Band-limited Extrapolation Research Articles

Fast algorithms for band-limited extrapolation by over sampling and Fourier series

In this paper, fast algorithms for the extrapolation of band-limited signals are presented by the sampling theorem and Fourier series in the case of over sampling. Assume the band-limited signal is known in a finite interval. We update the signal outside the interval by the Shannon sampling theorem in the case of over sampling. Then we obtain a fast algorithm in the form of Fourier series instead of the Fourier transform in the Papoulis–Gerchberg algorithm. Gibbs phenomena is analyzed in the method. An algorithm is presented to control the Gibbs phenomena, and some examples are given in the experimental results.

Spectral Super-resolution and Band-limited Extrapolation Using Slepian Series

Band-limited signal extrapolation and spectral super-resolutions are closely related. They both can be achieved using Slepian series. This method is often believed to be dependent on computing definite integrals. Instead of using numerical integration, a simple and reliable method, which can be implemented in MATLAB, is presented. This way, it is much easier to investigated the extent to which a signal can be extrapolated from its samples taken in a finite interval. The proposed method is tested with data from a real microphone array.

A Fast Convergence Algorithm for Band-Limited Extrapolation by Sampling

In this paper, an algorithm for band-limited extrapolation is presented. This uses Shannon's sampling theorem and Fourier series. Error analysis is given by proof. The convergence rate is much faster than that of the Papoulis-Gerchberg algorithm. The procedure of the fast convergence algorithm is described in detail. The result is demonstrated by numerical examples.

A Leapfrogging-in-Time Integral Equation Solver

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> An implicit leapfrogging-in-time surface-integral-equation solver for analyzing transient scattering from homogenous dielectric bodies is presented. Electric and magnetic current densities on dielectric interfaces are expanded in terms of space-time basis functions defined on staggered temporal grids. Band-limited extrapolation of these current densities enables a leapfrogging solution procedure that generates samples of electric and magnetic current densities separated by half time steps. Numerical results demonstrate that the proposed solver is more efficient than standard marching-on-in-time solvers that simultaneously generate electric and magnetic current samples separated by integer time steps. </para>

Improvement of resolution in equivalent currents reconstruction using Papoulis-Gerchberg algorithm and replicas of the spectrum

A new algorithm to increase the resolution in the reconstruction of equivalent currents from spherical near-field measurements is proposed. The algorithm obtains the non-visible part of the plane- wave spectrum used for the reconstruction. This is achieved by means of a band-limited extrapolation method, the Papoulis-Gerchberg algorithm, and the replication of the known part of the spectrum. The gain in resolution is from llambda in the method without the algorithm proposed, to 0.4-0.51 with this new algorithm. Results with real measurements are shown.

An Efficient Method for an Ill-Posed Problem — Band-Limited Extrapolation by Regularization

In this paper, a regularized spectral estimation formula and a regularized iterative algorithm for band-limited extrapolation are presented. The ill-posedness is taken into account. First the Fredholm equation is regularized. Then it is transformed to a differential equation in the case where the time interval is R. A fast algorithm to solve the differential equation is given by the finite difference and a regularized spectral estimation formula is obtained. Then a regularized iterative extrapolation algorithm is introduced and compared with the Papoulis and Gerchberg algorithm

A new one-step band-limited extrapolation procedure using empirical orthogonal functions

A one-step band-limited extrapolation procedure is systematically developed under an a priori assumption of bandwidth. The rationale of the proposed scheme is to expand the known signal segment based on a band-limited basis function set and then to generate a set of Empirical Orthogonal Functions (EOF’s) adaptively from the sample values of the band-limited function set. Simulation results indicate that, in addition to the attractive adaptive feature, this scheme also appears to guarantee a smooth result for inexact data, thus suggesting the robustness of the proposed procedure.

Image reconstruction: a unifying model for resolution enhancement and data extrapolation Tutorial

In reconstructing an object function F(r) from finitely many noisy linear-functional values integral of F(r)Gn(r)dr we face the problem that finite data, noisy or not, are insufficient to specify F(r) uniquely. Estimates based on the finite data may succeed in recovering broad features of F(r), but may fail to resolve important detail. Linear and nonlinear, model-based data extrapolation procedures can be used to improve resolution, but at the cost of sensitivity to noise. To estimate linear-functional values of F(r) that have not been measured from those that have been, we need to employ prior information about the object F(r), such as support information or, more generally, estimates of the overall profile of F(r). One way to do this is through minimum-weighted-norm (MWN) estimation, with the prior information used to determine the weights. The MWN approach extends the Gerchberg-Papoulis band-limited extrapolation method and is closely related to matched-filter linear detection, the approximation of the Wiener filter, and to iterative Shannon-entropy-maximization algorithms. Non-linear versions of the MWN method extend the noniterative, Burg, maximum-entropy spectral-estimation procedure.

Accelerated Iterative Band-Limited Extrapolation Algorithms

The iterative algorithms of Papoulis-Gerchberg and Cadzow are well-known algorithms for solving the band-limited extrapolation problem. These algorithms, however, are usually slowly convergent. We propose an acceleration scheme for both algorithms, based on finding best approximation in appropriate subspaces. The acceleration scheme can also be applied to other cases of signal restoration, which admit Youla's method of alternating orthogonal projections.

A linear, time-varying system framework for noniterative discrete-time band-limited signal extrapolation

A technique for the analysis and design of noniterative algorithms for discrete-time, band-limited signal extrapolation is described. The approach involves modeling the extrapolation process as a linear, time-varying (LTV) system, or filter. Together with a previously developed Fourier theory for LTV systems, this model provides a frequency-domain transfer function representation for the extrapolation system. This representation serves as a powerful tool for characterizing and comparing the reconstruction properties of several well-known least squares optimal algorithms for band-limited extrapolation. Moreover, the frequency-domain setting provides a conceptually attractive means for understanding the process of extrapolation itself. Additionally, a least squares approximation methodology for designing LTV filters for band-limited extrapolation is developed. The design technique is shown to unify a broad class of algorithms for extrapolating discrete-time data and, further, to provide a means for designing new and improved extrapolation algorithms. >

Energy concentration in band-limited extrapolation

The authors present a novel bandlimited extrapolation method that incorporates qualitative prior knowledge of the distribution of signal energy in both the time domain (spatial domain) and the frequency domain. The extrapolation is easily computed by simple noniterative algorithm. The prior information is incorporated by means of a weighted energy measure, and the resulting solution is derived by straightforward Hilbert space optimization. The algorithm accommodates complex-valued sequences, nonuniform sampling, and arbitrary spectral support and is applicable in one or several dimensions. Many existing extrapolation algorithms are noted as special cases. Examples of signal extrapolation and spectrum estimation using the new algorithm are given in one and two dimensions, algorithm modification for noisy data is investigated, and error bounds are presented. >

A general framework for developing and evaluating band-limited function extrapolation methods

It is shown that the bandlimited function extrapolation can be considered as an approximation problem in operator space. Within this framework, the Papoulis-Gerchberg method and two other methods for extrapolation are analyzed. Particular attention is given to approximation using Taylor and Chebyschev series, and to approximation using the Lagrange interpolation method. >

Resolution and robustness in image processing: a new regularization principle

As band-limited interpolation is possible to some extent, whereas band-limited extrapolation is strictly forbidden, it is natural to state the deconvolution problem in terms of weighted interpolation (in the frequency domain). The corresponding regularization principle, which is to be compared with that of Tikhonov, proves to be intimately related to the notion of resolution: the aperture to be synthesized explicitly appears in the definition of the stabilizing component of the objective function. The choice of this synthetic aperture of course depends on the error analysis. The study developed in this paper shows, in particular, how the amount and the nature of the interpolation to be performed play a central part in the error propagation. The corresponding analytic parameters are exhibited. This survey, which is illustrated with the aid of a simulated example, also indicates how to estimate the degree of confidence of any other deconvolution technique (clean, filtered singular-value decomposition, maximum-entropy, etc.).

Optimal reconstruction algorithm for holographic imaging of finite-size objects

This paper introduces the formulation of the optimal image reconstruction algorithm for finite-size objects. For practical data acquisition and computation considerations, the partial discrete (continuous–discrete) model is used for the formulation. The basic constraints utilized in this optimization process are the finite size of the object and the limited spatial-frequency bandwidth of the resultant wave field. The result indicates that the algorithm consists of a linear matrix transformation followed by backward propagation. It is also shown that the bandlimited wave field extrapolation method and backward propagation technique are degenerate cases of this optimal algorithm. In addition to image reconstruction, the formulation of this optimal algorithm can be used to determine the optimal wave field sampling for efficient data acquisition.

Measurement accuracies in band-limited extrapolation

It is shown that in band-limited (or visible angle limited) extrapolation the largest effective aperture L' that can be realized from a finite aperture L by over sampling is a function of the accuracy of measurements. It is shown that for sampling in the interval |x-L/2b|x<L/2, b > 1 the extrapolating series must be known within a relative error e L' (i.e. the ratio of smallest term over the dominant first term) given by ▪ where L is the physical aperture, L' is the extrapolated aperture, λ is the wavelength, L/b is the sample point closet to origin.

An improved version of Papoulis-Gerchberg algorithm on band-limited extrapolation

An iterative algorithm for extrapolating analog band-limited signals has been proposed by Papoulis and Gerchberg. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> By inserting a multiplication by a constant in the above algorithm, chosen to minimize the energy of the error in the extrapolation interval, aconsiderable speed up of its convergence has been achieved.

A unified approach to noniterative linear signal restoration

The main goal of this paper is to describe a unified framework for several noniterative algorithms for signal extrapolation reported in the literature. This unification is achieved through integral equation and Hilbert space theories. The importance of this unification is that we can bring to bear the vast body of techniques in these theories to the solution of the extrapolation problem. We will show that the so-called two-step procedures for extrapolation with different underlying models can be unified by means of noniterative algorithms for solving optimization problems in Hilbert spaces. In particular, we show that two-step procedures under a discrete-continuous model [1], [2] belong to a general class of well-known algorithms for solving linear integral equations of the first kind: given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x), x \in A</tex> find <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t), t \in \Omega</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x) = \int\min{\Omega} K(x,t)f(t)dt, x \in A</tex> (1) In addition, we will show that the prolate spheroidal expansion technique is also a special case of the well-known Picard's eigenfunction procedure for the general integral problem (1). This theoretical unification, together with that presented in [3] for iterative least-squares algorithms, demonstrates that most of the well-known procedures for band-limited extrapolation can be considered as special cases of standard techniques in integral equations and operator theory.

Some aspects of band-limited signal extrapolation: Models, discrete approximations, and noise

We present some theoretical results on the band-limited signal extrapolation problem. In Section I we describe four basic models for the extrapolation problem. These models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in [1] and [2]. One of these models was shown to approximate the continuous band-limited extrapolation problem [3]. Another model is obtained when the discrete Fourier transform (DFT) is used to implement the well-known iterative algorithm given in [4] and [5] which was designed for solving the continuous extrapolation problem; in Section II this model is related to the continuous model by means of an interesting approximation theorem. Also, an important conjecture is presented. Section III shows some approximation results. Specifically, we prove that some discrete-discrete and discrete-continuous extrapolations of noisy signals converge to solutions of a certain continuous-continuous noisy extrapolation problem when the noise η is bounded by a known number, max <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\eta(x)| leq \epsilon</tex> . This convergence is obtained by using normal families of entire functions in ¢ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and some other complex analysis tools. We also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired. This result indicates that in the presence of noise, extrapolation techniques should be used judiciously in order to obtain reasonable results.

Image restoration and resolution enhancement

The ill-posed problem of restoring object information from finitely many measurements of its spectrum can be solved by using the best approximation in Hilbert spaces appropriately designed to include a priori information about object extent and shape and noise statistics. The procedures that are derived are noniterative, the linear ones extending the minimum-energy band-limited extrapolation methods (and thus related to Gerchberg–Papoulis iteration) and the nonlinear ones generalizing Burg’s maximum-entropy reconstruction of nonnegative objects.

Nonuniqueness of superresolution techniques applied to sampled data

It is demonstrated that, given a finite number of samples of a function, the constraints of square integrability and band limitation are not sufficient to determine that function uniquely. Hence existing algorithms for band-limited interpolation and extrapolation (e.g., for super resolution) require additional constraints such as those provided by an appropriate model.