Sampling in the light of Wigner distribution

We propose a new method for analysis of the sampling and reconstruction conditions of real and complex signals by use of the Wigner domain. It is shown that the Wigner domain may provide a better understanding of the sampling process than the traditional Fourier domain. For example, it explains how certain non-bandlimited complex functions can be sampled and perfectly reconstructed. On the basis of observations in the Wigner domain, we derive a generalization to the Nyquist sampling criterion. By using this criterion, we demonstrate simple preprocessing operations that can adapt a signal that does not fulfill the Nyquist sampling criterion. The preprocessing operations demonstrated can be easily implemented by optical means.     

A compressive MUSIC spectral approach for identification of closely-spaced structural natural frequencies and post-earthquake damage detection

Motivated by practical needs to reduce data transmission payloads in wireless sensors for vibration-based monitoring of engineering structures, this paper proposes a novel approach for identifying resonant frequencies of white-noise excited structures using acceleration measurements acquired at rates significantly below the Nyquist rate. The approach adopts the deterministic co-prime sub-Nyquist sampling scheme, originally developed to facilitate telecommunication applications, to estimate the autocorrelation function of response acceleration time-histories of low-amplitude white-noise excited structures treated as realizations of a stationary stochastic process. Next, the standard multiple signal classification (MUSIC) spectral estimator is applied to the estimated autocorrelation function enabling the identification of structural natural frequencies with high resolution by simple peak picking in the frequency domain without posing any sparsity conditions to the signals. This is achieved by processing autocorrelation estimates without undertaking any (typically computationally expensive) signal reconstruction step in the time-domain, as required by various recently proposed in the literature sub-Nyquist compressive sensing-based approaches for structural health monitoring, while filtering out any broadband noise added during data acquisition. The accuracy and applicability of the proposed approach is first numerically assessed using computer-generated noise-corrupted acceleration time–history data obtained by a simulation-based framework examining white-noise excited structural systems with two closely-spaced modes of vibration carrying the same amount of energy, and a third isolated weakly excited vibrating mode. Further, damage detection potential of the developed method is numerically illustrated using a white-noise excited reinforced concrete 3-storey frame in a healthy and two damaged states caused by ground motions of increased intensity. The damage assessment relies on shifts in natural frequencies between the pre-earthquake and post-earthquake state. Overall, numerical results demonstrate that the considered approach can accurately identify structural resonances and detect structural damage associated with changes to natural frequencies as minor as 1% by sampling up to 78% below Nyquist rate for signal to noise ratio as low as 10dB. These results suggest that the adopted approach is robust and noise-immune while it can reduce data transmission requirements in acceleration wireless sensors for natural frequency identification and damage detection in engineering structures.     

Multirate asynchronous sampling of sparse multiband signals

Because optical systems have a huge bandwidth and are capable of generating low-noise short pulses, they are ideal for undersampling multiband signals that are located within a very broad frequency range. We propose a new scheme for reconstructing multiband signals that occupy a small part of a given broad frequency range under the constraint of a small number of sampling channels. The scheme, which we call multirate sampling (MRS), entails gathering samples at several different rates whose sum is significantly lower than the Nyquist sampling rate. The number of channels does not depend on any characteristics of a signal. In order to be implemented with simplified hardware, the reconstruction method does not rely on the synchronization between different sampling channels. Also, because the method does not solve a system of linear equations, it avoids one source of lack of robustness of previously published undersampling schemes. Our simulations indicate that our MRS scheme is robust both to different signal types and to relatively high noise levels. The scheme can be implemented easily with optical sampling systems.     

Signal-to-noise ratio trade-offs associated with coarsely sampled Fourier transform spectroscopy

We derive the spectral signal-to-noise ratio (SNR) trade-offs associated with coarsely sampled Fourier transform spectroscopy using a step-and-integrate measurement scheme. We show that there is no SNR penalty in the shot noise limit and a slight SNR benefit in the detector noise limit for the case of coarse sampling to achieve the same spectral resolution as a baseline Nyquist sampling scenario, where the total detector integration time is the same for both sampling cases.     

Correspondence

Abstract

According to the Nyquist sampling theorem, a large number of sampled images and small intervals between capturing cameras should be met for rendering high quality virtual views without aliasing, which is hard to realize in practice. Therefore, achieving a balance between multi-view data capturing and quality of the rendered view remains as open problems. To solve this problem, we analysed the spectral bounds of the scene and designed a reconstruction filter. A proper number for rendering and a three-dimensional surface describing the relation between multi-view data capturing and quality of the rendered view were derived. Experimental results for both the modelled scene and the real scene show that only about 20% of sample images are needed compared with Nyquist sampling, while the quality of the rendered view remains higher than that of a Nyquist sampled comparison.     

Determination of the optimum sampling frequency of noisy images by spatial statistics

In optical metrology the final experimental result is normally an image acquired with a CCD camera. Owing to the sampling at the image, an interpolation is usually required. For determining the error in the measured parameters with that image, knowledge of the uncertainty at the interpolation is essential. We analyze how kriging, an estimator used in spatial statistics, can generate convolution kernels for filtering noise in regularly sampled images. The convolution kernel obtained with kriging explicitly depends on the spatial correlation and also on metrological conditions, such as the random fluctuations of the measured quantity, and the resolution of the measuring devices. Kriging, in addition, allows us to determine the uncertainty of the interpolation, and we have analyzed it in terms of the sampling frequency and the random fluctuations of the image, comparing it with Nyquist criterion. By use of kriging, it is possible to determine the optimum-required sampling frequency for a noisy image so that the uncertainty at interpolation is below a threshold value.     

A lateral super-resolution imaging method using structured illumination without phase shift

Structured illumination microscopy has been a useful method for achieving lateral super-resolution, but it typically requires at least three precise phase shifts per orientation. In this paper, we propose a super-resolution method that utilizes structured illumination without phase shift. The reconstruction process requires only a conventionally illuminated image and an image with structured illumination. This method achieves the same effect as the traditional phase shift method, and more than doubles the resolution by synthesizing a few reconstructions at different illumination frequencies. We verify the resolution improvement process using a combination of theoretical derivations and diagrams, and demonstrate its effectiveness with numerical simulations.     

Notices

The expected number of maxima and level crossings of a continuous stationary Gaussian process and the discrete process obtained by sampling the continuous one are evaluated and compared. The ratio of these two values as a function of sampling rate for two classes of spectra is calculated. It is shown that the rule of thumb of sampling at twice the Nyquist frequency is a good one.     

基于多帧"差分"图像的分辨率提高算法

介绍了一种基于多帧"差分"图像重建高分辨率图像技术,并从CCD数字化采样能量的角度分析了多幅低分辨率的"差分"图像与高分辨率图像各像素灰度值的对应关系,给出了通过图像"差分"技术提高探测图像分辨率的倍数与所需CCD最少数目之间的一般规律.最后,采用Matlab软件以一幅黑白条纹均匀分布的图像作为目标物,模拟了图像"差分"技术的实现过程,仿真结果表明了该技术的可行性.     

Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy

A numerical reconstruction method believed to be new is proposed for hybrid holographic microscopy in which the hologram of a microscopic object is recorded by an image sensor and is then reconstructed by a computer. Because the Fresnel-Kirchhoff integral must be used for numerical reconstruction to achieve high resolution, we propose an approximation technique for reducing the calculation time. This approximation technique is suitable for microscopic application. The numerical reconstruction of 1-mum-sized objects was demonstrated with a He-Ne laser (lambda = 0.6328 mum).     

Direct-Fourier reconstruction in tomography and synthetic aperture radar

We investigate the use of direct-Fourier (DF) image reconstruction in computed tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolution-backprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobian-weighted two-dimensional periodic sinc-kernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms using optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the least-squares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentric-squares sampling scheme, DF interpolation can be performed accurately and efficiently, producing imagery that is superior to that obtainable by other algorithms. In the case of SAR, we show that the DF method performs very well with interpolators of low complexity. We also study DF reconstruction in SAR for trapezoidal grids. We conclude that the success of the DF method in SAR imaging is due to the nearly Cartesian shape of the sampling grid. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 1–13, 1998     

Sampling of the diffraction field

When optical signals, like diffraction patterns, are processed by digital means the choice of sampling density and geometry is important during analog-to-digital conversion. Continuous band-limited signals can be sampled and recovered from their samples in accord with the Nyquist sampling criteria. The specific form of the convolution kernel that describes the Fresnel diffraction allows another, alternative, full-reconstruction procedure of an object from the samples of its diffraction pattern when the object is space limited. This alternative procedure is applicable and yields full reconstruction even when the diffraction pattern is undersampled and the Nyquist criteria are severely violated. Application of the new procedure to practical diffraction-related phenomena, like in-line holography, improves the processing efficiency without creating any associated artifacts on the reconstructed-object pattern.     

Interpolation of Medical Ultrasound Images From Coherent and Noncoherent Signals

Medical ultrasound imaging instrumentation typically performs image interpolation using the signals acquired after envelope extraction, i.e., noncoherent signals. This operation is completely satisfying when the Nyquist condition for spatial sampling is fulfilled. However, the maximum spatial frequency of a signal increases as a consequence of the envelope extraction, making it more difficult to fulfill the Nyquist condition. For this reason, this paper suggests the interpolation of signals before envelope extraction, i.e., the use of coherent data, and discusses the applicability of this procedure to actual ultrasound scanners. Emphasis is given to the linear-array imaging technique, and the adoption of windowed sinc functions as interpolation kernels with limited support is assessed. Moreover, the authors investigated the situation in which coherent signals are also undersampled, and perfect reconstruction is not at all possible. The investigation was carried out using a numerical study and a simulated imaging test, and it was supported by subjective and objective evaluations. It is concluded that, at a moderate undersampling level, the interpolation of coherent signals allows significantly better performance than the interpolation of noncoherent signals.      

Research on Efficient Seismic Data Acquisition Methods Based on Sparsity Constraint

In actual exploration, the demand for 3D seismic data collection is increasing, and the requirements for data are becoming higher and higher. Accordingly, the collection cost and data volume also increase. Aiming at this problem, we make use of the nature of data sparse expression, based on the theory of compressed sensing, to carry out the research on the efficient collection method of seismic data. It combines the collection of seismic data and the compression in data processing in practical work, breaking through the limitation of the traditional sampling frequency, and the sparse characteristics of the seismic signal are utilized to reconstruct the missing data. We focus on the key elements of the sampling matrix in the theory of compressed sensing, and study the methods of seismic data acquisition. According to the conditions that the compressed sensing sampling matrix needs to meet, we introduce a new random acquisition scheme, which introduces the widely used Low-density Parity-check (LDPC) sampling matrix in image processing into seismic exploration acquisition. Firstly, its properties are discussed and its conditions for satisfying the sampling matrix in compressed sensing are verified. Then the LDPC sampling method and the conventional data acquisition method are used to synthesize seismic data reconstruction experiments. The reconstruction results, signal-to-noise ratio and reconstruction error are compared to verify the seismic data based on sparse constraints. The LDPC sampling method improves the current seismic data reconstruction efficiency, reduces the exploration cost and the effectiveness and feasibility of the method.     

基于生成对抗网络的压缩感知图像重构方法

目的为了解决传统压缩感知图像重构方法存在的重构时间长、重构图像质量不高等问题,提出一种基于生成对抗网络的压缩感知图像重构方法。方法基于生成对抗网络思想设计一种由具有稀疏采样功能的鉴别器和具有图像重构功能的生成器组成的深度学习网络模型,利用对抗损失和重构损失2个部分组成的新的损失函数对网络参数进行优化,完成图像压缩重构过程。结果实验表明,文中方法在12.5%的低采样率下重构时间为0.009s,相较于常用的OMP算法、CoSaMP算法、SP算法和IRLS算法,其峰值信噪比(PSNR)提高了10～12 dB。结论文中设计的方法应用于图像重构时重构时间短,在低采样率下仍能获得高质量的重构效果。     

Hybrid holographic microscopy: visualization of three-dimensional object information by use of viewing angles

One of the attractive features of hybrid holographic microscopy, in which the hologram of a microscopic object recorded by an image sensor is numerically reconstructed with a computer, is that the three-dimensional (3-D) information of a recorded object is obtained. The 3-D information has often been extracted by means of changing the reconstruction distance in the numerical reconstruction process, but here we describe an alternative technique that allows for variable viewing angles. That is, the perspective from which the object is viewed can be varied. The approximation used enables use of the fast-Fourier-transform algorithm for numerical reconstruction even in the high-resolution case in which the Fresnel approximation is no longer valid. The resolution of the proposed technique is also discussed.     

Eigenmode super-resolution imaging in arbitrary optical systems

We investigate optical super-resolution by means of eigenmode decomposition in arbitrary imaging systems. This technique is applicable for arbitrary objects but requires a knowledge of the eigenmodes of the imaging system. We outline a reconstruction technique that can be applied even to systems in which the eigenmodes are not orthogonal, and we present numerical simulations of eigenmode super-resolution in systems with resolution limited both by diffraction and by aberrations. Our results indicate that optical super-resolution by direct eigenmode decomposition provides a versatile method of sub-diffraction and distortion-free imaging in arbitrary optical systems.     

Efficient frequency-domain sample selection for recovering limited-support images

An image whose region of support is smaller than its bounding rectangle can, in principle, be reconstructed from a subset of the Nyquist samples. However, determining such a sampling set that gives a stable reconstruction is a difficult and computationally intensive problem. An algorithm is developed for determining periodic nonuniform sampling patterns that is orders of magnitude faster than existing algorithms. The speedup is achieved by using a sequential selection algorithm and heuristic metrics for the quality of sampling sets that are fast to compute, as opposed to the more rigorous linear algebraic metrics that have been used previously. Simulations show that the sampling sets determined using the new algorithm give image reconstructions that are of accuracy comparable with those determined by other slower algorithms.     

Statistical detection and imaging of objects hidden in turbid media using ballistic photons

We exploit recent advances in active high-resolution imaging through scattering media with ballistic photons. We derive the fundamental limits on the accuracy of the estimated parameters of a mathematical model that describes such an imaging scenario and compare the performance of ballistic and conventional imaging systems. This model is later used to derive optimal single-pixel statistical tests for detecting objects hidden in turbid media. To improve the detection rate of the aforementioned single-pixel detectors, we develop a multiscale algorithm based on the generalized likelihood ratio test framework. Moreover, considering the effect of diffraction, we derive a lower bound on the achievable spatial resolution of the proposed imaging systems. Furthermore, we present the first experimental ballistic scanner that directly takes advantage of novel adaptive sampling and reconstruction techniques.     

High topographical accuracy by optical shot noise reduction in digital holographic microscopy

In this work, we present a new method to reduce the shot noise in phase imaging of digital holograms. A spatial averaging process of phase images reconstructed at different reconstruction distances is performed, with the reconstruction distance range being specified by the numerical focus depth of the optical system. An improved phase image is attained with a 50% shot noise reduction. We use the integral of the angular spectrum as a reconstruction method to obtain a single-object complex amplitude that is needed to perform our proposal. We also show the corresponding simulations and experimental results. The topography of a homemade TiO2 stepwise of 100 nm high was measured and compared with the atomic force microscope results.