Shifting attention away from a visual stimulus reduces, but does not abolish, visual discrimination performance. This residual vision with 'poor' attention can be compared to normal vision with 'full' attention to reveal how attention alters visual perception. We report large differences between residual and normal visual thresholds for discriminating the orientation or spatial frequency of simple patterns, and smaller differences for discriminating contrast. A computational model, in which attention activates a winner-take-all competition among overlapping visual filters, quantitatively accounts for all observations. Our model predicts that the effects of attention on visual cortical neurons include increased contrast gain as well as sharper tuning to orientation and spatial frequency.