The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformly-spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

In this paper we analyze the performance of an iterative algorithm, similar to the discrete Paponiis-Gerchberg algorithm, and which can be used to recover missing samples in finite-length records of band-limited data. No assumptions are made regarding the distribution of the missing samples, in contrast with the often studied extrapolation problem, in which the known samples are grouped together. Indeed, it is possible to regard the observed signal as a sampled version of the original one, and to interpret the reconstruction result studied herein as a sampling result. We show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data. We give upper and lower bounds for the error as a function of the number of iterations, together with the signals for which the bounds are attained. Also, we analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix. From this analysis we infer the optimum value of the relaxation constant. We also point out, among all sampling sets with the same density, those for which the convergence rate of the recovery algorithm is maximum or minimum. For low-pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer. The worst convergence rates generally occur when the missing samples are contiguous.

We consider a class of interpolation algorithms, including the least-squares optimal Yen interpolator, and we derive a closed-form expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix which is specified for each set of sampling points. The error expression can be used to prove that the Yen interpolator is optimal. The implementation of the Yen algorithm suffers from numerical ill-conditioning, forcing the use of a regularized, approximate solution. We suggest a new, approximate solution, consisting of a sinc-kernel interpolator with specially chosen weighting coefficients. The newly designed sinc-kernel interpolator is compared with the usual sinc interpolator using Jacobian (area) weighting, through numerical simulations. We show that the sinc interpolator with Jacobian weighting works well only when the sampling is nearly uniform. The newly designed sinc-kernel interpolator is shown to perform better than ...

In some signal reconstruction problems, the observation equations can be used as a priori information for selecting the best combination of observations before acquiring them. In this paper, we define a selection criterion and propose efficient methods for optimizing the criterion with respect to the combination of observations. Our examples illustrate the value of optimized sampling using the proposed methods. I. Introduction Signal reconstruction uses measurements in one domain to estimate parameters or distributions in another domain. A common approach to the signal reconstruction problem is to model the observed signal y as a linear transformation of x observed in the presence of additive noise; that is, y = Ax + u ; (1) where u is additive noise, and A 2 C m\Thetan (m n) is full rank. For this problem, the goal is to reconstruct a good estimate of x given the observed signal y. In many applications, the relationship between the observation y i and the original signal x --- the...

Displaying a synthetic image on a computer display requires determining the colors of individual pixels. To avoid aliasing, multiple samples of the image can be taken per pixel, after which the color of a pixel may be computed as a weighted sum of the samples. The positions and weights of the samples play a major role in the resulting image quality, especially in real-time applications where usually only a handful of samples can be afforded per pixel. This paper presents a new error metric and an optimization method for antialiasing patterns used in image reconstruction. The metric is based on comparing the pattern against a given reference reconstruction filter in spatial domain and it takes into account psychovisually measured angle-specific acuities for sharp features. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Picture/Image Generation – Antialiasing

Abstract—Digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuoustime signal from a sequence of corrupted discrete-time samples. In this paper, a general formulation of this problem is developed that treats the interpolation problem from ideal, noisy samples, and the deconvolution problem in which the signal is filtered prior to sampling, in a unified way. The signal reconstruction is performed in a shift-invariant subspace spanned by the integer shifts of a generating function, where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter. Several alternative approaches to designing the correction filter are suggested, which differ in their assumptions on the signal and noise. The classical deconvolution solutions (least-squares, Tikhonov, and Wiener) are adapted to our particular situation, and new methods that are optimal in a minimax sense are also proposed. The solutions often have a similar structure and can be computed simply and efficiently by digital filtering. Some concrete examples of reconstruction filters are presented, as well as simple guidelines for selecting the free parameters (e.g., regularization) of the various algorithms. Index Terms—Deconvolution, interpolation, minimax reconstruction, sampling. I.

We investigate the use of direct-Fourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (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 2-D periodic sinc-kernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms utilizing 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...

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The eigenvalues of the matrices that occur in certain finite-dimensional interpolation problems are directly related to their well-posedness, and strongly depend on the distribution of the interpolation knots, that is, on the sampling set. We study this dependency as a function of the sampling set itself, and give accurate bounds for the eigenvalues of the interpolation matrices. The bounds can be evaluated in as few as four arithmetic operations, and so they greatly simplify the assessment of sampling sets regarding numerical stability. The accuracy and usefulness of the bounds are illustrated with examples. I. Introduction One problem commonly found in signal processing is that of recovering n lost samples of a band-limited discrete signal with a total of N samples. The Papoulis-Gerchberg algorithm [1, 2], although initially developed for extrapolation problems, is a well-known example of an iterative technique that can be used to solve this problem. It is related to a number ...

In this work we present a number of results concerning the finite dimensional band-limited interpolation problem. Our aim is to understand how the stability of the interpolation problem is affected by changes in the positions of the interpolation knots. We give upper and lower bounds for the eigenvalues of the interpolation matrix, as functions of the gaps between missing samples. The upper bound is especially useful, since it determines the convergence rate of an iterative interpolation procedure, and the conditioning of an equivalent noniterative method. The interplay between convergence rate, stability, normalized bandwidth of the data, and minimum gap between missing samples is clarified. 1. INTRODUCTION AND PRELIMINARIES This paper is concerned with finite-dimensional signals, that is, vectors x 2 C N . Such signals are called band-limited if their discrete Fourier transform x, defined by x k = 1 p N N \Gamma1 X i=0 x i e \Gammaj 2 N ik ; for 0 0 ! N , has a p...