We examine properties of perceptual image distortion models, computed as the mean squared error in the response of a 2-stage cascaded image transformation. Each stage in the cascade is composed of a linear transformation, followed by a local nonlinear normalization operation. We consider two such models. For the first, the structure of the linear transformations is chosen according to perceptual criteria: a center-surround filter that extracts local contrast, and a filter designed to select visually relevant contrast according to the Standard Spatial Observer. For the second, the linear transformations are chosen based on statistical criterion, so as to eliminate correlations estimated from responses to a set of natural images. For both models, the parameters that govern the scale of the linear filters and the properties of the nonlinear normalization operation, are chosen to achieve minimal/maximal subjective discriminability of pairs of images that have been optimized to minimize/maximize the model, respectively (we refer to this as MAximum Differentiation, or "MAD", Optimization). We find that both representations substantially reduce redundancy (mutual information), with a larger reduction occurring in the second (statistically optimized) model. We also find that both models are highly correlated with subjective scores from the TID2008 database, with slightly better performance seen in the first (perceptually chosen) model. Finally, we use a foveated version of the perceptual model to synthesize visual metamers. Specifically, we generate an example of a distorted image that is optimized so as to minimize the perceptual error over receptive fields that scale with eccentricity, demonstrating that the errors are barely visible despite a substantial MSE relative to the original image.