Progress on the two-dimensional filter diagonalization method. An efficient doubling scheme for two-dimensional constant-time NMR

An efficient way to treat two-dimensional (2D) constant-time (CT) NMR data using the filter diagonalization method (FDM) is presented. In this scheme a pair of N- and P-type data sets from a 2D CT NMR experiment are processed jointly by FDM as a single data set, twice as large, in which the signal effectively evolves in time for twice as long. This scheme is related to "mirror-image" linear prediction, but with the distinction that the data are directly used, without any preprocessing such as Fourier transformation along one dimension, or point-by-point reflection. As the signal has nearly perfect Lorentzian line shape in the CT dimension, it can be efficiently handled by the FDM approach. Applied to model and experimental signals, the scheme shows significant resolution improvement, and appears to tolerate noise reasonably well. Other complex aspects of multidimensional FDM are discussed and illustrated.     

Regularization of the two-dimensional filter diagonalization method: FDM2K

We outline an important advance in the problem of obtaining a two-dimensional (2D) line list of the most prominent features in a 2D high-resolution NMR spectrum in the presence of noise, when using the Filter Diagonalization Method (FDM) to sidestep limitations of conventional FFT processing. Although respectable absorption-mode spectra have been obtained previously by the artifice of “averaging” several FDM calculations, no 2D line list could be directly obtained from the averaged spectrum, and each calculation produced numerical artifacts that were demonstrably inconsistent with the measured data, but which could not be removed a posteriori. By regularizing the intrinsically ill-defined generalized eigenvalue problem that FDM poses, in a particular quite plausible way, features that are weak or stem from numerical problems are attenuated, allowing better characterization of the dominant spectral features. We call the new algorithm FDM2K.     

Phase-sensitive spectral estimation by the hybrid filter diagonalization method

A more robust way to obtain a high-resolution multidimensional NMR spectrum from limited data sets is described. The Filter Diagonalization Method (FDM) is used to analyze phase-modulated data and cast the spectrum in terms of phase-sensitive Lorentzian "phase-twist" peaks. These spectra are then used to obtain absorption-mode phase-sensitive spectra. In contrast to earlier implementations of multidimensional FDM, the absolute phase of the data need not be known beforehand, and linear phase corrections in each frequency dimension are possible, if they are required. Regularization is employed to improve the conditioning of the linear algebra problems that must be solved to obtain the spectral estimate. While regularization smoothes away noise and small peaks, a hybrid method allows the true noise floor to be correctly represented in the final result. Line shape transformation to a Gaussian-like shape improves the clarity of the spectra, and is achieved by a conventional Lorentzian-to-Gaussian transformation in the time-domain, after inverse Fourier transformation of the FDM spectra. The results obtained highlight the danger of not using proper phase-sensitive line shapes in the spectral estimate. The advantages of the new method for the spectral estimate are the following: (i) the spectrum can be phased by conventional means after it is obtained; (ii) there is a true and accurate noise floor; and (iii) there is some indication of the quality of fit in each local region of the spectrum. The method is illustrated with 2D NMR data for the first time, but is applicable to n-dimensional data without any restriction on the number of time/frequency dimensions.     

Ultra-high resolution 3D NMR spectra from limited-size data sets

The advantage of the filter diagonalization method (FDM) for analysis of triple-resonance NMR experiments is demonstrated by application to a 3D constant time (CT) HNCO experiment. With a 15N-,13C-labeled human ubiquitin sample (1.0 mM), high spectral resolution was obtained at 500 MHz in 25 min with only 6-8 increments in each of the CT dimensions. This data set size is about a factor of 50-100 smaller than typically required, yet FDM analysis results in a fully resolved spectrum with a sharp peak for each HNCO resonance. Unlike Fourier transform (FT) processing, in which spectral resolution in each dimension is inversely proportional to the acquisition time in this dimension, FDM is a true multi-dimensional method; the resolution in all dimensions is determined by the total information content of the entire signal. As the CT dimensions of the 3D HNCO signal have approximate time-reversal symmetry, they can each be doubled by combining the usual four hyper-complex data sets. This apparent quadrupling of the data is important to the success of the method. Thus, whenever raw sensitivity is not limiting, well-resolved n-dimensional spectra can now be obtained in a small fraction of the usual time. Alternatively, to maximize sensitivity, evolution periods of faster relaxing nuclei may be radically shortened, the total required resolution being obtained through chemical shift encoding of other, more slowly relaxing, spins. Improvements similar to those illustrated with a 3D HNCO spectrum are expected for other triple-resonance spectra, where CT evolution in the indirect dimensions is implemented.     

The multidimensional filter diagonalization method

The theory and numerical aspects of the recently developed multidimensional version of the filter diagonalization method (FDM) are described in detail. FDM can construct various "ersatz" or "hybrid" spectra from multidimensional time signals. Spectral resolution is not limited by the time-frequency uncertainty principle in each separate frequency dimension, but rather by the total joint information content of the signal, i.e., N(total) = N(1) x N(2) x vertical ellipsis x N(D), where some of the interferometric dimensions do not have to be represented by more than a few (e.g., two) time increments. It is shown that FDM can be used to compute various reduced-dimensionality projections of a high-dimensional spectrum directly, i.e., avoiding construction of the latter. A subsequent paper (J. Magn. Reson. 144, 357-366 (2000)) is concerned with applications of the method to 2D, 3D, and 4D NMR experiments.     

The Multidimensional Filter Diagonalization Method: II. Application to 2D Projections of 2D, 3D, and 4D NMR Experiments

The theory of the multidimensional filter diagonalization method (FDM) described in the previous paper (V. A. Mandelshtam, 2000, J. Magn. Reson. 144, 343–356 (2000)) is applied to NMR time signals with up to four independent time variables. Direct projections of the multidimensional time signals produce new kinds of 2D spectra. The resolution obtained by FDM can be far superior to that obtained by conventional phase-sensitive FT processing, and correlation peaks in heteronuclear and homonuclear experiments can be condensed to sharp singlets, removing all spin–spin couplings. Examples of singlet-HSQC and singlet-TOCSY spectra show big gains in resolution. It is not necessary to have a finely digitized spectrum, in which the individual multiplet components are resolved, for the methods to work. Examples of FDM spectra, ranging from simple organic molecules and steroids to metalloproteins, are shown.     

基于大规模并行计算的三维多群中子扩散方程有限差分方法

三维多群中子扩散方程的精确、高效求解是核动力堆芯设计及燃料管理的基础。应用有限差分方法求解该方程具有简便、精确、成熟的优点;然而,该方法的计算量和存储量均较大,极大地限制了它的计算规模和应用范围。本文基于大规模并行计算,研究三维多群中子扩散方程有限差分方法:采用中心有限差分格式离散中子扩散方程;基于MPI并行编程模型,采用空间区域分解的方式实现大规模并行计算;采用多群多区域耦合PGMRES算法进行并行加速。在集群服务器上开发了ParaFiDi程序,并采用IAEA3D,PHWR等多个基准题对该程序进行验证。数值结果表明,ParaFiDi程序具有较高的计算精度和计算效率。     

Enhanced spectral resolution by high-dimensional NMR using the filter diagonalization method and "hidden" dimensions

High-dimensional (HD) NMR spectra have poorer digital resolution than low-dimensional (LD) spectra, for a fixed amount of experiment time. This has led to "reduced-dimensionality" strategies, in which several LD projections of the HD NMR spectrum are acquired, each with higher digital resolution; an approximate HD spectrum is then inferred by some means. We propose a strategy that moves in the opposite direction, by adding more time dimensions to increase the information content of the data set, even if only a very sparse time grid is used in each dimension. The full HD time-domain data can be analyzed by the filter diagonalization method (FDM), yielding very narrow resonances along all of the frequency axes, even those with sparse sampling. Integrating over the added dimensions of HD FDM NMR spectra reconstitutes LD spectra with enhanced resolution, often more quickly than direct acquisition of the LD spectrum with a larger number of grid points in each of the fewer dimensions. If the extra-dimensions do not appear in the final spectrum, and are used solely to boost information content, we propose the moniker hidden-dimension NMR. This work shows that HD peaks have unmistakable frequency signatures that can be detected as single HD objects by an appropriate algorithm, even though their patterns would be tricky for a human operator to visualize or recognize, and even if digital resolution in an HD FT spectrum is very coarse compared with natural line widths.     

The single basis filter diagonalization method: a rapid multidimensional data processing scheme

A new way to apply the filter diagonalization method (FDM) that results in a large increase in the speed of calculation of multidimensional NMR spectra is presented. The speed increase is accompanied by slight differences in spectral lineshapes, although frequency estimates remain essentially identical. For contoured spectra, the method does not result in appreciable differences from the full FDM calculation. Optimal parameter sets for an FDM calculation can be estimated far more rapidly, which makes the FDM more straightforward to employ in practice. The performance of the method versus the full FDM was investigated for both model and experimental signals. The effect of noise on the method was also studied.     

Rapid 3D NMR using the filter diagonalization method: application to oligosaccharides derivatized with 13C-labeled acetyl groups

Rapid 3D NMR spectroscopy of oligosaccharides having isotopically labeled acetyl "isotags" was made possible with high resolution in the indirect dimensions using the filter diagonalization method (FDM). A pulse sequence was designed for the optimal correlation of acetyl methyl protons, methyl carbons, and carbonyl carbons. The multi-dimensional nature of the FDM, coupled with the advantages of constant-time evolution periods, resulted in marked improvements over Fourier transform (FT) and mirror-image linear prediction (MI-LP) processing methods. The three methods were directly compared using identical data sets. A highly resolved 3D spectrum was achieved with the FDM using a very short experimental time (28 min).     

基于自动线性拟合的快速拉曼基线校正算法

近年来拉曼光谱以其无创、灵敏度高等众多优点在化学表征、生物医药、材料等领域引起广泛关注,而基线漂移的存在为后续的定性定量分析带来严重困扰,因此设计高性能的基线校准算法以提高分析结果的有效性及准确性具有重要意义。针对传统算法在批量拉曼光谱数据基线校正方面的不足,基于自动线性拟合算法提出一种快速基线校正算法以校正具有相似背景的批量拉曼光谱数据并详细阐述了该算法的核心思想以及算法实现流程。该算法首先从批量拉曼光谱数据中自动选择一条拉曼光谱数据作为基准光谱,使用自动线性拟合算法对其进行基线校准得到其基线以及分段标记点,然后利用标记点快速计算出组内其他与基准光谱具有较高相关性的拉曼光谱数据的基线,对于组内与基准光谱相关性不满足阈值要求的拉曼光谱则使用自动线性拟合算法对其进行单独基线校正,这使得算法具有具有较强的鲁棒性,可以适应复杂的拉曼光谱基线校正情形。分别使用快速基线校正算法与单独基线校正算法对多组实际拉曼光谱数据进行基线校正以对比分析算法基线校正效果,结果表明该算法可以实现对批量拉曼光谱数据的快速校正,基线校正效果良好,并且相较于单独进行基线校正算法耗时减少了30%以上,算法无参,简单易行,无需额外人工干预,是一种切实可行的批量拉曼数据自动基线校正算法。     

Wavelet-based ultra-high compression of multidimensional NMR data sets

The application of a lossy data compression algorithm based on wavelet transform to 2D NMR spectra is presented. We show that this algorithm affords rapid and extreme compression ratios (e.g., 800:1), providing high quality reconstructed 2D spectra. The algorithm was evaluated to ensure that qualitative and quantitative information are retained in the compressed NMR spectra. Whilst the maximum compression ratio that can be achieved depends on the number of signals and on the difference between the most and the least intense peaks (dynamic range), a compression ratio of 80:1 is affordable even for the challenging case of homonuclear 2D experiments of large biomolecules.     

Rapid high-resolution four-dimensional NMR spectroscopy using the filter diagonalization method and its advantages for detailed structural elucidation of oligosaccharides

Four-dimensional nuclear magnetic resonance spectroscopy with high resolution of signals in the indirect dimensions is reported as an implementation of the filter diagonalization method (FDM). Using an oligosaccharide derivatized with 13C-labeled acetyl isotags, a four-dimensional constant-time pulse sequence was tailored for conjoint use with the FDM. Results demonstrate that high resolution in all dimensions can be achieved using a relatively short experimental time period (19 h), even though the spectrum is highly congested in the direct and all three indirect dimensions. The combined use of isotags, constant-time pulse sequences, and FDM permits rapid isolation of sugar ring proton spin systems in multiple dimensions and enables all endocyclic J-couplings to be simply measured, the key goal to assigning sugar stereochemistry and anomeric configuration. A general method for rapid, unambiguous elucidation of spin systems in oligosaccharides has been a long-sought goal of carbohydrate NMR, and isotags combined with the FDM now enable this to be easily performed. Additional general advantages of the FDM program for generating high-resolution 2D slices in any dimension from a 4D spectrum are emphasized.     

The Multidimensional Filter Diagonalization Method: I. Theory and Numerical Implementation

The theory and numerical aspects of the recently developed multidimensional version of the filter diagonalization method (FDM) are described in detail. FDM can construct various “ersatz” or “hybrid” spectra from multidimensional time signals. Spectral resolution is not limited by the time-frequency uncertainty principle in each separate frequency dimension, but rather by the total joint information content of the signal, i.e., N_total = N₁ × N₂ × × N_D, where some of the interferometric dimensions do not have to be represented by more than a few (e.g., two) time increments. It is shown that FDM can be used to compute various reduced-dimensionality projections of a high-dimensional spectrum directly, i.e., avoiding construction of the latter. A subsequent paper (J. Magn. Reson. 144, 357–366 (2000)) is concerned with applications of the method to 2D, 3D, and 4D NMR experiments.     

On Solving Three-Dimensional Electrostatic Field Distribution by Multigrid Method

A highly efficient numerical algorithm using the multigrid method (MGM) is introduced to solve a three-dimensional (3-D)field distribution. Taking advantage of the restriction and prolongation in MGM computation, a more accurate field distribution can be acquired rapidly. According to the MGM algorithm, a 3-D program is accomplished, which can solve the field distributions in electron optical systems for various electrostatic lenses. The 3-D field distribution in an electrostatic concentric spherical model is tested with the MGM algorithm and with an algorithm based on the finite difference method (FDM). Comparing these two results in terms of computational efficiency and computational accuracy, it appears that MGM is superior to FDM, which is now used the most in field computations. This paper shows that the 3-D field computation using MGM greatly improves the computational efficiency of field distributions in electron optical systems and shortens the computational time.     

基于近似l0范数最小化的NMR波谱稀疏重建算法

在核磁共振（NMR）波谱中,过长的数据采集时间会使很多化学以及分子生物学领域的高分辨率多维谱应用难以实现. 传统的解决办法是使用随机非均匀采样代替奈奎斯特采样,但这样会使谱图质量受损. 压缩传感的出现为此提供了更好的解决办法,合适的压缩传感重建算法可以通过很少的随机非均匀采样将谱图高质量的重建出来. 该文先介绍了一种可用于谱图重建的压缩传感重建算法,名为“平滑l₀范数最小化法”,然后针对该算法对采样噪声鲁棒性较差的缺点进行了改进. 通过将改进后的算法与原算法在一维实数域信号以及NMR波谱信号重建实验中进行对比后表明,改进后的算法对噪声的鲁棒性明显提高,并能获得更好的重建性能.     

数字图像识别在混合油类三维荧光光谱分析中的应用

海上溢油已成为全球环境污染的重要问题之一,溢油严重破坏了海洋生态的平衡,并导致人类健康受到危害。因此,研究高效的溢油检测方法对保护海洋生态环境具有重要意义。三维荧光光谱技术因能获得溢油的“指纹”图谱而成为溢油鉴别领域的有效分析手段,其与平行因子分析算法相结合获得了良好的溢油鉴别效果。但平行因子算法在使用过程中需要确定不同石油产品本身所适用的浓度范围,且其对预估计组分数敏感,组分数选择是否准确直接影响最终定性定量结果,这些问题都会对油类检测造成使用上的限制。油类组分极为复杂,其中各组分间不存在统一的线性浓度范围,其相互之间还受到荧光猝灭效应的影响。直接对未经稀释的油类样本进行光谱数据采集,所获得的三维荧光光谱会因样本中组分的种类及其含量不同而存在较大差异,导致对三维荧光光谱数据进行解析的平行因子分析算法不再适用。但组分的种类及含量相近的油样其光谱特征相似度较高,并且随着特定组分及其含量的改变,其光谱形状的变化规律也较为明显。基于此,将三维荧光光谱和数字图像识别相结合,提出一种针对混合油类样本的辨识方法。首先,利用五种矿物油（汽油、柴油、航空煤油、机油和润滑油）配制三类混合油样本,其中每类混合油是用其中两种不同矿物油以不同体积比直接混合配制而成;然后利用FS920荧光光谱仪获取样本的三维荧光光谱数据,并对该数据进行求导及灰度化预处理,进而得到三维荧光导数光谱灰度图;其次提取样本三维荧光导数光谱灰度图的颜色、纹理和形状等数字图像特征;最后,通过Fisher判别分析建立样本的分类模型,采用逐步回归建立混合油样本各组分相对体积的定量模型。分类模型对三类混合油样本的分类及识别效果良好。所建立的定量模型的线性相关性R大于0.99,显著性检验p值小于0.05。研究结果表明,三维荧光光谱的数字图像特征可以被本文所述方法有效提取并用于对油类样本的定性定量分析。该研究为海面溢油检测提供了一种简单、可靠的识别方法。     

A novel improved method for analysis of 2D diffusion-relaxation data--2D PARAFAC-Laplace decomposition

This paper demonstrates how the multi-linear PARAFAC model can with advantage be used to decompose 2D diffusion-relaxation correlation NMR spectra prior to 2D-Laplace inversion to the T(2)-D domain. The decomposition is advantageous for better interpretation of the complex correlation maps as well as for the quantification of extracted T(2)-D components. To demonstrate the new method seventeen mixtures of wheat flour, starch, gluten, oil and water were prepared and measured with a 300 MHz nuclear magnetic resonance (NMR) spectrometer using a pulsed gradient stimulated echo (PGSTE) pulse sequence followed by a Carr-Purcell-Meiboom-Gill (CPMG) pulse echo train. By varying the gradient strength, 2D diffusion-relaxation data were recorded for each sample. From these double exponentially decaying relaxation data the PARAFAC algorithm extracted two unique diffusion-relaxation components, explaining 99.8% of the variation in the data set. These two components were subsequently transformed to the T(2)-D domain using 2D-inverse Laplace transformation and quantitatively assigned to the oil and water components of the samples. The oil component was one distinct distribution with peak intensity at D=3 x 10(-12) m(2) s(-1) and T(2)=180 ms. The water component consisted of two broad populations of water molecules with diffusion coefficients and relaxation times centered around correlation pairs: D=10(-9) m(2) s(-1), T(2)=10 ms and D=3 x 10(-13) m(2) s(-1), T(2)=13 ms. Small spurious peaks observed in the inverse Laplace transformation of original complex data were effectively filtered by the PARAFAC decomposition and thus considered artefacts from the complex Laplace transformation. The oil-to-water ratio determined by PARAFAC followed by 2D-Laplace inversion was perfectly correlated with known oil-to-water ratio of the samples. The new method of using PARAFAC prior to the 2D-Laplace inversion proved to have superior potential in analysis of diffusion-relaxation spectra, as it improves not only the interpretation, but also the quantification.     

Algorithms for converting 3D surface roughness spectra when calculating frequency-angle reverberation characteristics

On the basis of an earlier developed model of sound scattering by surface disturbances characterized by 3D roughness spectra, we propose and test a sufficiently fast algorithm for calculating the spectral power density of surface reverberation that takes into account the propagation conditions, distance, and beam pattern of the receiver system. For a distance of up to 1000 m, a water area depth of ??20 m, and actually measured 3D roughness spectra in the Doppler frequency range of ±5 Hz, we have calculated the frequency-angle reverberation spectra for an acoustic background-radiation signal of 1.5 kHz. The obtained angle and frequency reverberation characteristics are compared with the results of acoustic measurements conducted using linear horizontal receiver antennas.     

Automatic frequency alignment and quantitation of single resonances in multiple magnetic resonance spectra via complex principal component analysis

Several algorithms for automatic frequency alignment and quantitation of single resonances in multiple magnetic resonance (MR) spectra are investigated. First, a careful comparison between the complex principal component analysis (PCA) and the Hankel total least squares-based methods for quantifying the resonances in the spectral sets of magnetic resonance spectroscopy imaging (MRSI) spectra is presented. Afterward, we discuss a method based on complex PCA plus linear regression and a method based on cross correlation of the magnitude spectra for correcting frequency shifts of resonances in sets of MR spectra. Their advantages and limitations are demonstrated on simulated MR data sets as well as on an in vivo MRSI data set of the human brain.