© 2015 ARVO. Shape is a critical cue to object identity. In psychophysical studies, radial frequency (RF) patterns, paths deformed from circular by a sinusoidal modulation of radius, have proved valuable stimuli for the demonstration of global integration of local shape information. Models of the mechanism of integration have focused on the periodicity in measures of curvature on the pattern, despite the fact that other properties covary. We show that patterns defined by rectified sinusoidal modulation also exhibit global integration and are indistinguishable from conventional RF patterns at their thresholds for detection, demonstrating some indifference to the modulating function. Further, irregular patterns incorporating four different frequencies of modulation are globally integrated, indicating that uniform periodicity is not critical. Irregular patterns can be handed in the sense that mirror images cannot be superimposed. We show that mirror images of the same irregular pattern could not be discriminated near their thresholds for detection. The same irregular pattern and a pattern with four cycles of a constant frequency of modulation completing 2π radians were, however, perfectly discriminated, demonstrating the existence of discrete representations of these patterns by which they are discriminated. It has previously been shown that RF patterns of different frequencies are perfectly discriminated but that patterns with the same frequency but different numbers of cycles of modulation were not. We conclude that such patterns are identified, near threshold, by the set of angles subtended at the center of the pattern by adjacent points of maximum convex curvature.