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.     

Deterministic schedules for robust and reproducible non-uniform sampling in multidimensional NMR

We show that a simple, general, and easily reproducible method for generating non-uniform sampling (NUS) schedules preserves the benefits of random sampling, including inherently reduced sampling artifacts, while removing the pitfalls associated with choosing an arbitrary seed. Sampling schedules are generated from a discrete cumulative distribution function (CDF) that closely fits the continuous CDF of the desired probability density function. We compare random and deterministic sampling using a Gaussian probability density function applied to 2D HSQC spectra. Data are processed using the previously published method of Spectroscopy by Integration of Frequency and Time domain data (SIFT). NUS spectra from deterministic sampling schedules were found to be at least as good as those from random schedules at the SIFT critical sampling density, and significantly better at half that sampling density. The method can be applied to any probability density function and generalized to greater than two dimensions.     

Theory of mirrored time domain sampling for NMR spectroscopy

A generalized theory is presented for novel mirrored hypercomplex time domain sampling (MHS) of NMR spectra. It is the salient new feature of MHS that two interferograms are acquired with different directionality of time evolution, that is, one is sampled forward from time t=0 to the maximal evolution time tmax, while the second is sampled backward from t=0 to -tmax. The sampling can be accomplished in a (semi) constant time or non constant-time manner. Subsequently, the two interferograms are linearly combined to yield a complex time domain signal. The manifold of MHS schemes considered here is defined by arbitrary settings of sampling phases ('primary phase shifts') and amplitudes of the two interferograms. It is shown that, for any two given primary phase shifts, the addition theorems of trigonometric functions yield the unique linear combination required to form the complex signal. In the framework of clean absorption mode (CAM) acquisition of NMR spectra being devoid of residual dispersive signal components, 'secondary phase shifts' represent time domain phase errors which are to be eliminated. In contrast, such secondary phase shifts may be introduced by experimental design in order to encode additional NMR parameters, a new class of NMR experiments proposed here. For generalization, it is further considered that secondary phase shifts may depend on primary phase shifts and/or sampling directionality. In order to compare with MHS theory, a correspondingly generalized theory is derived for widely used hypercomplex ('States') sampling (HS). With generalized theory it is shown, first, that previously introduced 'canonical' schemes, characterized by primary phases being multiples of π/4, afford maximal intensity of the desired absorptive signals in the absence of secondary phase shifts, and second, how primary phases can be adjusted to maximize the signal intensity provided that the secondary phase shifts are known. Third, it is demonstrated that theory enables one to accurately measure secondary phase shifts and amplitude imbalances. Application to constant time 2D [13C, 1H]-HSQC spectra recorded for a protein sample with canonical MHS/HS schemes showed that accurate CAM data acquisition can be readily implemented on modern spectrometers for experiments based on through-bond polarization transfer. Fourth, when moderate variations of secondary phase shifts with primary phase shift and/or sampling directionality are encountered, generalized theory allowed comparison of the robustness of different MHS/HS schemes for CAM data acquisition, and thus to identify the scheme best suited to suppress dispersive peak components and quadrature image peaks. Moreover, it is shown that for spectra acquired with several indirect evolution periods, the best suited scheme can be identified independently for each of the periods.     

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.     

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.     

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.     

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.     

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.     

压缩感知同步扫描重建及其采样方案的研究

压缩感知(compressed sensing,CS)-磁共振成像(magnetic resonance imaging,MRI)技术使用随机欠采样的k空间数据来重建图像,大大提高了成像速度.但典型的CS重建很费时,这也是CS-MRI临床应用的主要障碍之一.针对这一问题,该文提出了在扫描时同步进行CS图像重建的方案.在同步重建的过程中,可以实时显示重建图像的结果,用户可以根据图像质量来决定何时终止扫描,这样可以在节约扫描和重建时间的同时,更好地控制图像质量.由于预先无法确定最终的采样率,因此传统的变密度随机采样方法并不完全适用.该文设计了适用于同步重建过程的采样模式生成方案,同时提出了分段采样方法,把采样过程分为两个阶段,不同阶段使用不同的概率密度函数(probability density function,PDF)确定待采样的相位编码行.模拟实验的结果表明,与使用单一密度函数的采样方案相比,分段采样方案能够在整个同步扫描重建过程中始终获得更好的图像.     

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．     

Multichannel sampling theorem for bandpass signals in the linear canonical transform domain

The linear canonical transform (LCT) describes the effect of first-order quadratic phase optical system on a wave field. The classical multichannel sampling theorem for common bandlimited signals has been extended differently to bandlimited signals associated with LCT. However, a practical issue associated with the reconstruction of the original bandpass signal from multichannel samples in LCT domain still remains unresolved. The purpose of this paper is to introduce a practical multichannel sampling theorem for bandpass signals in LCT domain. The sampling expansion which is constructed by the ordinary convolution in the time domain can reduce the effect of spectral leakage and is easy to implement. The classical multichannel sampling theorem and the well-known sampling theorems for the LCT are shown to be special cases of it. Some potential applications of the multichannel sampling are also presented to show the advantage of the theory.     

面向全光物理随机数发生器的混沌实时光采样研究

基于混沌激光实现全光物理随机数发生器的物理基础是完成对混沌光信号的高速实时全光采样. 本文利用偏振无关的SOA构建出TOAD全光采样门, 以光反馈半导体激光器产生混沌激光, 对混沌激光的全光采样可行性进行了原理性实验论证, 实现了对光反馈半导体激光器产生的6.4 GHz带宽的混沌激光5 GSa/s的实时、高保真全光采样. 进一步研究显示, 光采样周期与外腔反馈时间成比例与否对混沌信号弱周期性的抑制水平影响显著. 当两者不成比例时, 可有效消除原始混沌信号的弱周期性, 有利于高质量物理随机数的产生.     

Adaptive k-space sampling design for edge-enhanced DCE-MRI using compressed sensing

The critical challenge in dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is the trade-off between spatial and temporal resolution due to the limited availability of acquisition time. To address this, it is imperative to under-sample k-space and to develop specific reconstruction techniques. Our proposed method reconstructs high-quality images from under-sampled dynamic k-space data by proposing two main improvements; i) design of an adaptive k-space sampling lattice and ii) edge-enhanced reconstruction technique. A high-resolution data set obtained before the start of the dynamic phase is utilized. The sampling pattern is designed to adapt to the nature of k-space energy distribution obtained from the static high-resolution data. For image reconstruction, the well-known compressed sensing-based total variation (TV) minimization constrained reconstruction scheme is utilized by incorporating the gradient information obtained from the static high-resolution data. The proposed method is tested on seven real dynamic time series consisting of 2 breast data sets and 5 abdomen data sets spanning 1196 images in all. For data availability of only 10%, performance improvement is seen across various quality metrics. Average improvements in Universal Image Quality Index and Structural Similarity Index Metric of up to 28% and 24% on breast data and about 17% and 9% on abdomen data, respectively, are obtained for the proposed method as against the baseline TV reconstruction with variable density random sampling pattern.     

High resolution dynamic cardiac MRI using partial separability of spatiotemporal signals with a novel sampling scheme

The partial separability (PS) of spatiotemporal signals has been exploited to accelerate dynamic cardiac MRI by sampling two datasets (training and imaging datasets) without breath-holding or ECG triggering. According to the theory of partially separable functions, the wider the range of spatial frequency components covered by the training dataset, the more accurate the temporal constraint imposed by the PS model. Therefore, it is necessary to develop a new sampling scheme for the PS model in order to cover a wider range of spatial frequency components. In this paper, we propose the use of radial sampling trajectories for collecting the training dataset and Cartesian sampling trajectories for collecting the imaging dataset. In vivo high resolution cardiac MRI experiments demonstrate that the proposed data sampling scheme can significantly improve the image quality. The image quality using the PS model with the proposed sampling scheme is comparable to that of a commercial method using retrospective cardiac gating and breath-holding. The success of this study demonstrates great potential for high-quality, high resolution dynamic cardiac MRI without ECG gating or breath-holding through use of the PS model and the novel data sampling scheme.     

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.     

Gray-Level Image Encryption Scheme Using Full Phase Encryption and Phase-Encoded Exclusive-OR Operations

A new gray-level image encryption scheme based on phase-encoded exclusive-OR (XOR) operations and a full phaseencoding method is presented. The gray-level image can be sliced into binary images which have the same pixel level, and each of them is encrypted by phase-encoded XOR operations with each phase-encoded binary random image. We combine XORed images and phase-encoded binary random images into an encrypted image and a key image, respectively. Gray-level encrypted data and key data are obtained by the phase-encoding process of the encrypted image and the key image in a space domain. The decryption process is simply implemented by using a phasevisualization system and the performance of the proposed scheme is evaluated by computer simulation.     

深穿透粒子输运问题的自适应抽样方法

屏蔽计算中的深穿透问题一直是蒙特卡罗计算的一个难题,研究了一种发射点作为驿站的随机游动机制,推导了相应的自适应抽样方法。其主要优势在于,在蒙特卡罗方法求解粒子输运的同时,利用已经获得的信息,自适应地控制各次抽样数,不断完善计算进程。通过对碰撞点引进重要性函数,实现发射点作为驿站的重要性抽样,并结合自适应控制达到最佳抽样状态。数值结果表明:基于发射点作为驿站的自适应抽样方法,在一定程度上克服了深穿透计算中估计值偏低现象。相应的重要函数抽样方法获得了满意的结果。     

Linear exchanging operation and random phase encoding in gyrator transform domain for double image encryption

We present an optical method for double image encryption by using linear exchanging operation and double random phase encoding (DRPE) in the gyrator transform (GT) domain. In the linear exchanging operation, two primitive images are linearly recombined via a random orthogonal transform matrix. The resultant blended images are employed to constitute a complex-valued image, which is then encoded into a noise-like encrypted image by a DRPE structure in the GT domain. One can recover the primitive images exactly with all decryption keys correctly applied, including the transform orders, the random phase masks and random angle function used for linear exchanging operation. Computer simulations have been given to demonstrate that the proposed scheme eliminates the difference in key spaces between the phase-based image and the amplitude-based image encountered in the previous schemes. Moreover, our scheme has considerably high security level and certain robustness against data loss and noise disturbance.     

基于低采样率模数转换器的延时复用频分多址无源光网络

基于正交频分复用技术的无源光网络中,光网络单元为了获得其所属小部分下行数据,需高采样率模数转换器将所有频宽的信号恢复才能分出其所需要数据.同时正交频分信号峰均比很高,传输中容易引起非线性效应.为此,本文提出一种基于低采样模数转换器的延时复用频分多址无源光网络.在光线路终端将数据序列交错排序并在时域映射为正交幅度调制信号;再通过离散傅里叶变换扩频技术,将信号转换为频域信号并映射到子载波上.通过预先发送和回传训练信号,估测包括延时采样和低采样接收在内的信道频响;再将频域信号利用估测信息在光线路终端做预处理,从而使信号传输中的失真得到有效预补偿.本文实验演示了含有多个光网络单元的系统,对于含有M个光网络单元的无源光网络,模数转换器的采样率可以降低到1/M Nyquist采样率,实验中模数转换器的采样率可以降低到1/32 Nyquist采样率;由于下行信号通过光线路终端预处理实现失真预补偿,光网络单元接收到的信号不需要均衡,不需要傅里叶变换和傅里叶逆变换,避免了与之对应的相关计算量,降低了光网络单元的计算复杂度;由于使用了扩频技术,信号波形具有更低的峰均比,从而降低了非线性对信号的影响,增加了功率预算.此外,随着光网络单元的增加,信号的误码率几乎没有增加,光网络单元个数增加到32时,向前纠错极限为10~(-3)的功率代价小于0.5 dB;系统对光网络单元采样时刻偏离具有一定容限;25 km光纤传输的功率代价大约0.5 dB.理论和实验均证明本方案能够简化光网络单元,降低无源光网络的成本;与传统的无源光网络相比具有明显优势.