Local image statistics are important for visual analysis of textures, surfaces, and form. There are many kinds of local statistics, including those that capture luminance distributions, spatial contrast, oriented segments, and corners. While sensitivity to each of these kinds of statistics have been well-studied, much less is known about visual processing when multiple kinds of statistics are relevant, in large part because the dimensionality of the problem is high and different kinds of statistics interact. To approach this problem, we focused on binary images on a square lattice – a reduced set of stimuli which nevertheless taps many kinds of local statistics. In this 10-parameter space, we determined psychophysical thresholds to each kind of statistic (16 observers) and all of their pairwise combinations (4 observers). Sensitivities and isodiscrimination contours were consistent across observers. Isodiscrimination contours were elliptical, implying a quadratic interaction rule, which in turn determined ellipsoidal isodiscrimination surfaces in the full 10-dimensional space, and made predictions for sensitivities to complex combinations of statistics. These predictions, including the prediction of a combination of statistics that was metameric to random, were verified experimentally. Finally, check size had only a mild effect on sensitivities over the range from 2.8 to 14 min, but sensitivities to second- and higher-order statistics was substantially lower at 1.4 min. In sum, local image statistics form a perceptual space that is highly stereotyped across observers, in which different kinds of statistics interact according to simple rules.