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.     

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.     

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.     

New convolution theorem for the linear canonical transform and its translation invariance property

The linear canonical transform (LCT), which is a generalization of the Fourier transform (FT), has many applications in several areas, including signal processing and optics. Many properties for this transform are already known, but an extension of convolution theorem of FT is still not having a widely accepted closed form expression. In the literature of recent past different authors have tried to formulate convolution theorem for LCT, but none have received acclamation because their definition do not generalize very nicely the classical result for the FT. Moreover, those definitions exhibit only partial invariance properties which prevent their actual use in many applications of signal processing. The purpose of this paper is to introduce a new convolution structure for the LCT that preserves the translation invariance property. Indeed, an effective translation invariance is obtained by slightly modifying the former definitions and by introducing linear canonical translation operators.     

Non-Iterative Regularized reconstruction Algorithm for Non-CartesiAn MRI: NIRVANA

We introduce a novel noniterative algorithm for the fast and accurate reconstruction of nonuniformly sampled MRI data. The proposed scheme derives the reconstructed image as the nonuniform inverse Fourier transform of a compensated dataset. We derive each sample in the compensated dataset as a weighted linear combination of a few measured k-space samples. The specific k-space samples and the weights involved in the linear combination are derived such that the reconstruction error is minimized. The computational complexity of the proposed scheme is comparable to that of gridding. At the same time, it provides significantly improved accuracy and is considerably more robust to noise and undersampling. The advantages of the proposed scheme makes it ideally suited for the fast reconstruction of large multidimensional datasets, which routinely arise in applications such as f-MRI and MR spectroscopy. The comparisons with state-of-the-art algorithms on numerical phantoms and MRI data clearly demonstrate the performance improvement.     

Measurement of pulsatile flow using MRI and a Bayesian technique of probability analysis

This work shows that complete spatial information of periodic pulsatile fluid flows can be rapidly obtained by Bayesian probability analysis of flow encoded magnetic resonance imaging data. These data were acquired as a set of two-dimensional images (complete two-dimensional sampling of k-space or reciprocal position space) but with a sparse (six point) and nonuniform sampling of q-space or reciprocal displacement space. This approach enables more precise calculation of fluid velocity to be achieved than by conventional two q-sample phase encoding of velocities, without the significant time disadvantage associated with the complete flow measurement required for Fourier velocity imaging. For experimental comparison with the Bayesian analysis applied to nonuniformly sampled q-space data, a Fourier velocity imaging technique was used with one-dimensional spatial encoding within a selected slice and a uniform sampling of q-space using 64 values of the pulsed gradients to encode fluid flow. Because the pulsatile flows were axially symmetric within the resolution of the experiment, the radial variation of fluid velocity, in the direction of the pulsed gradients, was reconstructed from one-dimensional spatial projections of the velocity by exploiting the central slice theorem. Data were analysed for internal consistency using linearised flow theories. The results show that nonuniform q-space sampling followed by Bayesian probability analysis is at least as accurate as the combined uniform q-space sampling with Fourier velocity imaging and projection reconstruction method. Both techniques give smaller errors than a two-point sampling of q-space (the conventional flow encoding experiment).     

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.     

One step hologram calculation for multi-plane objects based on nonuniform sampling (Invited Paper)

The nonuniform sampling method in hologram plane is proposed to reconstruct objects on multi-plane simultaneously. The hologram is nonuniformly sampled by decomposing it into several parts with various sampling rates. The hologram is calculated based on the nonuniform fast Fourier transform （NUFFT） algorithm. In the experiment, we load this nonuniformly sampled hologram on phases-only spatial light modulator （SLM）, and by illumination with collimated light objects with different sampling rates are reconstructed at different distant planes simultaneously. Both of the numerically simulation and optical experiments are performed to demonstrate the feasibility of our method. The experiment also shows that our proposed nonuniform sampled hologram for multi-plane objects is calculated by only one step, better than conventional method that needs several steps of calculation proportional to the numbers of object planes.     

基于多项式调频Fourier变换的信号分量提取方法

为了从含有噪声的混合信号中有效提取各个信号分量, 提出一种基于多项式调频Fourier变换的分量提取方法. 通过研究Fourier变换和分数阶Fourier变换的信号能量积累方式及变换基函数的时频表示, 提出利用时频平面上的多项式调频曲线族代替Fourier变换和分数阶Fourier 变换的调频直线族, 将变换的适用范围扩展到非线性调频信号. 采用粒子群智能优化算法搜索调频曲线族的最优多项式参数, 使混合信号中的某一分量在多项式调频Fourier域上能量谱集中. 最后对能量谱集中的分量进行窄带滤波, 并利用多项式调频逆Fourier变换重构信号分量. 仿真实验结果表明, 该方法不仅能够提取混合信号中的线性调频分量, 还能够实现非线性调频分量的能量谱集中、信号分离和时频特征提取.     

基于离散Fourier变换的内源全息图重构计算方法

对基于离散Fourier变换的内源全息图重构计算方法作了深入系统的分析，讨论了如何计算整个球面全息图对原子重构像的贡献，克服了以往该方法只能计算半个球面全息图的不足，并运用采样定理知识，分析了全息图的采样率、重构像的空间范围和分辨率等问题. 关键词： 内源全息术 离散Fourier变换 采样定理 同步辐射     

Novel spectrum properties of the periodic π-phase-shifted fiber Bragg grating

A special periodic π-phase-shifted (π-PS) fiber Bragg grating (FBG) is proposed to achieve multichannel reflections. The Fourier analysis and the simulations based on the transfer matrix method show that this simple structure can generate multichannel response with high reflectivity. The reflectivity is much greater than that of the traditional sampled FBG for the same value of κL. Therefore the length of the fiber or the amplitude of refractive index modulation can be greatly reduced if such a periodic π-PS FBG is used instead of the traditional sampled FBG in the applications of comb filtering and optical wavelength division multiplexing. Besides, when the duty cycle is 0.5, this periodic π-PS FBG can create dual wavelength spectrum with high reflectivity.     

Generalized wavelet transform based on the convolution operator in the linear canonical transform domain

The wavelet transform (WT) and linear canonical transform (LCT) have been shown to be powerful tool for optics and signal processing. In this paper, firstly, we introduce a novel time-frequency transformation tool coined the generalized wavelet transform (GWT), based on the idea of the LCT and WT. Then, we derive some fundamental results of this transform, including its basis properties, inner product theorem and convolution theorem, inverse formula and admissibility condition. Further, we also discuss the time-fractional-frequency resolution of the GWT. The GWT is capable of representing signals in the time-fractional-frequency plane. Last, some potential applications of the GWT are also presented to show the advantage of the theory. The GWT can circumvent the limitations of the WT and the LCT.     

Generalized analytic signal associated with linear canonical transform

Analytic signal is tightly associated with Hilbert transform and Fourier transform. The linear canonical transform is the generalization of many famous linear integral transforms, such as Fourier transform, fractional Fourier transform and Fresnel transform. Based on the parameter (a, b)-Hilbert transform and the linear canonical transform, in this paper, we develop some issues on generalized analytic signal. The generalized analytic signal can suppress the negative frequency components in the linear canonical transform domain. Furthermore, we prove that the kernel function of the inverse linear canonical transform satisfies the generalized analytic condition and get the generalized analytic pairs. We show the generalized Bedrosian theorem is valid in the linear canonical transform domain.     

Sampling of fractional bandlimited signals associated with fractional Fourier transform

Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of reconstructing a fractional bandlimited signal with FRFT. We propose a novel reconstruction method for fractional bandlimited signals using the fractional Fourier series (FRFS). The advantage is that the sampling expansion can be deduced directly not based on the Shannon theorem. By utilizing the generalized form of Parseval’s relation for complex FRFS, we obtain the sampling expansion for fractional bandlimited signals with FRFT. We show that the sampling expansion for fractional bandlimited signals with FRFT is a special case of Parseval’s relation for complex FRFS.     

Papoulis-like generalized sampling expansions in fractional Fourier domains and their application to superresolution

We present a generalized convolution theorem in the fractional Fourier domains that preserves the convolution theorem of the conventional Fourier transform. The Papoulis-like generalized sampling expansions in the fractional Fourier domains using this generalized convolution theorem are also derived and it is shown that the classical generalized Papoulis sampling expansion is a special case of it. Its application in the context of the image superresolution is also discussed.     

基于非均匀周期采样的傅里叶望远镜时域信号采集方法

为了在低信噪比条件下清晰重构深空暗弱目标,提出了一种基于非均匀周期采样（NUPS）的傅里叶望远镜（FT）时域信号采集方法。对提出的方法进行了模拟实验并与均匀采样方法重构的图像进行了对比。基于NUPS方法,用1 MHz和5 MHz的采样频率分别采集100个点,对两个序列信号分别进行快速傅里叶变换,并对关心频率信息进行平均;传统的均匀采样方法则分别用1 MHz和5 MHz的采样频率采集200个点,再进行解调平均。对比结果显示：当信噪比（SNR）为50时,本文重构图像与衍射极限图像的斯托里尔比（Strehl）相比原方法提升了0.03,SNR为20时,Strehl比为0.531 1,较均匀采样提高了0.223 3。实验结果表明：NUPS方法在低信噪比条件下成像质量较高,可降低对激光功率的要求,为FT工程系统的实施奠定了技术基础。     

一种面向信息带宽的频谱感知方法研究

本文利用频带宽度先验信息,提出一种面向信息带宽的自适应调制宽带转换器结构.该结构的总采样率为信号信息带宽的四倍,远小于信号的奈奎斯特采样频率,从而更有效利用采样资源,降低采样数据量,提高处理实时性.通过对该结构中随机波形函数周期的选择,可以实现对系统采样率和系统物理实现复杂度的权衡取舍,从而适应不同场合中的应用.本文通过理论分析给出了该结构实现信号精确重构的充分条件.引入多重信号分类算法,分析了该结构适用此算法的充分条件.本文通过仿真实验对上述分析进行了有效性验证.该系统可以应用于隐形装备的吸波材料的前端特性分析、认知无线电的频谱感知.     

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.     

Signal reconstruction from the undersampled signal samples

It is well-known from the celebrated Shannon sampling theorem for bandlimited signals that if the sampling rate is below the Nyquist rate, aliasing takes place and the original signal cannot be reconstructed back by simply passing the signal samples through an ideal lowpass filter. However, researchers such as Stern and Gori have shown the existence of some classes of signals for which the signals are sampled below the Nyquist rate but perfect signal reconstruction is still possible from the given signal samples. Here, we present a generalized lowpass sampling theorem and show that Stern’s and Gori’s lowpass sampling theorems are special cases of it. A sampling theorem for the bandpass signals in the linear canonical transform domains is also presented and its special cases are discussed. Using a modification of the conventional natural sampling waveform with a specific width of the pulses, it is shown that the sampling rate in our generalized lowpass sampling theorem and hence in the Stern’s and the classical Shannon sampling theorems can be further reduced by a factor of two, while for the bandpass signals, the reduction in the sampling rate by some factor is possible only under some restricted conditions.