Illusory bending of a rigidly moving line segment:
Effects of image motion and smooth pursuit eye movements
Lore Thaler; James T. Todd; Miriam Spering; Karl R. Gegenfurtner

Abstract
Four experiments in which observers judged the apparent “rubberiness” of a line segment undergoing different types of rigid motion are reported. The results reveal that observers perceive illusory bending when the motion involves certain combinations of translational and rotational components and that the illusion is maximized when these components are presented at a frequency of approximately 3 Hz with a relative phase angle of approximately 120°. Smooth pursuit eye movements can amplify or attenuate the illusion, which is consistent with other results reported in the literature that show effects of eye movements on perceived image motion. The illusion is unaffected by background motion that is in counterphase with the motion of the line segment but is significantly attenuated by background motion that is in-phase. This is consistent with the idea that human observers integrate motion signals within a local frame of reference, and it provides strong evidence that visual persistency cannot be the sole cause of the illusion as was suggested by J. R. Pomerantz (1983). An analysis of the motion patterns suggests that the illusory bending motion may be due to an inability of observers to accurately track the motions of features whose image displacements undergo rapid simultaneous changes in both space and time. A measure of these changes is presented, which is highly correlated with observers' numerical ratings of rubberiness.

Introduction
Since the seminal work of Wallach (1935), it has long been recognized that the motions of smooth contours can be perceptually ambiguous. Consider, for example, the rotating ellipse that is presented in Auxiliary Movie 1. Although the ellipse is rotating rigidly in the image plane, it appears perceptually to be undergoing a nonrigid deformation (see Hildreth, 1984; Weiss & Adelson, 2000). The reason for this effect is that all points along the contour are visually indistinguishable so that it is not possible to measure the component of motion that is parallel to the contour at any given location. If, however, the pattern contains some distinct identifiable points, as in Auxiliary Movie 2, then the unambiguous motions of those features can constraint the interpretation of the contour motion, resulting in the perception of rigid rotation.
The experiments described in Wallach's (1935) original monograph all involved the translatory motions of straight-line contours. The perceptual ambiguity in that case is typically quite constrained. Although observers may perceive an illusory direction of motion, the moving contour always appears rigid. Indeed, this should not be surprising, given that the collinearity of the contour is never altered.
There is an interesting parlor demonstration called “the rubber pencil illusion” that is especially compelling because it violates this basic intuition. If a pencil is held loosely off center and wiggled up and down, it can appear to undergo a nonrigid bending motion (see Figure 1), although the pencil remains physically straight at all times. Note that this illusion occurs despite the presence of trackable features at the endpoints of the moving pencil and the absence of any contour curvature in its optical projection.

The first scientific investigation of the rubber pencil illusion was performed by Pomerantz (1983). He presented observers with computer-generated displays of a rigid line segment undergoing various combinations of translation and rotation, and he asked them to rate the apparent “rubberiness” of each display on a 100-point scale. Figure 2 shows a static representation of four of the conditions used in that study. Each panel depicts a superposition of all of the discrete frames of a particular motion sequence. Figure 2A shows a horizontal line segment whose vertical position varies sinusoidally over time; Figure 2B shows a line segment whose orientation varies sinusoidally over time; and Figures 2C and 2D show different combinations of these basic translational and rotational components. Note in the latter two conditions how the motion trace produces a smoothly curved envelope. Pomerantz suggested that it is the curvature of the densest motion trace that leads to the illusory perception of bending, and he argued that this may be due to visual persistence at early levels of processing, perhaps even in the retina.

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