Which geometric model for the curvature of 2-D shape contours?
Résumé
We investigated the geometric representations underlying the perception of 2-D contour curvature. 88 arcs representing lower and upper halves of concentric circles, or halves of ellipses derived mathematically through planar projection by affinity with the circles, a special case of Newton's transform, were generated to produce curved line segments with negative and positive curvature and varying sagitta (sag) and/or aspect ratio. Aspect ratio is defined here as the ratio between the sagitta and the chord-length of a given arc. The geometric properties of the arcs suggest a regrouping into four structural models. The 88 stimuli were presented in random order to 16 observers eight of whom were experienced in the mathematical and visual analysis of 2-D curvature ('expert observers'), and eight of whom were not ('non-expert observers'). Observers had to give a number, on a psychophysical scale from 0 to 10, that was to reflect the magnitude of curvature they perceived in a given arc. The results show that the subjective magnitude of curvature increases exponentially with the aspect ratio and linearly with the sagitta of the arcs for both experts and nonexperts. Statistical analysis of the correlation coefficients of linear fits to individual data represented on a logarithmic scale reveals significantly higher correlation coefficients for aspect ratio than for sagitta. The difference is not significant when curves with the longest chords only (7 degrees -10 degrees ) are considered. The geometric model that produces the best psychometric functions is described by a combination of arcs of vertically and horizontally oriented ellipses, indicating that perceptual sensations of 2-D contour curvature are based on geometric representations that suggest properties of 3-D structures. A 'buckled bar model' is shown to optimally account for the perceptual data of all observers with the exception of one expert. His perceptual data can be linked to a more analytical, less 'naturalistic' representation originating from a specific perceptual experience, which is discussed. It is concluded that the structural properties of 'real' objects are likely to determine even the most basic geometric representations underlying the perception of curvature in 2-D images. A specific perceptual learning experience may engender changes in such representations.