Locating and Correcting Errors in Images

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(n) or even O(n logn) 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. 1 Notation The complex N -dimensional space, with the usual inner product and norm, is denoted by C . The conjugate transpose is denoted by ∗. The Fourier matrix F is the unitary N × N matrix whose elements Fab are given by Fab = 1 √ N e−j 2π N , where j denotes the imaginary unit. The discrete Fourier transform (DFT) of a signal or image x is denoted by x̂. 2 Outline Many of the known image interpolation and extrapolation methods have one common characteristic: the positions of the unknown pixels are assumed to be known. In practice, this is not always the case, a fact that renders these methods of limited practical usefulness under some circumstances. In this paper we attempt to bridge this gap, and propose an algorithm for locating the positions of the incorrect pixels in an image, assuming only partial knowledge of its Fourier transform. Note that this is a nonlinear problem, in which the unknown quantities are the values and positions of the erroneous pixels. We show that the positions of n incorrect pixels in a N ×N image can be evaluated in O(n) or O(n logn) time by solving a set of n equations and computing a FFT. It is not necessary to know n beforehand: the computation of n is part of the algorithm. After locating the position of the incorrect pixels, it remains to interpolate them — any of the known interpolation methods can be in principle used for this purpose. The stability of the algorithm, which naturally depends on the pattern of the incorrect pixels, is also briefly mentioned. 3 Related work There is a vast literature on image interpolation, extrapolation, and related issues — they include implementation aspects, stability analysis, convergence acceleration, noniterative algorithms, the sampled analog of the problem, convergence criteria, the effect of stabilizing constraints, and much more. See [1–17], among many others. The problem discussed in this paper is also related to error-control codes: it is known that oversampling [18] is an alternative to error-control coding. A discussion concerning the relations between these two and a few other relevant issues, in the context of signal and image reconstruction, can be found in [19]. 4 Locating the incorrect pixels This section describes a simple technique that enables the location of n incorrect pixels in any row or column of a N ×N image (in the following we assume that we are dealing with a row). The constraint upon which the method depends is the vanishing of 2n elements of the DFT of the row. Given a unconstrained row of N − 2n pixels this can readily be accomplished 0-8186-8183-7/97 $10.00 @ 1997 IEEE 691 Ferreira: Locating and Correcting Errors in Images by computing its DFT, padding 2n zeros, and computing the inverse DFT. The set U = {i0, i1, . . . in−1} is used to describe the positions of the n incorrect pixels in a row x of a N ×N image. We denote by y the observed row, and by e the error