Defining perceptual metrics in shape space

PURPOSE: The goal of our research is to define a perceptually meaningful metric for measuring shape similarity. For comparing point sets, two measures have been proposed in the mathematics/statistics literature. The Procrustes distance measures the sum of squares of Euclidean distances between corresponding points, following the best similarity transformation for aligning the two sets. Kendall (1984) proposed a different measure, which has the advantage of being a Riemannian metric in shape space. We conducted experiments to evaluate whether these metrics are perceptually meaningful. METHOD: White 2D convex polygons were randomly generated and presented parafoveally on a mid-gray background. In a 2-IFC paradigm, subjects were shown shapes on either side of a fixation cross and they indicated which interval contained two different shapes. Shape discrimination thresholds were measured in terms of Procrustes and Kendall's distances, using the polygon vertices. First, we expect human discrimination thresholds to remain constant for both random and systematic shape changes. Thresholds were measured for random jitter in the vertices and for affine transformations. Second, we expect that size changes will not affect human thresholds for shape discrimination. Third, we expect that when the rotation between two shapes is increased, human thresholds will increase. RESULTS: Thresholds measured in Kendall's distance were consistent for shape changes with random jitter and affine transformations, while thresholds measured in Procrustes distance were much less consistent. The thresholds in both metrics were robust to size changes but were found to increase for increasingly rotated shapes. CONCLUSIONS: Kendall's distance provides a reliable perceptual shape similarity metric in these experiments.