The extent to which visual inference is shaped by attentional goals is unclear. Voluntary attention may simply modulate the priority with which information is accessed by the higher cognitive functions involved in perceptual decision making. Alternatively, voluntary attention may influence fundamental visual processes, such as those involved in segmenting an incoming retinal signal into a structured scene of coherent objects, thereby determining perceptual organization. Here we tested whether the segmentation and integration of visual form can be determined by an observer's goals, by exploiting a novel variant of the classical Kanizsa figure. We generated predictions about the influence of attention with a machine classifier and tested these predictions with a psychophysical response classification technique. Despite seeing the same image on each trial, observers' perception of illusory spatial structure depended on their attentional goals. These attention-contingent illusory contours directly conflicted with other, equally plausible visual forms implied by the geometry of the stimulus, revealing that attentional selection can determine the perceived layout of a fragmented scene. Attentional goals, therefore, not only select precomputed features or regions of space for prioritized processing, but under certain conditions also greatly influence perceptual organization, and thus visual appearance.