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

The stereokinetic phenomena of the tilted disk and of the ellipsoid are visual illusions of depth elicited by a flat figure with elliptic contour rotating at uniform speed in the frontal plane of an observer. Strictly related to the appearance of the ellipsoid is the stereokinetic phenomenon of the tilted bar, elicited by a line segment of constant length rotating at uniform speed in the frontal plane. We present a mathematical model of these phenomena, based on an assumption of minimization by the Visual System of the differences between the lengths of the velocity vectors of the stimulus (minimum relative motion assumption): the "rigidity hypothesis" is able to explain the appearance of the tilted disk but not the appearance of the ellipsoid and of the tilted bar.

A Mathematical Model of Depth Displacement with a Contracting Bar.

When a vertical line segment contracts at both ends according to a (decelerating) time law of hyperbolic type, it appears to rotate around its midpoint. The phenomenon is quite surprising from the point of view of projective geometry as the segment should rather appear to recede along the sagittal plane. Apparent displacement in depth is however obtained when the bar simultaneously contracts and is displaced laterally on the frontal plane.

Serum tumor markers in monitoring patients: interpretation of results using analytical and biological variation.

During cancer monitoring, data on biological and analytical variation are required in order to define the critical difference which provides an objective means to interpret serial values. We evaluated four tumor markers on serial samples collected from healthy subjects and patients. Analytical coefficients of variation (CV(A)), were obtained from "precision profiles" based on the differences between duplicates cumulated from assay runs in the laboratory.

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

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".

Analytic determination of the depth effect in stereokinetic phenomena without a rigidity assumption.

When a circular disk with an eccentric dot is set in slow rotary motion a compelling impression of a three-dimensional cone is observed. Similarly a line segment of constant length, a bar, rotating on the frontal plane appears slanted in depth. The two stereokinetic phenomena cannot be explained on the basis of Ullman's method of extracting depth from 2-D moving stimuli i.