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

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 band-limited finite-dimensional interpolation problem (ii) the problem of estimating a signal that is the superposition of a finite number of harmonics (iii) an error-control coding problem in the real field, and (iv) certain techniques that occur in algorithm-based 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 n-dimensional space, with the usual inner product and norm, is denoted by C n . A signal is a n-dimensional 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 band-limited if a subset of the samples of x vanish, and is low-pass if the nonzero DFT Fax +351-34-370545, e-mails vieira@inesca.pt and pjf@inesca.pt. This work was supported by JNICT.

We consider the problem of restoring randomly distributed sets of missing pixels in band-limited discrete images, and give non-iterative and iterative algorithms capable of error-free 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 Papoulis-Gerchberg 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 band-limited discrete images. This problem could be solved using the two-dimensional version of the well-known Papoulis-Gerchberg algorithm, at the cost of essentially a direct and an inverse discrete Fourier transform of the whole imag...

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.

. 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 fault-tolerant 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 finite-dimensional theory because of i...

Discrete Fourier transform codes (DFT codes) or real number codes have been studied and recognized as useful (as joint source-channel codes, for example) but are not stable under bursty losses. This letter introduces a two-channel DFT code with an interleaver and shows that its numerical stability far exceeds that of the corresponding single-channel DFT code (the ratio of the frame bounds for the two-channel system can be smaller by many orders of magnitude). This leads to a stable way of dealing with bursts of errors using DFT codes.

In this paper we consider the following problem: to identify a subset of the samples of a band-limited 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 band-limited). This is a nonlinear problem which is closely connected to several other signal processing problems. I. Introduction Throughout this paper, a signal is a N-dimensional 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 pass-band B if the samples x(i) with i = 2 B vanish. Consider the following two problems, both of which can be...

This paper describes a new fast, iterative algorithm for interactive image noise removal. Given the locations of noisy pixels and a prototype image, the noisy pixels are to be restored in a natural way. Most existing image noise removal algorithms use either frequency domain information (e.g low pass filtering) or spatial domain information (e.g median filtering or stochastic texture generation). However, for good noise removal, both spatial and frequency information must be used. The existing algorithms that do combine the two domains (e.g Gerchberg-Papoulis and related algorithms) place the limitation that the image be band limited and the band limits be known. Also, some of these may not work well when the noisy pixels are contiguous and numerous. Our algorithm combines the spatial and frequency domain information by using projection onto convex sets (POCS). But unlike previous methods it does not need to know image band limits and does not require the image to be band limited. Resu...

There are many algorithms for the interpolation or extrapolation of band-limited one or two dimensional signals from nonuniform samples. The limitations of some of these algorithms, which are acutely felt under certain circumstances, lead several authors to seek alternative and better ways of accomplishing these goals. A recent and welcome addition to this family of methods is a class of algorithms that spring from the work of Grochenig and Strohmer on interpolation and extrapolation, and which seem to have a number of very desirable characteristics. Since this approach is relatively recent, it is perhaps not yet widely known. In this work we discuss certain aspects of these ideas and compare them against another recent approach, also perhaps not yet widely known. Both approaches yield efficient algorithms for interpolation and extrapolation, which are often orders of magnitude more efficient than methods which have been proposed in the recent past. We stress the similarities between t...

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