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Interpolation and the Discrete PapoulisGerchberg Algorithm
 IEEE Trans. Signal Processing
, 1994
"... In this paper we analyze the performance of an iterative algorithm, similar to the discrete PaponiisGerchberg algorithm, and which can be used to recover missing samples in finitelength records of bandlimited data. No assumptions are made regarding the distribution of the missing samples, in cont ..."
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Cited by 32 (20 self)
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In this paper we analyze the performance of an iterative algorithm, similar to the discrete PaponiisGerchberg algorithm, and which can be used to recover missing samples in finitelength records of bandlimited 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 lowpass 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.
Analysis And Design Of MinimaxOptimal Interpolators
 IEEE Trans. Signal Proc
, 1998
"... We consider a class of interpolation algorithms, including the leastsquares optimal Yen interpolator, and we derive a closedform 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 sa ..."
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Cited by 13 (3 self)
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We consider a class of interpolation algorithms, including the leastsquares optimal Yen interpolator, and we derive a closedform 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 illconditioning, forcing the use of a regularized, approximate solution. We suggest a new, approximate solution, consisting of a sinckernel interpolator with specially chosen weighting coefficients. The newly designed sinckernel 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 sinckernel interpolator is shown to perform better than ...
Dualization of signal recovery problems
, 2009
"... In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenab ..."
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Cited by 8 (3 self)
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In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature FenchelMoreauRockafellar dual problems that can be solved by forwardbackward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing dualitybased signal recovery methods and to be applicable to a variety of new problems beyond their scope.
Duality and Convex Programming
, 2010
"... We survey some key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging. ..."
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Cited by 4 (2 self)
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We survey some key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging.
Signal Processing Issues In Synthetic Aperture Radar And Computer Tomography
, 1998
"... This paper also proposed another reconstruction method based on a direct approximation of the Fourier inversion formula using a twodimensional (2D) trapezoidal rule. In addition, the possibility of reconstruction from a concentricsquares raster was discussed. Numerous simple interpolators have bee ..."
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Cited by 1 (0 self)
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This paper also proposed another reconstruction method based on a direct approximation of the Fourier inversion formula using a twodimensional (2D) trapezoidal rule. In addition, the possibility of reconstruction from a concentricsquares raster was discussed. Numerous simple interpolators have been tried in DF reconstruction with the results compared with CBP [33]. In [34] and [35], the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In [36], a DF reconstruction using bilinear interpolation for diffraction tomography provided image quality that was comparable to that produced by the CBP algorithm. Very good reconstruction quality was obtained in [37] and [38] using a spline interpolator, or a hybrid type of spline interpolator. The notion of "gridding" was introduced in [39] as a method of obtaining optimal inversion of Fourier data. An optimal gridding function was proposed, and successful results were obtained when applied to the tomographic reconstruction problem. In [40], several different gridding functions were tried for DF reconstruction, and the performances were compared. In [41, 42], the linogram reconstruction method was proposed as a form of DF reconstruction. The data collection grid in the linogram method is the same as in the concentricsquares sampling scheme. The inversion of the Fourier data in [41, 42] was accomplished by first applying the chirpz transform in one direction and then computing FFTs in the other direction. In CT, many of these attempts at DF reconstruction have given a poorer result than the CBP algorithm, due to the error incurred in the process of the polartoCartesian interpolation. The attraction of DF reconstruction, however, is that it is thought to require less computation than ...
unknown title
, 907
"... In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in signal recovery. These problems are not easily amenable to solution ..."
Abstract
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In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in signal recovery. These problems are not easily amenable to solution by current methods but they feature FenchelMoreauRockafellar dual problems that can be solved reliably by forwardbackward splitting and allow for a simple construction of primal solutions from dual solutions. The proposed framework is shown to capture and extend several existing dualitybased signal recovery methods and to be applicable to a variety of new problems beyond their scope.
unknown title
, 907
"... In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in signal recovery. These problems are not easily amenable to solution ..."
Abstract
 Add to MetaCart
In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in signal recovery. These problems are not easily amenable to solution by current methods but they feature FenchelMoreauRockafellar dual problems that can be solved reliably by forwardbackward splitting and allow for a simple construction of primal solutions from dual solutions. The proposed framework is shown to capture and extend several existing dualitybased signal recovery methods and to be applicable to a variety of new problems beyond their scope.