The general goal of vision is to identify the perceptual and physiological mechanisms that allow human observers to successfully navigate through their environment, discriminate objects that populate it, and gather these objects for survival. Such actions are possible only if the appropriate information for carrying them out is somehow picked up from the environment by the visual system. What makes performing these actions difficult is that the same retinal images are consistent with the projection of many 3-D Euclidean scenes and objects. Given this problem, recent empirical and theoretical investigations have attempted to establish the appropriate spatial primitives for describing objects, images, and percepts. In particular, Lappin and Craft (2000) have addressed this problem by introducing a new formulation of visual information. They defined visual information as a set of spatial relationships specified intrinsically among texture elements belonging to discrete positions within a smooth 2-D manifold, without making reference to any extrinsic frame of reference. The central problem in surface perception, according to Lappin and Craft, is to identify the relationships that preserve one-to-one mappings between environmental objects, retinal images, and visual perception. Thus, they examined spatial relationships of different levels of complexity, depending on the number of texture elements involved 1 and the number of spatial dimensions along which these relationships are measured.Special emphasis is given by Lappin and Craft (2000) to fourth-order structures-that is, to second-order spatial differentials taken in the 2-D neighborhood surrounding a given point (see also Koenderink & van Doorn, 1992). These higher order differential structures map one-to-one to local shapes, of which there are four alternatives: planar, parabolic, elliptic, or hyperbolic shapes (see Koenderink's, 1990, concept of shape index). The local shape at any point on a smooth surface depends only on the relative magnitude of two curvatures (i.e., second-order differentials) measured conjointly along two spatial dimensions in the local neighborhood surrounding the point, suggesting that local shape is the primitive information for spatial vision (see also Koenderink, 1990;Koenderink & van Doorn, 1992;Perotti, Todd, Lappin, & Phillips, 1998). That is, local shape is not derived from lower order spatial information (positions of texture elements, 2-D distances between texture elements, 3-D orientations, and so on).Lappin and Craft (2000) tested their analysis by disrupting lower order structures while maintaining (higher) fourth-order structures. Here, we focus on their third experiment, which partially motivated our present work. In that experiment, Lappin and Craft measured hyperacuity on a task in which observers adjusted a probe dot to lie Investigators have proposed that qualitative shapes are the primitive information of spatial vision: They preserve an approximately one-to-one mapping between surfaces, images, and perception. Given ...