Humans can rapidly recognize a multitude of objects despite differences in their appearance. The neural mechanisms that endow high-level sensory neurons with both selectivity to complex stimulus features and "tolerance" or invariance to identity-preserving transformations, such as spatial translation, remain poorly understood. Previous studies have demonstrated that both tolerance and selectivity to conjunctions of features are increased at successive stages of the ventral visual stream that mediates visual recognition. Within a given area, such as visual area V4 or the inferotemporal cortex, tolerance has been found to be inversely related to the sparseness of neural responses, which in turn was positively correlated with conjunction selectivity. However, the direct relationship between tolerance and conjunction selectivity has been difficult to establish, with different studies reporting either an inverse or no significant relationship. To resolve this, we measured V4 responses to natural scenes, and using recently developed statistical techniques, we estimated both the relevant stimulus features and the range of translation invariance for each neuron. Focusing the analysis on tuning to curvature, a tractable example of conjunction selectivity, we found that neurons that were tuned to more curved contours had smaller ranges of position invariance and produced sparser responses to natural stimuli. These trade-offs provide empirical support for recent theories of how the visual system estimates 3D shapes from shading and texture flows, as well as the tiling hypothesis of the visual space for different curvature values.A lthough object recognition feels effortless, it is in fact a challenging computational problem (1). There are two important properties that any system that mediates robust object recognition must have. The first property is known as "invariance": the ability of the system to respond similarly to different views of the same object. The second property is known as "selectivity." Selectivity requires that systems' components, such as neurons within the ventral visual stream, produce different responses to potentially quite similar objects (such as different faces) even when presented from similar viewpoints. It is straightforward to make detectors that are invariant but not selective or selective but not invariant. The difficulty lies in how to make detectors that are both selective and invariant.To address this problem, both computer object recognition algorithms (2) and neural systems use a series of hierarchical stimulus representations, increasing both in complexity and the range of invariance (1, 3). For example, in each successive area of visual processing, neurons become selective for increasingly complex stimulus features (4-9) and grow more tolerant to identity-preserving transformations, such as image translation, scaling, and, to some degree, rotation and the presence of "clutter" from other objects in the scene (3,(10)(11)(12). This has led to the idea that high-level sensory neurons are...