SEnD NMR: Sensitivity Enhanced n-Dimensional NMR

Sparse sampling offers tremendous potential for overcoming the time limitations imposed by traditional Cartesian sampling of indirectly detected dimensions of multidimensional NMR data. However, in many instances sensitivity rather than time remains of foremost importance when collecting data on protein samples. Here we explore how to optimize the collection of radial sampled multidimensional NMR data to achieve maximal signal-to-noise. A method is presented that exploits a rigorous definition of the minimal set of radial sampling angles required to resolve all peaks of interest in combination with a fundamental statistical property of radial sampled data. The approach appears general and can achieve a substantial sensitivity advantage over Cartesian sampling for the same total data acquisition time. Termed Sensitivity Enhanced n-Dimensional or SEnD NMR, the method involves three basic steps. First, data collection is optimized using routines to determine a minimal set of radial sampling angles required to resolve frequencies in the radially sampled chemical shift evolution dimensions. Second, appropriate combinations of experimental parameters (transients and increments) are defined by simple statistical considerations in order to optimize signal-to-noise in single angle frequency domain spectra. Finally, the data is processed with a direct multidimensional Fourier transform and a statistical artifact and noise removal step is employed.     

Sampling of the NMR time domain along concentric rings

We present a novel approach to sampling the NMR time domain, whereby the sampling points are aligned on concentric rings, which we term concentric ring sampling (CRS). Radial sampling constitutes a special case of CRS where each ring has the same number of points and the same relative orientation. We derive theoretically that the most efficient CRS approach is to place progressively more points on rings of larger radius, with the number of points growing linearly with the radius, a method that we call linearly increasing CRS (LCRS). For cases of significant undersampling to reduce measurement time, a randomized LCRS (RLCRS) is also described. A theoretical treatment of these approaches is provided, including an assessment of artifacts and sensitivity. The analytical treatment of sensitivity also addresses the sensitivity of radially sampled data processed by Fourier transform. Optimized CRS approaches are found to produce artifact-free spectra of the same resolution as Cartesian sampling, for the same measurement time. Additionally, optimized approaches consistently yield fewer and smaller artifacts than radial sampling, and have a sensitivity equal to Cartesian and better than radial sampling. We demonstrate the method using numerical simulations, as well as a 3D HNCO experiment on protein G B1 domain.     

Two-dimensional Fourier transform of arbitrarily sampled NMR data sets

A new procedure for Fourier transform with respect to more than one time variable simultaneously is proposed for NMR data processing. In the case of two-dimensional transform the spectrum is calculated for pairs of frequencies, instead of conventional sequence of one-dimensional transforms. Therefore, it enables one to Fourier transform arbitrarily sampled time domain and thus allows for analysis of high dimensionality spectra acquired in a short time. The proposed method is not limited to radial sampling, it requires only to fulfill the Nyquist theorem considering two or more time domains at the same time. We show the application of new approach to the 3D HNCO spectrum acquired for protein sample with radial and spiral time domain sampling.     

Optimized angle selection for radial sampled NMR experiments

Sparse sampling offers tremendous potential for overcoming the time limitations imposed by traditional Cartesian sampling of indirectly detected dimensions of multidimensional NMR data. Unfortunately, several otherwise appealing implementations are accompanied by spectral artifacts that have the potential to contaminate the spectrum with false peak intensity. In radial sampling of linked time evolution periods, the artifacts are easily identified and removed from the spectrum if a sufficient set of radial sampling angles is employed. Robust implementation of the radial sampling approach therefore requires optimization of the set of radial sampling angles collected. Here we describe several methods for such optimization. The approaches described take advantage of various aspects of the general simultaneous multidimensional Fourier transform in the analysis of multidimensional NMR data. Radially sampled data are primarily contaminated by ridges extending from authentic peaks. Numerical methods are described that definitively identify artifactual intensity and the optimal set of sampling angles necessary to eliminate it under a variety of scenarios. The algorithms are tested with both simulated and experimentally obtained triple resonance data.     

Phasing arbitrarily sampled multidimensional NMR data

The recent re-introduction of the two-dimensional Fourier transformation (2D-FT) has allows for the transformation of arbitrarily sampled time domain signals. In this respect, radial sampling, where two incremented time dimensions (t(1) and t(2)) are sampled such that t(1)=taucosalpha and t(2)=tausinalpha, is especially appealing because of the relatively small leakage artifacts that occur upon Fourier transformation. Unfortunately radially sampled time domain data results in a fundamental artifact in the frequency domain manifested as a ridge of intensity extending through the peak positions perpendicular to +/- the radial sampling angle. Successful removal of the ridge artifacts using existing algorithms requires absorptive line shapes. Here we present two procedures for retrospective phase correction of arbitrarily sampled data.     

窄带高光谱干涉成像的压缩采样复原方法

利用干涉成像光谱仪对目标进行窄带高光谱成像探测具有高光通量、高光谱分辨率和高目标分辨率等优点。按照尼奎斯特定理对窄带光谱干涉信息进行采样存在较大的数据冗余,增加了后期傅里叶变换的数据处理量,影响光谱的复原效率。在分析窄带光谱傅里叶变换特性的基础上,提出了基于滤光片光谱透射率函数的窄带光谱压缩采样方法。引入滤光片参数和混叠参数,可以复原不同精度的窄带光谱信息。配以符合要求的多带通窄带滤光片,可对目标进行压缩采样获取多个谱段的窄带光谱信息,从而避免了逐个谱段探测,提高了探测效率。对该方法进行了仿真分析和实验验证,得到了与目标光谱相吻合的复原窄带光谱。     

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.     

Nonuniform sampling and maximum entropy reconstruction applied to the accurate measurement of residual dipolar couplings

Residual dipolar couplings (RDC) provide important global restraints for accurate structure determination by NMR. We show that nonuniform sampling in combination with maximum entropy reconstruction (MaxEnt) is a promising strategy for accelerating and potentially enhancing the acquisition of RDC spectra. Using MaxEnt-processed spectra of nonuniformly sampled data sets that are reduced up to one fifth relative to uniform sampling, accurate 13C'-13Calpha RDCs can be obtained that agree with an RMS of 0.67 Hz with those derived from uniformly sampled, Fourier transformed spectra. While confirming that frequency errors in MaxEnt spectra are very slight, an unexpected class of systematic errors was found to occur in the 6th significant figure of 13C' chemical shifts of doublets obtained by MaxEnt reconstruction. We show that this error stems from slight line shape perturbations and predict it should be encountered in other nonlinear spectral estimation algorithms. In the case of MaxEnt reconstruction, the error can easily be rendered systematic by straightforward optimization of MaxEnt reconstruction parameters and self-cancels in obtaining RDCs from nonuniformly sampled, MaxEnt reconstructed spectra.     

Ising耦合体系中量子傅里叶变换的优化

量子傅里叶变换是量子计算中一种重要的量子逻辑门. 任意量子位的傅里叶变换可以分解为一系列普适的单比特量子逻辑门和两比特量子逻辑门, 这种分解方式使得傅里叶变换的实验实现简单直观, 但所用的实验时间显然不是最短的. 本文利用优化控制和数值计算方法对Ising耦合体系中多量子位傅里叶变换的实验时间进行优化, 优化后的实现方法明显短于传统方法. 优化方法的核磁共振实验实现验证了其有效性.     

基于低秩矩阵的非均匀采样NMR波谱重建进展

多维核磁共振（Nuclear Magnetic Resonance,NMR）利用多维波谱来分析分子结构,被广泛用于化学、生物学和医学等领域,但信号采样时间随波谱维度和采样点数增加而迅速增长．非均匀采样通过降低间接维采样点数来加速数据采集,并引入合理的重建方法获得完整的NMR波谱．如何快速重建高质量的波谱,是NMR信号处理研究的前沿．本文主要综述近年来基于低秩矩阵的NMR波谱重建方法的发展．首先介绍了低秩矩阵的相关数学基础;然后从一般低秩矩阵和结构化低秩汉克尔矩阵两个角度来论述重建模型,并讨论相关的NMR波谱应用;最后分析了该技术存在的不足,并展望其未来发展的趋势．     

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.     

A new general-purpose fully automatic baseline-correction procedure for 1D and 2D NMR data

A new procedure for automatic baseline correction of NMR data sets is presented. It is based on an improved automatic recognition of signal-free regions that uses a Continuous Wavelet transform derivative calculation, followed by a baseline modelling procedure based on the Whittaker smoother algorithm. The method has been proven to automatically flatten 1D and 2D NMR spectra with large baseline distortions arising from different sources, is tolerant to low signal-to-noise ratio spectra, and to signals of varying widths in a single spectrum. Even though this procedure has so far only been applied to NMR spectra, we believe it to also be applicable to other spectroscopies having relatively narrow peaks (e.g., mass spectrometry), and potentially to those with broad peaks (e.g., near infrared or ultraviolet).     

Modified oscillating gradient pulses for direct sampling of the diffusion spectrum suitable for imaging sequences

A variation of the oscillating gradient spin echo method had been developed, which isolates temporal frequencies of the dephasing spectrum. This allows sampling of the diffusion spectrum, the Fourier transform of the velocity correlation function (VCF). It has been shown that restriction and flow alter this function in ways that can be mathematically characterized, yielding quantitative information on restriction geometry and flow parameters. It is demonstrated that in many systems of interest, dispersion of velocity will produce a peak in the VCF spectrum near omega=0, while restricted diffusion will manifest itself in the spectrum at higher frequencies. The method, therefore, may be useful for decoupling their effects on the apparent diffusion coefficient (ADC), as well as in revealing the physics of both phenomena. This method has been implemented in model systems of packed beads, yielding data consistent with theoretical models of restricted diffusion spectra and data from one previous study. The method may have significant application to biology and medicine, as well as the study of transport phenomena in porous media and complex flow.     

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.     

成像型任意反射面速度干涉仪数据处理方法

 采用傅里叶变换方法,列出了对成像型任意反射面速度干涉仪得到的干涉条纹图进行处理的过程。利用文献结果,处理了冲击波整形时产生的有间断干涉条纹图,对处理结果进行了分析。结果表明：条纹图要干净,条纹间距清晰、均匀,应使用１维傅里叶变换的方法处理条纹图,减小滤波的难度。研究了冲击波测试当中条纹丢失的问题,提出了间断条纹图间断起始点相位确定的方法。对透明和不透明靶产生条纹图的不同进行了讨论,得出对透明靶产生的条纹图应采用适当的条纹外延技术进行预处理后再进行常规处理。     

非均匀采样的功率谱反演大气湍流相位屏的快速模拟

对大气湍流功率谱非均匀采样可以有效改善传统功率谱反演法低频采样严重不足的缺陷, 实现高精度的大气湍流相位屏的模拟. 但采用的直接求和运算计算复杂度高, 相位屏的模拟速度极慢. 将非均匀快速傅里叶变换(NUFFT)引入到大气湍流相位屏的模拟, 可以实现相位屏的快速模拟. 从随机过程的谱分解出发, 将大气湍流相位随机过程表示为有限谐波分量叠加和的均方极限. 通过一个高斯核函数的卷积, 将非均匀分布的谐波复振幅映射到均匀网格空间, 进而利用快速傅里叶变换, 降低计算复杂度, 加快大气湍流相位屏的模拟速度. 以大气湍流的Kolmogorov 谱为例, 利用NUFFT仿真得到大气湍流相位屏, 并对相位屏的模拟精度、模拟速度和误差进行统计分析. 结果表明, NUFFT的引入可以实现快速、高精度的大气湍流相位屏的模拟.     

Narrow peaks and high dimensionalities: Exploiting the advantages of random sampling

Level of artifacts in spectra obtained by Multidimensional Fourier Transform has been studied, considering randomly sampled signals of high dimensionality and long evolution times. It has been shown theoretically and experimentally, that this level is dependent on the number of time domain samples, but not on its relation to the number of points required in appropriate conventional experiment. Independence of the evolution time domain size (in the terms of both: dimensionality and evolution time reached), suggests that random sampling should be used rather to design new techniques with large time domain than to accelerate standard experiments. 5D HC(CC-TOCSY)CONH has been presented as the example of such approach. The feature of Multidimensional Fourier Transform, namely the possibility of calculating spectral values at arbitrary chosen frequency points, allowed easy examination of resulting spectrum. We present the example of such approach, referred to as Sparse Multidimensional Fourier Transform.     

Selective excitation in Fourier transform nuclear magnetic resonance. 1978

The applications of frequency-selective excitation methods in Fourier transform NMR are discussed, and a simple technique is described for selective excitation of a narrow frequency region of a high-resolution NMR spectrum in a Fourier transform spectrometer. A regular sequence of identical radiofrequency pulses of small flip angle exerts a strong cumulative effect on magnetizations close to resonance with the transmitter frequency or one of a set of equally spaced sidebands separated by the pulse repetition rate. All other magnetizations precess through an incomplete number of full rotations between pulses, and are caught by successive pulses at an ever changing phase of their precession, which destroys the cumulative effect. The motion of the various nuclear magnetization vectors may be described pictorially according to the Bloch equations, neglecting relaxation during the pulse sequence. A general theory is presented for selective or “tailored” excitation by an arbitrary modulation of the radiofrequency transmitter signal. It confirms earlier conclusions that the frequency-domain excitation spectrum corresponds to the Fourier transform of the transmitter modulation pattern, provided that the NMR response remains linear. The excitation spectra calculated for the selective pulse sequence by these two alternative approaches show good agreement within their respective limitations. A number of practical applications of selective excitation are explored, including solvent peak suppression, the detection of partial spectra from individual chemical sites, selective studies of relaxation and slow chemical exchange, and holeburning or localized saturation.     

非均匀采样干涉数据光谱反演技术研究

干涉光谱成像仪获取的干涉数据是一种中间数据,需要进行光谱反演才能够得到目标光谱数据,傅里叶变换方法是常规的光谱反演方法。由于干涉数据中存在非均匀采样问题,若忽略光谱混叠,直接采用快速傅里叶变换会导致反演光谱的失真,难以满足实时处理需求。针对非均匀采样干涉数据的光谱反演需求,将插值及非均匀快速傅里叶变换(NUFFT)方法应用到光谱反演处理中,对过采样及部分欠采样情况下的非均匀采样干涉数据,提出了相应的光谱反演方法,并分析了方法的适用性。最后对过采样及部分欠采样情况下的光谱反演方法进行计算机仿真,过采样情况下采用NUFFT方法反演光谱的精度要明显高于插值方法,而部分欠采样情况下插值方法反演光谱的精度要明显高于NUFFT方法,并对欠采样造成的光谱混叠有一定的修正,验证了方法的有效性。