The normalized compression distance and image distinguishability

We use an information-theoretic distortion measure called the Normalized Compression Distance (NCD), first proposed by M. Li et al., to determine whether two rectangular gray-scale images are visually distinguishable to a human observer. Image distinguishability is a fundamental constraint on operations carried out by all players in an image watermarking system. The NCD between two binary strings is defined in terms of compressed sizes of the two strings and of their concatenation; it is designed to be an effective approximation of the noncomputable but universal Kolmogorov distance between two strings. We compare the effectiveness of different types of compression algorithms? in predicting image distinguishability when they are used to compute the NCD between a sample of images and their watermarked counterparts. Our experiment shows that, as predicted by Li’s theory, the NCD is largely independent of the underlying compression algorithm. However, in some cases the NCD fails as a predictor of image distinguishability, since it is designed to measure the more general notion of similarity. We propose and study a modified version of the NCD to model the latter, which requires that not only the change be small but also in some sense random with respect to the original image.