Quantitative determination of the three-dimensional appearances of a rotating ellipse without a rigidity assumption

Abstract

When a flat figure of uniform color and with an elliptic contour is slowly rotated around an axis orthogonal to the plane of the image, an observer set in the frontal position will perceive it first as a rotating ellipse. After a few seconds of inspection, the ellipse appears to deform with an amoeba-like movement until it appears as a rigid, circular disk tilting back and forth in 3-D space; finally it is seen as a rotating ellipsoid tilted in depth at a constant inclination angle with respect to the rotating platform. In an attempt to provide an explanation for the apparent ellipsoid, the authors present a mathematical model based on an hypothesis of velocity differences minimization, successfully used to describe other stereokinetic phenomena, such as "the rotating cone" and the "tilted bar". This technique does not make use of Ullman's "rigidity hypothesis" for extracting depth from 2-D moving stimuli. A comparison with experimental results supports the validity of the model.

A Mathematical Model of Depth Displacement with a Contracting Bar

“…So far the velocity minimization model has been applied to various other phenomena (Beghi et al, 1991a(Beghi et al, , 1991b(Beghi et al, , 1997a(Beghi et al, , 1997bXausa et al, 1997;Zanforlin, 1988;Zanforlin 6 Vallortigara, 1988) besides those here presented, and it appears more valid in explaining the perceived structure from motion than any of those based on``rigidity assumption.'' APPENDIX A Case 1: Uniform vertical contraction and uniform horizontal translation.…”

“…The hypothesis, derived from the gestalt theory of minimum principle (Koffka, 1933;Musatti, 1931); 2 Hatfield 6 Epstein, 1985), has already been successfully applied to other phenomena concerning the recovery of structure from motion (Beghi, Xausa, 6 Zanforlin, 1991a, 1997bBeghi, Xausa, De Biasio, 6 Zanforlin, 1991b;Beghi, Xausa, Tomat, 6 Zanforlin, 1997a;Xausa, Beghi, 6 Zanforlin, 1997;Zanforlin, 1988;Zanforlin 6 Vallortigara, 1988.…”

Modelling stereokinetic phenomena by a minimum relative motion assumption: the tilted disk, the ellipsoid and the tilted bar

“…We skip this part of our modelling procedure for the sake of simplicity: a more detailed explanation is contained in (Beghi et al 1991b).…”

“…(a) at first the ellipse appears simply as a flat rigid figure; (b) after approximately 80 s of inspection, the ellipse appears to deform and take on an amoeba-like movement and at the same time the figure appears to maintain its orientation in space during rotation; (c) on continuing inspection, the elastic appearance vanishes and the ellipse appears as a rigid circular disk tilting back and forth in depth (Musatti 1924(Musatti , 1955Mefferd 1968a;Vallortigara et al 1988;Todorovic 1993); (d) after some time the rigid disk vanishes, the ellipse appears again rigid or deforming and then it may appear as a solid ellipsoid, an egg-like object, tilted in depth and of well-defined length, longer than the stationary ellipse (Mefferd 1968a;Vallortigara 1986, Vallortigara et al 1988;Beghi et al 1991b;Zanforlin 2000);…”

Form and Movement in Stereokinetic Cycloids: Motion Lost and Found

“…Following earlier researches into the stereokinetic effect, in which flat displays rotated in the frontal plane are perceived as having three-dimensional structure (Musatti 1924(Musatti , 1975; and many others), we published studies of the part played in the effect by motion along contours, which we chose to call`lost motion' since it is a type of motion which, because it does not change retinal stimulation, is not sensed and therefore can be included in the percept only by inference (Wilson et al 1983(Wilson et al , 1986Robinson et al 1985). Since that time several theories have been developed to explain the illusions, largely in geometrical terms (Zanforlin and Vallortigara 1988;Beghi et al 1991aBeghi et al , 1991bProfitt et al 1992;Caudek and Profitt 1993). These theories work well in that they have clearly specified the projection of the display at the eye, but there are some effects which they do not predict.…”