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27
Combining Frequency and Spatial Domain Information for Fast Interactive Image Noise Removal
, 1996
"... Scratches on old films must be removed since these are more noticeable on higher definition and digital televisions. Wires that suspend actors or cars must be carefully erased during post production of special effects shots. Both of these are time consuming tasks but can be addressed by the followin ..."
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Cited by 44 (1 self)
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Scratches on old films must be removed since these are more noticeable on higher definition and digital televisions. Wires that suspend actors or cars must be carefully erased during post production of special effects shots. Both of these are time consuming tasks but can be addressed by the following image restoration process: given the locations of noisy pixels to be replaced and a prototype image, restore those noisy pixels in a natural way. We call it image noise removal and this paper describes its fast iterative algorithm. Most existing algorithms for removing image noise use either frequency domain information (e.g low pass filtering) or spatial domain information (e.g median filtering or stochastic texture generation). The few that do combine the two domains place the limitation that the image be band limited and the band limits be known. Our algorithm works in both spatial and frequency domains
Interpolation, Spectrum Analysis, ErrorControl Coding, and FaultTolerant Computing
 In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 97, volume III
, 1997
"... This paper uncovers relations between the topics mentioned in the title, relations that we believe to have gone nearly unnoticed so far. More precisely, we show that four often studied problems in signal processing, spectrum analysis, information theory, and computing are closely related or even equ ..."
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Cited by 16 (7 self)
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This paper uncovers relations between the topics mentioned in the title, relations that we believe to have gone nearly unnoticed so far. More precisely, we show that four often studied problems in signal processing, spectrum analysis, information theory, and computing are closely related or even equivalent in a certain sense (if one of them can be solved, so can any of the others, and using essentially the same algorithms). The problems are (i) a nonlinear bandlimited finitedimensional interpolation problem (ii) the problem of estimating a signal that is the superposition of a finite number of harmonics (iii) an errorcontrol coding problem in the real field, and (iv) certain techniques that occur in algorithmbased fault tolerant computing. The advantages of recognizing these problems as equivalent are obvious: the techniques commonly used in one field can be imported to the others, the duplication of research e#orts is prevented, and the overall degree of understanding of the four problems increases. New algorithms are suggested as a result of these investigations. 1. NOTATION The complex ndimensional space, with the usual inner product and norm, is denoted by C n . A signal is a ndimensional complex vector x, with components, or samples, x(0), x(1), . . . , x(n 1). The Fourier matrix F is the n n matrix whose elements F ab are given by F ab = e j n ab where j denotes the imaginary unit. The discrete Fourier transform (DFT) of x, denoted by x, is defined by x = Fx. A signal x is bandlimited if a subset of the samples of x vanish, and is lowpass if the nonzero DFT Fax +35134370545, emails vieira@inesca.pt and pjf@inesca.pt. This work was supported by JNICT.
Errorless Restoration Algorithms for BandLimited Images
 in Proceedings of the First IEEE International Conference on Image Processing, ICIP94
, 1994
"... We consider the problem of restoring randomly distributed sets of missing pixels in bandlimited discrete images, and give noniterative and iterative algorithms capable of errorfree restoration. The methods discussed have minimum dimension, that is, the size of the matrices and vectors which appea ..."
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Cited by 14 (7 self)
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We consider the problem of restoring randomly distributed sets of missing pixels in bandlimited discrete images, and give noniterative and iterative algorithms capable of errorfree restoration. The methods discussed have minimum dimension, that is, the size of the matrices and vectors which appear in the algorithm is determined by the number of unknown pixels. This is a characteristic which an alternative iterative formulation, based on the PapoulisGerchberg iteration, does not have. Convergence proofs for both the basic algorithms and a number of accelerated iterative methods are included as well. The performance of the methods is demonstrated with examples. 1. INTRODUCTION In this paper we consider the problem of restoring sets of lost pixels in bandlimited discrete images. This problem could be solved using the twodimensional version of the wellknown PapoulisGerchberg algorithm, at the cost of essentially a direct and an inverse discrete Fourier transform of the whole imag...
Interpolation in the Time and Frequency Domains
 IEEE SIGNAL PROCESSING LETTERS
, 1996
"... In this letter, we clarify the connections between two recently proposed and apparently unrelated approaches to bandlimited interpolation by showing that, in a certain sense made precise below, they are the dual of each other. The advantages of recognizing this duality are discussed. ..."
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Cited by 14 (11 self)
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In this letter, we clarify the connections between two recently proposed and apparently unrelated approaches to bandlimited interpolation by showing that, in a certain sense made precise below, they are the dual of each other. The advantages of recognizing this duality are discussed.
Mathematics for Multimedia Signal Processing II: Discrete Finite Frames and Signal Reconstruction
 in Signal Processing for Multimedia
, 1999
"... . Certain signal reconstruction problems can be understood in terms of frames and redundant representations. The redundancy is useful because it leads to robust signal representations, that is, representations in which partial loss of data can be tolerated without misbehavior or adverse effects. ..."
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Cited by 10 (3 self)
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. Certain signal reconstruction problems can be understood in terms of frames and redundant representations. The redundancy is useful because it leads to robust signal representations, that is, representations in which partial loss of data can be tolerated without misbehavior or adverse effects. This chapter begins by presenting a few engineering problems in which robust data representations play a central role. It turns out that these problems, which occur in signal processing, spectrum analysis, information theory, and faulttolerant computing, are closely related or even equivalent. However, perhaps surprisingly, the connections between them have gone nearly unnoticed so far. Frames, and in particular discrete finite frames, provide one of the ways of understanding certain of these problems, including the important missing data problem. Some of the methods that can be used to recover from missing data errors are examined, emphasizing finitedimensional theory because of i...
Iterative and Noniterative Recovery of Missing Samples for 1D BandLimited Signals
, 2000
"... ..."
Stable DFT Codes and Frames
 IEEE Signal Processing Letters, vol.10 No.2,(2003
, 2003
"... Discrete Fourier transform codes (DFT codes) or real number codes have been studied and recognized as useful (as joint sourcechannel codes, for example) but are not stable under bursty losses. This letter introduces a twochannel DFT code with an interleaver and shows that its numerical stability f ..."
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Cited by 6 (0 self)
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Discrete Fourier transform codes (DFT codes) or real number codes have been studied and recognized as useful (as joint sourcechannel codes, for example) but are not stable under bursty losses. This letter introduces a twochannel DFT code with an interleaver and shows that its numerical stability far exceeds that of the corresponding singlechannel DFT code (the ratio of the frame bounds for the twochannel system can be smaller by many orders of magnitude). This leads to a stable way of dealing with bursts of errors using DFT codes.
Locating and Correcting Errors in Images
 IN PROC. ICIP
, 1997
"... Most image interpolation or extrapolation algorithms assume that the locations of the unknown pixels are known. In this paper we attempt to remove this restriction. More precisely, we propose an algorithm for locating the incorrect pixels of an image, assuming only partial knowledge of its Fourier t ..."
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Cited by 4 (1 self)
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Most image interpolation or extrapolation algorithms assume that the locations of the unknown pixels are known. In this paper we attempt to remove this restriction. More precisely, we propose an algorithm for locating the incorrect pixels of an image, assuming only partial knowledge of its Fourier transform. Note that this is a nonlinear problem: the unknown quantities are the positions and values of the (say) n erroneous pixels. We show that the positions can be evaluated in O(n²) or even O(n log n) flops by solving a set of n linear equations and computing a FFT. The determination of n is part of the algorithm, whose stability is also briefly discussed. The values of the n incorrect pixels can then be estimated using any of the interpolation methods known.
Eldar, “Nonuniform sampling of periodic bandlimited signals: Part I–Reconstruction theorems,” submitted to
 IEEE Trans. Signal Processing
, 2004
"... Abstract—Digital processing techniques are based on representing a continuoustime signal by a discrete set of samples. This paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples. In practical applications, only a finite number of val ..."
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Cited by 4 (0 self)
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Abstract—Digital processing techniques are based on representing a continuoustime signal by a discrete set of samples. This paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples. In practical applications, only a finite number of values are given. Extending the samples periodically across the boundaries, and assuming that the underlying continuous time signal is bandlimited, provides a simple way to deal with reconstruction from finitely many samples. Two algorithms for reconstructing a periodic bandlimited signal from an even and an odd number of nonuniform samples are developed. In the first, the reconstruction functions constitute a basis while in the second, they form a frame so that there are more samples than needed for perfect reconstruction. The advantages and disadvantages of each method are analyzed. Specifically, it is shown that the first algorithm provides consistent reconstruction of the signal while the second is shown to be more stable in noisy environments. Next, we use the theory of finite dimensional frames to characterize the stability of our algorithms. We then consider two special distributions of sampling points: uniform and recurrent nonuniform, and show that for these cases, the reconstruction formulas as well as the stability analysis are simplified significantly. The advantage of our methods over conventional approaches is demonstrated by numerical experiments. Index Terms—Interpolation, nonuniform sampling, periodic signals, reconstruction, recurrent nonuniform sampling, stability, uniform sampling. I.
Detection and Correction of Missing Samples
 In Proceedings of the 1997 Workshop on Sampling Theory and Applications
, 1997
"... In this paper we consider the following problem: to identify a subset of the samples of a bandlimited signal that has been corrupted by noise. It is assumed that the Fourier transform of the signal is partially known (typically because the signal is bandlimited). This is a nonlinear problem which ..."
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Cited by 3 (2 self)
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In this paper we consider the following problem: to identify a subset of the samples of a bandlimited signal that has been corrupted by noise. It is assumed that the Fourier transform of the signal is partially known (typically because the signal is bandlimited). This is a nonlinear problem which is closely connected to several other signal processing problems. I. Introduction Throughout this paper, a signal is a Ndimensional complex vector x 2 C N , whose elements (or samples) are denoted by x(0); x(1); : : : ; x(N \Gamma 1). The conjugate transpose of a matrix M will be denoted by M . The Fourier matrix F is the unitary N \Theta N matrix whose elements F ab are given by F ab = 1 p N e \Gammai 2 N ab ; where i denotes the imaginary unit. The discrete Fourier transform (DFT) of x, denoted by x, is defined by x = Fx. A signal x 2 C N is bandlimited with passband B if the samples x(i) with i = 2 B vanish. Consider the following two problems, both of which can be...