Everyone knows what convexity is, yet no single convexity measure captures its effect on figure/ground perception

Among geometric cues to figure/ground, “convexity” is one of the strongest, with a long history in vision science. However, a literature review reveals that researchers have been using the term “convex” for decades without a unified formal definition. This results in the concept being used inconsistently across studies. Our goal here was to 1) document the various definitions used in the literature to measure convexity (contour-based/region-based; local/global); 2) run computational experiments applying these measures to stimuli from the literature (and their variants) to see how well they track the effects of “convexity” on figure-ground judgments. We started with creating graded versions of commonly-used highly “convex” stimuli and incrementally turned them into unbiased stimuli by manipulating the relative strengths of their part boundaries. For each, we computed the values of various convexity measures on each side of the contour. Finally, we compared the results of these convexity computations with figure-ground judgments from existing studies. Our results showed that certain versions of region-based definitions (i.e., percentage of pairs of dots that can “see” each other) yielded convexity values that contradict what is generally judged as “convex” in the literature. Whereas different versions of curvature-based measurements (i.e., arc-length ratio of positive-to-negative curvature, or the integral of signed curvature), as well as global region-based measures (i.e., ratio of region’s area to its convex hull), yielded values consistent with the side judged as more “convex”; however, the computed magnitudes of convexity did not parallel their effects on figure-ground judgments. Our results demonstrate that no single formal measure of convexity consistently captures the strong effects of “convexity” on figure/ground perception. We suggest that these effects are closely related to the organization of visual shape in terms of parts and axes—specifically to the visual salience of the parts that result from viewing each side as potentially figural.