Shannon Interpolation Research Articles

Two‐dimensional adaptive Whittaker–Shannon Sinc‐based zooming

AbstractIn this work, we introduce a novel image zooming methodology that transitions from a nonadaptive Sin‐based approach to an adaptive Sinc‐based zooming technique. The two techniques base their theoretical foundation on the Whittaker–Shannon interpolation formula and the Nyquist theorem. The evolution into adaptive Sinc‐based zoom is accomplished through the use of two novel concepts: (1) the pixel‐local scaled k‐space and (2) the k‐space filtering sigmoidal function. The pixel‐local scaled k‐space is the standardized and scaled k‐space magnitude of the image to zoom. The k‐space filtering sigmoidal function scales the pixel‐local scaled k‐space values into the numerical interval [0, 1]. Using these two novel concepts, the Whittaker–Shannon interpolation formula is elaborated and used to zoom images. Zooming is determined by the shape of the Sinc functions in the Whittaker–Shannon interpolation formula, which, in turn, depends on the combined effect of the pixel‐local scaled k‐space, the sampling rate, and the k‐space filtering sigmoidal function. The primary outcome of this research demonstrates that the Whittaker–Shannon interpolation formula can achieve successful zooms for values of the sampling rate significantly greater than the bandwidth. Conversely, when the sampling rate is much greater than the bandwidth, the nonadaptive technique fails to perform the zoom correctly. The conclusion is that the k‐space filtering sigmoidal function is identified as the crucial parameter in the adaptive Sinc‐based zoom technique. The implications of this research extend to Sinc‐based image zooming applications.

Two-dimensional image zooming by adaptive Sinc-based technique

This article reports a novel adaptive Sinc-based zooming technique. The theoretical basis of the zooming technique is the Whittaker–Shannon interpolation formula and the Nyquist–Shannon sampling theorem. The technique can be used to zoom out, zoom in, and reconstruct two-dimensional images with no zoom. The technique introduces a novel concept called pixel-local-scaled k-space, which is calculated as follows. The k-space magnitude and the bandwidth of the image to zoom are calculated. The bandwidth of the frequency components of the image to zoom is estimated by the difference between the maximum and the minimum numerical value of the k-space magnitude. The ratio between the standardised k-space magnitude of the image and the bandwidth is calculated pixel-by-pixel, and is called pixel-local-scaled k-space. The signal is reconstructed using a finite convolution between image pixel intensities and Sinc functions. The zooming factor is consequential to the numerical difference between the pixel-local-scaled k-space and the sampling rate. The novelty of this zooming technique is to be adaptive because the pixel-local-scaled k-space is a pixel-by-pixel map, that is, an image with the same number of pixels as the departing image. Adaptive Sinc-based zooming is validated through the comparison with non-adaptive Sinc-based zooming, and its smoothing effect is compared to the smoothing of a variety of image space and z-space filters. The adaptive Sinc-based zooming technique achieves image zooming reconstruction using sampling rates within an extended range versus the range allowed by non-adaptive Sinc-based zooming.

Full reconstruction of acoustic wavefields by means of pointwise measurements

A novel method for full wavefield reconstruction from pointwise measurements data collected by a vertical receiver array is proposed for geoacoustic waveguides of a shallow sea. This method is based on a generalization of the Whittaker–Shannon interpolation formula to the case of a waveguide cross-section. The generalization is obtained by means of the so-called discrete variable representation. The method can be implemented both for tonal and pulse signals. It is shown that the method is robust against array distortions and ambient noise of moderate amplitude. Efficiency of reconstruction is verified by means of numerical simulation with a model of a shallow-sea waveguide. It is shown that the presence of sound attenuation in the seabottom results in the possibility to accurately reconstruct the field using a relatively sparse receiver array.

Parametric Algorithm for Predicting the Channel Characteristics for High-Mobility Users in the LTE Communication System

We consider the problem of predicting the channel characteristics for the high-mobility users of the LTE communication system. An algorithm is proposed for predicting the channel coefficients on the basis of a parameterized autoregressive model whose parameters should be estimated using the superresolution methods of spectral analysis. Using numerical simulation and the system-level simulator, we estimate the efficiency of the proposed algorithm and compare it with the prediction algorithm on the basis of the autocorrelation approach and the Whittaker–Shannon interpolation formula. It is shown that the proposed algorithm allows one to achieve high accuracy of predicting the characteristics of the communication channel of the high-mobility users, which is comparable with the accuracy of estimating these characteristics for the users moving with low velocities.

Continuous limits of linear and nonlinear quantum walks

In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove that the walker (solution) of the NLQW on a lattice [Formula: see text] uniformly converges (in Sobolev space [Formula: see text]) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as [Formula: see text]. Here, to compare the walker defined on [Formula: see text] and the solution to the NLD defined on [Formula: see text], we use Shannon interpolation.

Superposition, reduction of multivariable problems, and approximation

We study reduction schemes for functions of “many” variables into system of functions in one variable. Our setting includes infinite dimensions. Following Cybenko–Kolmogorov, the outline for our results is as follows: We present explicit reduction schemes for multivariable problems, covering both a finite, and an infinite, number of variables. Starting with functions in “many” variables, we offer constructive reductions into superposition, with component terms, that make use of only functions in one variable, and specified choices of coordinate directions. Our proofs are transform based, using explicit transforms, Fourier and Radon; as well as multivariable Shannon interpolation.

Compressed sensing based sub-Nyquist sampling of multiple sinusoids with dual rate channels.

In this paper, we propose a reconstruction approach for a multiple-sinusoidal signal. The signal reconstruction requires a small number of samples and is based on a sub-Nyquist sampling scheme with dual rate channels. In the proposed sampling scheme, the samples are grouped into multiple cosets. To obtain enough different cosets to reconstruct a signal, the sampling rates of channels are required to be relative coprime. For each coset, the Whittaker-Shannon interpolation formula is employed to construct the relation between the sub-Nyquist samples and the original signal, which is used to construct the measurement matrix. Since the multiple-sinusoidal signal is sparse in the frequency domain, compressed sensing theory can be adopted to reconstruct the signal. Simulation results are reported to demonstrate the feasibility and effectiveness of the proposed approach.

Computational efficient inverse groundwater modeling using Random Mixing and Whittaker–Shannon interpolation

Geostatistical inverse modeling problems can potentially be very high-dimensional and computationally expensive. Using the Random Mixing approach the dimensionality can be reduced, and any optimization algorithm can be applied to solve the inverse problem in this lower dimensional space. In order to reduce computational costs the standard optimization is replaced by a sequence of one-dimensional optimizations. In this one-dimensional space a simplified calculation of the objective function using Whittaker–Shannon interpolation is carried out. This procedure requires a significantly reduced number of forward model runs and it also guarantees monotonic convergence of the objective function. A synthetic and a real world example will be used to demonstrate the procedure and to evaluate the quality of the Whittaker–Shannon interpolation in comparison to the actual objective functions.

The Shannon Total Variation

Discretization schemes commonly used for total variation regularization lead to images that are difficult to interpolate, which is a real issue for applications requiring subpixel accuracy and aliasing control. In the present work, we reconciliate total variation with Shannon interpolation and study a Fourier-based estimate that behaves much better in terms of grid invariance, isotropy, artifact removal and subpixel accuracy. We show that this new variant (called Shannon total variation) can be easily handled with classical primal–dual formulations and illustrate its efficiency on several image processing tasks, including deblurring, spectrum extrapolation and a new aliasing reduction algorithm.

Stability analysis in milling of the thin walled part considering multiple variables of manufacturing systems

In order to improve the precision of the milling stability analysis, the more exact milling dynamic model considering multiple variables of manufacturing systems is proposed, and in order to reduce the calculation time for the stability analysis of the milling dynamic model, the fast calculation of the stability analysis is proposed. Firstly, the contact region of the milling cutter and the thin walled part is discretized into many small intervals, the milling dynamic model considering helix angle of the milling cutter, two degrees of freedom of the thin-walled part, and coupling dynamic characteristic of the tool–spindle system is established in a discrete small interval. The dynamic model in the small discrete interval considering multiple variables is transformed into equation of states, and the equation of states is integrated in the range of the milling depth to establish the milling dynamic model considering multiple variables of manufacturing systems. Secondly, in order to enhance the calculation speed of the stability analysis for the established dynamic model considering multiple variables, and to avoid the loss of information, the sampling rule of the delay term is defined based on the sampling rule of Shannon sampling theorem, and the delayed term is reconstructed in the discrete interval by using the Shannon interpolation function, and the validity of the fast calculation of the stability analysis is verified by two degrees of the freedom milling dynamic model. Finally, the validity of the established milling dynamic model considering multiple variables is verified by milling experiment, and the results show that the established dynamic model is effective.

The reconstruction of a semi-discretization method for milling stability prediction based on Shannon standard orthogonal basis

In order to increase the calculation speed of the semi-discretization method (SDM) without accuracy loss, this paper reconstructs the SDM for predicting the stability lobes of the dynamic milling process, mainly considering the regenerative effect. The model of the dynamic milling process is expressed as the linear delay-differential equations (DDE). The fast calculation method is established by reconstructing the SDM based on the Shannon standard orthogonal basis (SSOB). First, the delay term of DDE is constructed without information loss based on Shannon interpolation functions, and SSOB is derived. Secondly, the closed form expression for the transition matrix of the system is constructed based on the SSOB, and the stability limit is predicted based on the Floquet theory. The transition matrix-based SDM and SSOB are theoretically compared, and it shows that the SDM is a special case of the method based on SSOB when the SSOB is regarded as the average in the sampling interval. The fast calculation method is established by using the variable sampling numbers during the period of the delay time in which the variable sampling numbers are determined by the condition which is used to construct the SSOB. Finally, this proposed fast method is used to the one and two degrees of freedom milling model, and the results show that the calculation accuracy is not reduced, and the calculation speed based on the proposed method can be improved nearly five times on the one degree of freedom model and 2.6 times on the two degrees of freedom model, compared to the semi-discretization method.

Статистичне моделювання сейсмічного шуму в багатовимірній області змінних для визначення частотних характеристик геологічного середовища

The paper deals with the theory and methods of statistical simulation of random processes and fields based on their spectral decomposition and Kotelnikov-Shennon modified interpolation sums, as well as applying these methods for environmental geophysical monitoring. Statistical simulation of multivariate random fields (those homogeneous in time and homogeneous isotropic in n other variables) are considered to be essential for seismological research into frequency characteristics of geological media. A statistical model and a numerical algorithm of simulating random fields are built on the basis of Kotelnikov-Shennon modified interpolation decomposition to generate adequate realizations of seismic noise. The paper examines real-valued random fields ξ(t,x),tϵÎR,xÎRn, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. It also considers approximation of random fields by the random fields with a bounded spectrum. There is made an analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum. Besides, there are obtained estimates of the mean-square approximation of random fields in the space R´Rn by a model constructed with the help of spectral decomposition and Kotelnikov–Shannon interpolation formula. The paper provides a mechanism for statistical simulation of Gaussian random fields with a bounded spectrum; namely, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. Proved have been the theorems of the mean-square approximation of random fields (those homogeneous in time and homogeneous isotropic ones relative to n- other variables) by special partial sums. A simulation method was used to formulate an algorithm of numerical simulation by means of these theorems. There are also considered ways to carry out spectral analysis of generated seismic noise realizations. Finally, there have been developed universal methods of statistical simulation (Monte Carlo methods) of multi-parameter seismology data for generating seismic noise on 2D and 3D grids of the required detail and regularity.

Accurate Discretization of Analog Audio Filters With Application to Parametric Equalizer Design

This paper is concerned with accurate discretization of linear analog filters such that the frequency response of the discrete time filter accurately matches that of the continuous time filter. The approach is based on formal reconstruction of the continuous time signal using Shannon's interpolation theorem and numerical solving of the differential equation corresponding to the analog filter. When the formal continuous time system is sampled, the resulting filter reduces to discrete linear filter, which can be realized either as a state space model or as an infinite impulse response (IIR) filter. The proposed methodology is applied to design of filters for parametric equalizers.

Plate impulse response spatial interpolation with sub-Nyquist sampling

Impulse responses of vibrating plates are classically measured on a fine spatial grid satisfying the Shannon–Nyquist spatial sampling criterion, and interpolated between measurement points. For homogeneous and isotropic plates, this study proposed a more efficient sampling and interpolation process, inspired by the recent paradigm of compressed sensing. Remarkably, this method can accommodate any star-convex shape and unspecified boundary conditions. Here, impulse responses are first decomposed as sums of damped sinusoids, using the Simultaneous Orthogonal Matching Pursuit algorithm. Finally, modes are interpolated using a plane wave decomposition. As a beneficial side effect, these algorithms can also be used to obtain the dispersion curve of the plate with a limited number of measurements. Experimental results are given for three different plates of different shapes and boundary conditions, and compared to classical Shannon interpolation.

Handbook of Fourier analysis & its applications

Fourier analysis has many scientific applications - in physics, number theory, combinatorics, signal processing, probability theory, statistics, option pricing, cryptography, acoustics, oceanography, optics and diffraction, geometry, and other areas. In signal processing and related fields, Fourier analysis is typically thought of as decomposing a signal into its component frequencies and their amplitudes. This practical, applications-based professional handbook comprehensively covers the theory and applications of Fourier Analysis, spanning topics from engineering mathematics, signal processing and related multidimensional transform theory, and quantum physics to elementary deterministic finance and even the foundations of western music theory. As a definitive text on Fourier Analysis, Handbook of Fourier Analysis and Its Applications is meant to replace several less comprehensive volumes on the subject, such as Processing of Multifimensional Signals by Alexandre Smirnov, Modern Sampling Theory by John J. Benedetto and Paulo J.S.G. Ferreira, Vector Space Projections by Henry Stark and Yongyi Yang and Fourier Analysis and Imaging by Ronald N. Bracewell. In addition to being primarily used as a professional handbook, it includes sample problems and their solutions at the end of each section and thus serves as a textbook for advanced undergraduate students and beginning graduate students in courses such as: Multidimensional Signals and Systems, Signal Analysis, Introduction to Shannon Sampling and Interpolation Theory, Random Variables and Stochastic Processes, and Signals and Linear Systems.

Application of a nonuniform spectral resampling transform in Fourier-transform spectrometry.

We describe a nonuniform spectral resampling transform (NUSRT) that resamples a frequency-scaled spectrum that has been measured by a Fourier-transform spectrometer (FTS). Frequency scaling of a spectrum can arise from measurements made with off-axis detectors and Doppler shift induced by motion of a spaceborne FTS relative to an input radiation source. In addition, a spectrum may need to be rescaled in frequency to match spectral lines for applications such as the retrieval of atmospheric state parameters. The NUSRT is cast as a linear algebraic expression that relates a nonuniformly sampled interferogram to an input spectrum. A polynomial approximation is applied to this expression that reduces the inverse of the NUSRT to a series of Fourier transforms that can be implemented as fast Fourier transforms (FFTs). We show that this NUSRT algorithm requires on the order of 6N log N flops, which reduces the computational cost of rescaling by more than 1 order of magnitude compared with conventional FFT-based Shannon interpolation techniques while comparable accuracy is maintained.

Sampling and reconstruction schemes for biomagnetic sensor arrays

In this paper we generalize the approach of Ahonen et al (1993 IEEE Trans. Biomed. Eng. 40 859–69) to two-dimensional non-uniform sampling. The focus is on two main topics: (1) searching for the optimal sensor configuration on a planar measurement surface; and (2) reconstructing the magnetic field (a continuous function) from a discrete set of data points recorded with a finite number of sensors. A reconstruction formula for Bz is derived in the framework of the multidimensional Papoulis generalized sampling expansion (Papoulis A 1977 IEEE Trans. Circuits Syst. 24 652–4, Cheung K F 1993 Advanced Topics in Shannon Sampling and Interpolation Theory (New York: Springer) pp 85–119) in a particular case. Application of these considerations to the design of biomagnetic sensor arrays is also discussed.

A Fourier-Shannon approach to closed contours modelling

This paper describes a modelling method for continuous closed contours. The initial input data set consists of two-dimensional (2-D) points, which may be represented as a discrete function in a polar coordinate system. The method uses the Shannon interpolation between these data points to obtain the global continuous contour model. A minimal description of the contour is obtained using the link between the Shannon interpolation kernel and the Fourier series of polar development (FSPD) for periodic functions. The Shannon interpolation kernel allows the direct interpretation of the contour smoothness in terms of both samples and Fourier frequency domains. In order to deal with deformation point sources, often encountered in active modelling techniques, a method of local deformation is proposed. Each local deformation is performed in an angular sector centred on the deformation point source. All the neighbouring characteristic samples are displaced in order to minimize the oscillations of the newly created model outside the deformation sector. This deformation technique preserves the frequency characteristics of the contour, regardless of the number and the intensity of deformation sources. In this way, the technique induces a frequency modelling constraint, which may be subsequently used in an active detection and modelling environment. Experiments on synthetic and real data prove the efficiency of the proposed technique. The method is currently used to model contours of the left ventricle of the heart obtained from ultrasound apical images. This work is part of a larger project, the aim of which is to analyse the space and time deformations of the left ventricle. The 2-D Fourier–Shannon model is used as a basis for more complex three-dimensional and four-dimensional Fourier models, able to recover automatically the movement and deformation of the left ventricle of the heart during a cardiac cycle.

Scale periodicity and its sampling theorem

The scalar transform is a new representation for signals, offering a perspective that is different from the Fourier transform. We introduce the notion of a scalar periodic function. These functions are then represented through the discrete scale series. We also define the notion of a strictly scale-limited signal. Analogous to the Shannon interpolation formula, we show that such signals can be exactly reconstructed from exponentially spaced samples of the signal in the time domain. As an interesting, practical application, we show how properties unique to the scale transform make it very useful in computing depth maps of a scene.

SPARK, a kernel of software programs for spatial reconstruction in electron microscopy

SummarySPARK, an acronym for ‘SPAtial Reconstruction Kernel’, is the nucleus of a software library being developed for the three‐dimensional (3‐D) reconstruction of objects observed by the electron microscope. A unifying concept is used: the Fourier transform, known in several central sections, is resampled to obtain a 3‐D Cartesian grid, which is inverted by a fast Fourier transform. This technique is used for both single‐axis tilting (of 2‐D periodic layers or of isolated objects) and for the random conical‐tilt technique. The principles that make it possible to recover Cartesian grids in the two different geometries are illustrated and some preliminary results are reported. SPARK resamples the Cartesian grids with the use of a fast and efficient algorithm of Shannon interpolation developed by the authors.Compared to back‐projection techniques the method shows a considerable improvement in execution time with no sacrifice in accuracy; it therefore allows the effects of a variety of parameters in a given reconstruction to be scrutinized in a reasonable time. Some new possibilities and future extensions of the library are briefly outlined.