneous scattering and polarization effects are investigated. A polarization sensitive Monte Carlo (MC) simulation model was used to generate Mueller matrices from turbid media exhibiting simultaneous linear birefringence, optical activity and multiple scattering effects. Mueller matrix decomposition was performed with different selected multiplication orders of the constituent basis matrices, which were further analyzed to derive quantitative individual polarization medium properties. The results show that for turbid medium having weak diattenuation (differential attenuation of two orthogonal polarization states), the decomposition-derived polarization parameters are independent of the multiplication order. Importantly, the values for the extracted polarization parameters were found to be in excellent agreement with the controlled inputs, showing self-consistency in inverse decomposition analysis and successful decoupling of the individual polarization effects. These results were corroborated further by selected experimental results from phantoms having optical (scattering and polarization) properties similar to those used in the MC model. Results from tissue polarimetry confirm that the magnitude of diattenuation is generally lower compared to other polarization effects, so that the demonstrated self-consistency of the decomposition formalism with respect to the potential ambiguity of ordering of the constituent matrices should hold in biological applications.     

Multivariate data analysis for Raman spectroscopic imaging

This article reviews the analytic techniques for Raman spectroscopic imaging with emphasis on chemometrics. Key information included in Raman spectra is often distributed broadly throughout the dataset. It is possible to condense the information into a very compact matrix representation by a chemometric technique of factor analysis such as principal component analysis (PCA) or self‐modeling curve resolution (SMCR). PCA yields two matrices called scores and loadings which complementarily represent the entire features broadly distributed in the dataset. This concept can be further extended to other forms of data transformation schemes, including bilinear data decomposition based on SMCR analysis. SMCR offers a firmer model which is chemically or physically interpretable. The information derived from these techniques readily brings useful insight into building a mechanistic model for understanding complex phenomena studied by Raman spectroscopy. Illustrative examples are given for applications of both PCA and SMCR to Raman imaging of pharmaceutical tablets. Copyright © 2009 John Wiley & Sons, Ltd.     

MCR-BANDS用于高光谱成像分辨中旋转模糊度的评估

自建模曲线分辨用来将双线性光谱数据矩阵分解成具有明确物理或化学意义的曲线,一方面反映了复杂体系中各个主成分对应的纯光谱,同时也能够解析得到其对应的相对浓度。这一方法在高光谱解析中充分发挥了其优势,成为了高光谱分析中的重要方法之一。然而,双线性结构数据多元分辨模型在约束不充分的条件下往往不能获得唯一解,这一问题是由顺序模糊、尺度模糊和旋转模糊引起,其中旋转模糊最难消除。在高光谱成像中,如果不能明确浓度分布情况,将导致在成像领域难以确认目标的准确位置或形貌。为了充分了解旋转模糊带来的影响,评估非唯一解情况下可行解的范围,并进一步为实际应用中需要解析获得的每个主成分的纯光谱或波谱信号,以及其对应的浓度信息提供客观的评估依据,在以往的研究中,研究人员分别使用网格法、蒙特卡洛法进行抽样,以计算曲线分辨结果中可行解的范围,也有科研人员使用了几何多边形内部和外围面积的形式表示结果,但是这些结果往往面临运算时间过长,或者无法实现高维可视化而不适用于大于四个主成分的数据等问题,而且通常这些方法很难将曲线分辨过程中施加的除非负约束之外的其他约束方法加以利用,导致可行解范围计算不准确。为了解决以上问题,采用MCR-BANDS对MCR-ALS（多元曲线分辨-交替最小二乘）分辨获得的结果进行了旋转模糊程度的评估,并将其应用到遥感高光谱成像的解析中。首先以美国地质勘探局矿物光谱库中的纯光谱为基础的模拟数据集对MCR-ALS和MCR-BANDS的结果进行了评测,在模拟数据中能够方便地控制噪声的影响,控制选取主成分之间的纯光谱差异、仿照真实环境中浓度渐变特征等因素,考查了特定条件下MCR结果中旋转模糊的水平。随后为了证实所用方法的可行性,进一步采用MCR-ALS分析了机载可见/红外成像光谱仪（AVIRIS）获得的遥感高光谱图像数据,并首次采用MCR-BANDS对MCR-ALS的分辨结果的旋转模糊进行了分析,实现了对遥感高光谱数据成像浓度分布的受到旋转模糊影响的可视化表示。可以发现真实解和MCR-ALS获得的可行解均在MCR-BANDS计算得到的可行解范围之内。结果表明,MCR-BANDS方法基于最大和最小的信号贡献对旋转模糊的范围进行计算,能够适用于不同主成分的体系中,并且完美对接MCR-ALS中使用到的诸如非负、单峰、封闭和选择性约束等。MCR-BANDS的分析结果可以为MCR-ALS的解析结果提供相应的旋转模糊水平估计,有利于对MCR-ALS结果的解释;在充分约束条件下,能够有效减少甚至消除旋转模糊对MCR-ALS分辨结果的影响,为精确确定遥感高光谱中解析得到的目标物位置提供了客观的范围。     

Universal correlations and power-law tails in financial covariance matrices

We investigate whether quantities such as the global spectral density or individual eigenvalues of financial covariance matrices can be best modelled by standard random matrix theory or rather by its generalisations displaying power-law tails. In order to generate individual eigenvalue distributions a chopping procedure is devised, which produces a statistical ensemble of asset-price covariances from a single instance of financial data sets. Local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets. They are in good agreement with the universal Tracy-Widom distribution and Wigner surmise, respectively. This suggests a strong degree of robustness especially in the low-lying sector of the spectra, most relevant for portfolio selections. Conversely, the global spectral density of a single covariance matrix as well as the average over all unfolded nearest-neighbour spacing distributions deviate from standard Gaussian random matrix predictions. The data are in fair agreement with a recently introduced generalised random matrix model, with correlations showing a power-law decay.     

基于有约束非负矩阵分解的原稿基色色料光谱预测方法

针对直接在光谱反射率空间,对原稿颜色样本光谱的主成分分析会导致特征向量的数目超过真实物理维度(原稿所用基色色料)的数量,以及特征向量和对应系数存在负值,不能直接表示原稿基色色料的光谱特性和对应浓度等情况。创新性的提出需根据原稿色料的光学特性建立一个完全线性的光谱空间,并在该空间中使用带约束条件的非负矩阵分解实现对原稿基色数量和光谱形状进行预测的方法。对此,首先设计了一个实现对原稿基色色料光谱预测方法的总体研究方案和实现步骤,再以透明色料原稿为例,研究如何选择和构建一个符合其光学特性的光谱线性空间,然后再在基本非负矩阵分解(BNMF)基础上提出针对基色色料光谱预测的有约束非负矩阵分解算法(SCNMF)。针对BNMF算法会出现多重最优解,为了提高预测精度,使矩阵分解结果有明确的物理意义,所提出的SCNMF算法需要满足4个约束条件：非负性约束;全加性约束;平滑性约束;稀疏性约束。建立了满足约束条件的目标函数和迭代算法。预测结果表明本文提出的新方法能有效的实现对原稿基色物理维度和基色色料光谱的准确预测。     

Adaptive non-negative bases for reconstruction of spectral data from colorimetric information

The positive basis functions of the reflectance spectra of Munsell color chips are extracted by using the classical nonnegative matrix factorization method. Different numbers of basis, i.e., 3 to 5, are determined and used as projection spaces. The spectral reflectances of samples are defined in the desired compact spectral spaces and the performances of the spaces are evaluated through the cost of the “lost data” by computing of the root mean square error between the actual and the reconstructed spectra as well as the corresponding color difference values under different viewing conditions. The method is also compared with the most welcomed technique, i.e., principal component analyzing, and its priority is shown to some extent in the three dimensional space. To show the importance of the spectral behaviors of samples in the dataset on the extracted basis and consequently the error of the spectral reconstruction trial, the adaptive non-negative matrix factorization method is introduced and examined in the reconstruction of spectral data from the colorimetric tristimulus values. The suggested method weights the samples in the database proportional to the colorimetric differences between the specimens in the dataset and the sample whose spectral has been aimed, prior to the extraction of all positive bases. The root mean square errors between the actual and the reconstructed spectra as well as the metamerism indices under A and F11 illuminants are calculated and the results show considerable improvement over the classic non-negative matrix factorization method as well as the principal component analyzing technique.     

The importance of the privilege for appearance of inverse-power solutions in Ito equations

It is shown that the true cause of inverse-power distributions in the Ito equation is some kind of privilege which is hidden in the course of evolution of the system. Connections between Ito equations with additive noise or/and multiplicative noise with additive processes, multiplicative processes, multiplication of probabilities and return-to-the-origin problem are found. On the basis of two toy models, the appearance of particular functions for deterministic and stochastic forces in the Ito equation is explained. The paper stands as the next contribution confirming the hypothesis that the adequate privilege is the cause for the origin of inverse-power distributions in many phenomena.     

Eigenvalue spectra of random matrices for neural networks

The dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix describing their synaptic connectivity. In large networks, elements of the synaptic connectivity matrix can be chosen randomly from appropriate distributions, making results from random matrix theory highly relevant. Unfortunately, classic results on the eigenvalue spectra of random matrices do not apply to synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or inhibitory. Therefore, we compute eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances.     

Spectral resolution enhancement by chemical shift imaging

Three-dimensional chemical shift imaging (3D CSI) with appropriate data postprocessing can be used as a tool to improve spectral resolution in samples where large susceptibility differences and limited shim capabilities prevent good sample shimming. Data postprocessing is reduced to the realignment of individual 3D voxel spectra. As a result, the line broadening due to the field inhomogeneity over the sample's volume is reduced to the broadening by inhomogeneity within individual voxels. We compared this method with the resolution enhancement by window multiplication. We demonstrated, theoretically and experimentally, that in the presence of large, lower-order gradients, 3D CSI achieves better resolution enhancement with smaller sensitivity losses. An application of the method to a simple biological system is presented as well.     

Deconvolution of pure component FT‐Raman spectra from thermal emission of barium sulfate and graphite samples using the BTEM algorithm

Band‐target entropy minimization (BTEM) was applied for the extraction of pure component Raman spectra from samples exhibiting a significant thermal background due to sample emission. In this method, singular value decomposition was first used to calculate the right singular vectors of the spectroscopic data matrix. Then the non‐noise right singular vectors were examined for localized spectral features, which were subsequently used for spectral recovery. These local features were targeted with the BTEM algorithm to recover the pure component Raman spectra. Accordingly, the interfering thermal background was removed. This method of analysis is applied to graphite and barium sulfate Raman spectroscopic data. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Accuracy of spectrum estimate in fluorescence spectral microscopy with spectral filters

We establish the accuracy of the spectrum that is estimated with an inexpensive fluorescence spectral microscope utilizing a small set of spectral filters [Soriano et al, Opt. Exp. 10, 1458–1464 (2002)]. The spectrum at an arbitrary image location of the fluorescent sample is estimated as a linear superposition of basis spectra that are derived by singular value decomposition (SVD) or principal component analysis (PCA) from a spectral library of fluorescence spectra. Estimation performance is analyzed as a function of library statistics, filter selection and sequencing, minimum negativity constraint and signal to noise ratio (SNR) of fluorescence image. We consider image SNR degradations that arise from weakening of image intensity, additive Gaussian noise, intensity-dependent Poisson noise and quantization errors. The recovery of specific spectral features like spectral resolution, general similarity and peak alignments, is assessed using Linfoot’s criteria of fidelity, structural content, and correlation. We found that estimation with SVD basis spectra is more robust against image noise than that with PCA basis spectra. However for high SNR images, accurate estimation is achieved more quickly with PCA basis spectra and with better response to the application of minimum negativity constraint.     

Harmonic Interpolation and Lie Groups

A harmonic interpolation of a polygon (for odd and even numbers of points forming the polygon) used in computer graphics is derived from the primary permutation matrix using the spectral decomposition of the matrix. This matrix can be used to generate an orthonormal basis in the Hilbert space of all n × n matrices over C. The connection with Lie groups is discussed.     

An image compressed sensing algorithm based on adaptive nonlinear network

Traditional compressed sensing algorithm is used to reconstruct images by iteratively optimizing a small number of measured values. The computation is complex and the reconstruction time is long. The deep learning-based compressed sensing algorithm can greatly shorten the reconstruction time, but the algorithm emphasis is placed on reconstructing the network part mostly. The random measurement matrix cannot measure the image features well, which leads the reconstructed image quality to be improved limitedly. Two kinds of networks are proposed for solving this problem. The first one is Recon Net's improved network IRecon Net, which replaces the traditional linear random measurement matrix with an adaptive nonlinear measurement network. The reconstruction quality and anti-noise performance are greatly improved.Because the measured values extracted by the measurement network also retain the characteristics of image spatial information, the image is reconstructed by bilinear interpolation algorithm(Bilinear) and dilate convolution. Therefore a second network USDCNN is proposed. On the BSD500 dataset, the sampling rates are 0.25, 0.10, 0.04, and 0.01, the average peak signal-noise ratio(PSNR) of USDCNN is 1.62 d B, 1.31 d B, 1.47 d B, and 1.95 d B higher than that of MSRNet. Experiments show the average reconstruction time of USDCNN is 0.2705 s, 0.3671 s, 0.3602 s, and 0.3929 s faster than that of Recon Net. Moreover, there is also a great advantage in anti-noise performance.     

Covariance NMR spectroscopy by singular value decomposition

Covariance NMR is demonstrated for homonuclear 2D NMR data collected using the hypercomplex and TPPI methods. Absorption mode 2D spectra are obtained by application of the square-root operation to the covariance matrices. The resulting spectra closely resemble the 2D Fourier transformation spectra, except that they are fully symmetric with the spectral resolution along both dimensions determined by the favorable resolution achievable along omega2. An efficient method is introduced for the calculation of the square root of the covariance spectrum by applying a singular value decomposition (SVD) directly to the mixed time-frequency domain data matrix. Applications are shown for 2D NOESY and 2QF-COSY data sets and computational benchmarks are given for data matrix dimensions typically encountered in practice. The SVD implementation makes covariance NMR amenable to routine applications.     

Influence of impurities on the de Haas-van Alphen effect

A general theory is given for the effects of impurities or other localized defects in metals on the amplitudes (“Dingle factor”) and the periods of the de Haas-van Alphen oscillations. Starting from the Green's function for the crystal with defects, after configuration averaging a simple expression for the spectral density and the level broadening is obtained, expressed in terms of the transition matrix for a single defect. The density of states leading to the de Haas-van Alphen oscillations is gained by summing the spectral density over the electron states in the presence of the magnetic field. Using the transition matrices obtained earlier, the changes of the oscillations may be calculated for general localized defects and Fermi surfaces. A previous paper by Brailsford on the same topic is critically discussed.     

Transfer-matrix for the multichannel scattering problem

A transfer-matrix for the multichannel scattering problem is obtained. The elements of this matrix are expressed in terms of transmission and reflection amplitudes. On the basis of the matrix for a system of N localized and nonoverlapped scattering centers the recurrent equations for the transfermatrix elements are derived and the initial conditions are defined.     

Polarization bremsstrahlung at collisions of fast ions with multiatomic targets

The processes of polarization bremsstrahlung at collisions of fast ions with linear chains consisting of isolated atoms have been considered. The intensities and angular distributions of radiation spectra have been obtained for an arbitrary number of atoms in a chain. It has been shown that the interference of the photon emission amplitudes leads to a noticeable change in the spectral angular distributions of polarization bremsstrahlung as compared to the distributions at collisions with an isolated atom. The results allow standard generalization to the cases of polarization bremsstrahlung at channeling of fast ions over surfaces and in solid lattices.     

Numerical simulation of PRESS localized MR spectroscopy

Numerical simulations of NMR spectra can provide a rapid and convenient method for optimizing acquisition sequence parameters and generating prior spectral information required for parametric spectral analysis. For spatially resolved spectroscopy, spatially dependent variables affect the resultant spectral amplitudes and phases, which must therefore be taken into account in any spectral simulation model. In this study, methods for numerical simulation of spectra obtained using the PRESS localization pulse sequence are examined. A comparison is made between three different simulation models that include different levels of detail regarding the spatial distributions of the excitation functions, and spin evolution during application of the pulses. These methods were evaluated for measurement of spectra from J-coupled spin systems that are of interest for in vivo proton spectroscopy and results compared with experimental data. It is demonstrated that for optimized refocusing pulses it is sufficient to account for chemical shift effects only, although there is some advantage to implementing a more general numerical simulation approach that includes information on RF pulse excitation profiles, which provides sufficient accuracy while maintaining moderate computational requirements and flexibility to handle different spin systems.     

Theoretical Evaluation and Comparison of Fast Chemical Shift Imaging Methods

Magnetic resonance chemical shift imaging (CSI) is becoming the method of choice for localized NMR spectroscopic examinations, allowing simultaneous detection of NMR spectra from a large number of voxels. The main limitation of these methods is their long experimental duration. A number of fast CSI experiments have been presented, promising to reduce that duration. In this contribution the criteria for evaluating and optimizing the sensitivity of fast CSI experiments are elaborated. For a typical experiment in the human brain, the performance of various methods is compared. While conventional CSI provides optimal sensitivity per unit time, it is shown in which circumstances fast sequences allow a shorter experimental duration. Using these results, the best method for any experimental requirements can be selected.