Binary‐image restoration

We consider the problem of binary‐image restoration. The image being restored is not random, and we make no assumption about the nature of its contents. The estimate of the colour at each site is a fixed (the same for all sites) function of the data available in a neighbourhood of that site. Under this restriction, the estimate minimizing the overall mean squared error of prediction is the conditional expectation of the true colour given the observations in the neighbourhood of a site. The computation of this conditional expectation leads to the formal definition of the local characteristics of an image, namely, the frequency with which each pattern appears in the true unobserved image. When the “true” distribution of the patterns is unknown, it can be estimated from the records. The conditional expectation described above can then be evaluated using the estimated distribution of the patterns, and this procedure leads to a very natural estimate of the colour at each site. We propose two unbiased and consistent estimates for the distribution of patterns when the noise is a Gaussian white noise. Since the size of realistic images is very large, the estimated pattern distribution is usually close to the true one. This suggests that the estimated conditional expectation can be expected to be nearly optimal. An interesting feature of the proposed restoration methods is that they do not require prior knowledge of the local or global properties of the true underlying image. Several examples based on synthetic images show that the new methods perform fairly well for a variety of images with different degrees of colour continuity or textures.