The influence function for interpolation of sampled parabolic contours

Purpose. Observers can accurately interpolate an (invisible) parabolic contour when they see as few as 5-7 points sampled at approximately regular intervals along the contour (Koh & Maloney, Inv. Ophth. & Vis. Sci. [abstract], 1988, 29, 408). We measured the influence of each of the contour sampled points on the observer's setting of the interpolated point. We plot it as a function of the relative positions of sampled and interpolated points on the contour. We tested whether this influence function is rotation- and scale-invariant. Methods. Stimuli were presented by means of a spatially-calibrated projection CRT driven by a CRS board. The observer always saw eight sampled points that fell on or near an invisible contour. On each trial, the observer moved a ninth point, initially displaced off of the curve, along an invisible constraint line until it was judged to be on the contour. Each of two observers made 6 settings for all possible combinations of (a) two parabolic contours, (b) two scalings of the entire stimulus array, (c) three rotations of the entire stimulus array, (d) two magnitudes of perturbation, and (e) choice of point to be perturbed. Results. The influence functions of both observers resembled a difference of Gaussians curve, falling rapidly to zero. They were scale-invariant, rotation-invariant, and invariant under scaling of the magnitude of perturbation. Conclusions. These results indicate that the "human visual spline" is scale- and rotation-invariant over the range considered, and computable locally on a sampled contour. We will computed the measured influence functions of observers to computed influence functions for local spline algorithms.