Generalized Sampling Theorem Research Articles

Comparative study of generalized-sampling-theorem-based digital super-resolution for 2D data

The size of pixels in a digital recording device, such as a CCD array, limits the spatial resolution in images obtained by an optical imaging system, thereby degrading the image quality. Digital super-resolution (DSR) techniques are used to reconstruct a high-resolution (HR) image from multiple sub-pixel-shifted low-resolution images in order to improve the image quality. In this article, we formulate a mathematical framework for DSR using the generalized sampling theorem (GST). The GST-based DSR method’s performance is evaluated by comparing it to existing resolution enhancement methods on the basis of evaluation metrics like percentage mean square error (%MSE), structural similarity index measure (SSIM), and peak signal-to-noise ratio (PSNR). The GST DSR method exhibits an overall superior quality image reconstruction based on quantitative analysis (with a near zero %MSE, SSIM around one, and improved PSNR in dB) and qualitative comparison. The robustness of the GST DSR method is further demonstrated in the presence of frame-to-frame shift estimation error using %MSE and SSIM by comparing it with multi-frame interpolation approaches.

A PSF Approach to Far Field Discretization for Conformal Sources

Field sampling should be devised to preserve the information required for the knowledge of the radiation of an antenna. In this paper, we introduce a sampling scheme based on an inverse source problem approach to the far field radiated by a conformal current source. The regularized solution of the problem requires the computation of the Singular Value Decomposition (SVD) of the relevant linear operator, leading to introduce the Point Spread Function in the observation domain, which can be related to the capability of the source to radiate a focusing beam. Then, the application of the Kramer generalized sampling theorem allows introducing a non-uniform discretization of the angular observation domain, tailored to each source geometry. The nearly optimal property of the scheme is compared with the best approximation achievable under a regularized inversion of the pertinent SVD. Numerical results for different two-dimensional curve sources show the effectiveness of the approach with respect to standard sampling approaches with uniform spacing, since it allows to reduce the number of sampling points of the far field.

A Review of Streamline Calibration Approaches for Digital Storage Oscilloscopes with Time-Interleaved Channels

The digital storage oscilloscope (DSO) has a vital role in measurement practice, since its resourcefulness fits both general purpose aims and requirements of more specific applications. A deep knowledge of DSO operation principles and available features is fundamental for technicians to best exploit the possibilities of the instrument. Operation principles regard digitization, acquisition, processing and visualization of the input waveform and concern a number of issues. For performant DSO digitizing systems, one of the most relevant issues is the calibration of the time-interleaved structure at the very heart of the instrument. Unfortunately, the calibration techniques adopted by the manufacturers are never disclosed in plain form with application notes or demos, and limited information can be read from patents, if available. In this article, the state-of-the-art of DSO technologies is reviewed; a general framework to face the calibration issue in time-interleaved systems, which is based on the generalized sampling theorem, is presented; and the case of some DSOs, which require special calibration approaches, implemented through the analysis of the input-output crosscorrelation of the individual acquisition channels, is discussed. Concluding remarks underline the value of the contribution, that provides a significant example of how more ideal acquisition channels can be realized by the combination of analog circuitry with digital processors.

On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms

In this paper, we establish generalized sampling theorems, generalized stability theorems and new inequalities in the setting of shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed-norms. A convergence theorem of general iteration algorithms for sampling in some shift-invariant subspaces of Lp→(Rd) are also given.

Performance Limits of Generalized Sampling Based 2-Channel Analog-to-Digital Converter

This brief proposes an analog-to-digital converter (ADC) based on Generalized Sampling Theorem and derives a closed form expression of its performance, viz., signal-to-quantization noise ratio (SQNR) and alias-suppression (AS). The ADCs in the 2-channels sample the signal and its filtered version, respectively, at half the Nyquist rate. The outputs of the two ADCs are upsampled by a factor of 2, passed through appropriate reconstruction filters, and then added to give an effective sample-rate which is double that of the sample-rate of each channel ADC. Performance of the proposed ADC has been summarized for simple filters like differentiator, integrator, passive RC low-pass filter, and passive CR high-pass filters. The proposed ADC could therefore be used as an alternative to other popular ADC structures like time-interleaved, frequency-interleaved, and filter-bank ADCs, to achieve high sample-rate with minimal analog circuit overhead.

Compact full-color holographic 3-D display based on undersampled computer-generated holograms and oblique projection imaging.

A compact full-color electro-holographic three-dimensional (3-D) display with undersampled computer-generated holograms (US-CGHs) and oblique projection imaging (OPI) is proposed. For its realization, undersampling conditions of the CGH enabling the complete recovery of image information are derived, and the OPI-based longitudinal-to-lateral depth conversion (LTL-DC) scheme allowing the simple reconstruction of full-color images is also proposed. Three-color off-axis US-CGHs are generated with their center-shifted principle fringe patterns (CS-PFPs) of the novel look-up table (NLUT) method, where center-shifts are calculated with the derived undersampling conditions of the CGH based on the generalized sampling theorem, and then multiplexed into the color-multiplexed hologram (CMH). The CMH is loaded on a SLM (spatial light modulator) and reconstructed by being illuminated with a multi-wavelength light source, where an original full-color image is reconstructed being spatially separated from the other color-dispersed images on the projected image plane with the OPI-based LTL-DC process, which enables us to view the original full-color image just with a simple filter mask. Performance analysis and successful experiments with the test 3-D objects in motion confirm the feasibility of the proposed system.

FORMULATION AND SOLUTION OF OPTIMIZATION PROBLEM REDUNDANCY SYSTEM COMPRISING SITUATION CENTER

The analysis of the concept, properties and features of heterogeneous redundancy in modern complex ergatic systems, including those included in the situation centers (SC). On the basis of the qualimetric paradigm, the generalized analytical model of quality and optimization of quality by private, group, summary and aggregated quality indicators is justified. Practical ways of realization of the model and methods of optimization of the objects which are a part of SC and them as a whole at the expense of reduction of structural, functional and other types of redundancy under the obligatory condition of non-reduction of the required value of quality are given. On the example of the generalized sampling theorem when choosing the optimal value of the sampling frequency of the real bandpass signal, the criticality and significant influence on the redundancy of data in their further processing in the SC is shown.

Using a stereo-modeling operator to reduce aliasing during migration

The sampling theorem requires minimum two points per wavelength to ensure the proper reconstruction of a wavefield. Violation of this limit will result in data aliasing. With the increased demand for high-frequency seismic processing, data recorded in a field may not have sufficient spatial sampling. Reverse time migration (RTM) using a conventional scheme will not be able to produce a high-quality image from such an aliased input data set. However, when the wavefield and its gradients are available, the generalized sampling theorem relaxes the requirement to only one point per wavelength. Stereo-modeling methods use a system of wave equations to compute a vector containing the particle displacement and its spatial gradients in wavefield propagation. Therefore, in cases in which a wavefield and its spatial gradients are available, such as a multicomponent data set, RTM using the stereo-modeling methods can be potentially much more accurate and efficient than those commonly used ones, such as Lax-Wendroff correction. We investigated the merits of one such method, called the nearly analytic central difference, in RTM to produce high-quality images. Numerical experiments on different data sets produce consistent results with the generalized sampling theorem, which indicates that the stereo-modeling method can yield high-quality images from insufficiently sampled data compared with conventional finite-difference methods.

Optical fiber imaging based tomography reconstruction from limited data

The paper discusses tomography reconstruction of distributed physical field parameters by means of fiber optical measuring system based on distributed fiber optical measuring network. Measuring network represents a set of fiber-optical measuring lines stacked in accordance with a certain setup on the surface studied. This work considers the case of parallel non-equidistant measuring lines stacking schemes which requires application of generalized sampling theorems. Two restoration methods are proposed, algebraic and neural-like. The approach whose novelty involves measuring network data structurization for computing process optimization and further application of neural or algebraic technologies to restore a full image of the functions studied is presented.

Markovian statistics on evolving systems

A novel framework for the analysis of observation statistics on time discrete linear evolutions in Banach space is presented. The model differs from traditional models for stochastic processes and, in particular, clearly distinguishes between the deterministic evolution of a system and the stochastic nature of observations on the evolving system. General Markov chains are defined in this context and it is shown how typical traditional models of classical or quantum random walks and Markov processes fit into the framework and how a theory of quantum statistics (sensu Barndorff-Nielsen, Gill and Jupp) may be developed from it. The framework permits a general theory of joint observability of two or more observation variables which may be viewed as an extension of the Heisenberg uncertainty principle and, in particular, offers a novel mathematical perspective on the violation of Bell’s inequalities in quantum models. Main results include a general sampling theorem relative to Riesz evolution operators in the spirit of von Neumann’s mean ergodic theorem for normal operators in Hilbert space.

Sampling and Reconstruction in Arbitrary Measurement and Approximation Spaces Associated With Linear Canonical Transform

The linear canonical transform (LCT), which generalizes many classical transforms, has been shown to be a powerful tool for signal processing and optics. Sampling theory of the LCT for bandlimited signals has blossomed in recent years. However, in practice signals are never perfectly bandlimited, and in many cases measurement devices are nonideal. The objective of this paper is to develop a sampling theorem for the LCT from general measurements, which can provide a suitable and realistic model of sampling and approximation for real-world applications. We first describe a general class of approximation spaces for the LCT and provide a full characterization of their basis functions. Then, we propose a generalized sampling theorem for arbitrary measurement and approximation spaces associated with the LCT. Several properties of the proposed sampling theorem are also discussed. Furthermore, the approximation error is estimated. Finally, numerical results and several applications of the derived results are presented.

Generalized Sampling and Infinite-Dimensional Compressed Sensing

We introduce and analyze a framework and corresponding method for compressed sensing in infinite dimensions. This extends the existing theory from finite-dimensional vector spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary by demonstrating that existing finite-dimensional techniques are ill suited for solving a number of key problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. A conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. Central to this work is the introduction of two novel concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize an infinite-dimensional problem.

A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases

We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be viewed as an extension of the well-known consistent reconstruction technique (Eldar et al.). However, whilst the latter imposes stringent assumptions on the reconstruction basis, and may in practice be unstable, our framework allows for recovery in any (Riesz) basis in a manner that is completely stable. Whilst the classical Shannon Sampling Theorem is a special case of our theorem, this framework allows us to exploit additional information about the approximated vector (or, in this case, function), for example sparsity or regularity, to design a reconstruction basis that is better suited. Examples are presented illustrating this procedure.

Optimal Design of Optical Filter for White Light Interferometry based on Sampling Theory

Vertical-scanning white light interferometry is a technique of profiling the surface topography of objects such as semiconductors, liquid crystal displays (LCDs), and so on. The profile is given by the “squared-envelope function” of the interference signal, which is called the interferogram. The world’s fastest surface profiling algorithm exploits a generalized sampling theorem that directly reconstructs the squared-envelope function from an infinite number of samples of the interferogram. The optical system of the white light interferometer is equipped with an optical filter. So far there were no guidelines for designing the optical filter’s characteristics. Field engineers empirically chose some combination of commercial optical filters by checking the corresponding interferograms. The main purpose of this paper is to provide a guideline for designing the optical filter’s characteristics. That is, we devise the optimal characteristics of the optical filter in the sense that, with a fixed passband of the optical filter, the second moment of the squared interferogram is minimized. Simulation results confirm the effectiveness of the optimal characteristics.

The Resonance Mode Theory for Exterior Problems of Electrodynamics and Its Application to Discrete Antenna Modeling in a Frequency Range

The problem under consideration here is that of building a versatile discrete antenna model in a frequency range. The eigenfunction (mode) expansion or singularity expansion method (SEM) is used in working out the problem. A detailed substantiation of SEM is provided and an expression determining the dependence of mode amplitudes on exterior source fields is derived. A simple example is provided in support of the theory. The SEM is used in inquiring into the frequency dependence of antenna-generalized matrix entries. It is intimated that all of the matrix entries contain resonance factors and factors that are Hankel transforms of radial coordinate functions. A sampling theorem, which is a generalized extension of the classic sampling theorem, is used for the Hankel transform. The application of the generalized sampling theorem completes the process of building a versatile discrete antenna model in a frequency range.

Reconstruction of optical fields with the Quasi-discrete Hankel transform

Unlike the FFT, the Quasi Discrete Hankel Transform (QDHT) is not sampled on a uniform grid; in particular the field may no longer be sampled on axis. We demonstrate how the generalised sampling theorem may be applied to optical problems, evaluated with the QDHT, to efficiently and accurately reconstruct the optical field at any point. Without sacrificing numerical accuracy this is demonstrated to be typically 50x faster than using an equivalent 2D FFT.

Regular and Irregular Sampling Theorem for Multiwavelet Subspaces

We study the sampling theorem for frames in multiwavelet subspaces. Firstly, a sufficient condition under which the regular sampling theorem holds is established. Then, notice that irregular sampling is also useful in practice; we consider the general cases of the irregular sampling and establish a general irregular sampling theorem for multiwavelet subspaces. Finally, using this generalized irregular sampling theorem, we obtain an estimate for the perturbations of regular sampling in shift-invariant spaces.

Time-Interleaved Analog-to-Digital-Converter Compensation Using Multichannel Filters

Published methods that employ a filter bank for compensating the timing and bandwidth mismatches of an M-channel time-interleaved analog-to-digital converter (TIADC) were developed based on the fact that each sub-ADC channel is a downsampled version of the analog input. The output of each sub-ADC is filtered in such a way that, when all the filter outputs are summed, the aliasing components are minimized. If each channel of the filter bank has N coefficients, the optimization of the coefficients requires computing the inverse of an MN times MN matrix if the weighted least squares (WLS) technique is used as the optimization tool. In this paper, we present a multichannel filtering approach for TIADC mismatch compensation. We apply the generalized sampling theorem to directly estimate the ideal output of each sub-ADC using the outputs of all the sub-ADCs. If the WLS technique is used as the optimization tool, the dimension of the matrix to be inversed is N times N. For the same number of coefficients (and also the same spurious component performance given sufficient arithmetic precision), our technique is computationally less complex and more robust than the filter-bank approach. If mixed integer linear programming is used as the optimization tool to produce filters with coefficient values that are integer powers of two, our technique produces a saving in computing resources by a factor of approximately (100.2N(M- 1)/(M-1) in the TIADC filter design.

Artifacts suppression in limited data problem for parallel fiber optical measuring systems

The paper discusses tomography reconstruction of distributed physical fields by means of fiber optical measuring systems (FOMN) [1] for parallel schemes of measuring lines stacking with a small number of scanning directions. This work considers the sampling theory for a case with sampling sets which are unions of two co-sets. The case of non-equidistant sampling requires application of new approaches, namely the generalized sampling theorems with group-theory basis. An explicit reconstruction formula for extracting additional (missing or distorted) projection data values are provided, using approximating of a projection function on a non regular grid. Furthermore, standard restoration algorithms are applied to this function. Two restoration methods are proposed, UQC 1 (union of quotient classes), and UQC 2. Comparative analysis of the operating efficiency of UQC 1 and UQC 2 methods versus standard analytical FBP method for fiber-optical tomography reconstruction is reported.

Sampling theorem and multi-scale spectrum based on non-linear Fourier atoms

This study concerns some new developments of unit analytic signals with non-linear phase. It includes ladder-shaped filter, generalized Sinc function based on non-linear Fourier atoms, generalized sampling theorem for non-bandlimited signals and the notion of multi-scale spectrum for discrete sequences. We first introduce the ladder-shaped filter and show that the impulse response of its corresponding linear time-shift invariant system is the generalized Sinc function as a product of periodic Poisson kernel and Sinc function. Secondly, we establish a Shannon-type sampling theorem based on generalized Sinc function for this type of non-bandlimited signal. We further prove that this type of signal may be holomorphically extended to strips in the complex plane containing the real axis. Finally, we introduce the notion of multi-scale spectrums for discrete sequences and develop the related fast algorithm.