Visual Depth Perception

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Visual Depth Perception

Depth from Binocular Vision

The 2D Motion/Pursuit Formula at t=-5

Asymptotic Approximation

The preceding side-by-side comparison of binocular disparity and motion parallax is intended to show the similar underlying geometry. In fact there is a strong (asymptotic) approximation connecting the two and the “Motion/Pursuit Law” of depth from lateral translation can be seen as an asymptotic case of the formula for relative depth from binocular disparity and convergence,

Dynamic Approximation of Static Quantities

Binocular disparity

Motion parallax

Wolfram Demonstration Site Playlist

Motion/Pursuit Law in 1D

Motion, Pursuit, Fixate & Distraction

Motion/Pursuit Law in 2 D

Motion/Pursuit Law on Invariant Circles

The different positions of our eyes means that images of objects in physical space lie at different positions on our two eyes. A mathematical model of this “binocular disparity” is given in Fixation & Distraction.

The amount of disparity of different distractor objects for one fixation point is the same on a circle thru one distractor and the eye nodes as shown in the second demonstration Vieth-Müller Circles.

Binocular disparity alone does NOT determine the depth of a distractor beyond the fixate even in the case that they are on the perpendicular line centered on the interocular segment as shown in Binocular Disparity vs Depth.

The convergence angle needed to maintain the fixate point on the fovea together with disparity does determine symmetric depth as shown in Disparity, Convergence & Depth

Images of objects in rigid physical space appear to a translating observer to move. For example, if you look out the side window of a moving car at a farmhouse, the trees beyond the house seem to move with you and the mailbox in front of the house seems to move opposite your translation. Just as in depth from binocular disparity and convergence, there are two motions described in Tracking & Separation.

The changing separation angle gives the appearance of the moving trees and mailbox. At a fixed position, the points with the same retinal motion lie on a circle through one distractor tangent to the line of translation as shown in Invariant Circles of Retinal Motion.

Retinal motion alone does NOT determine the depth of a distractor beyond the fixate as shown in Motion Parallax vs Depth, 2D, 3D.

The the rate of rotation of the eye (“pursuit”) needed to center the fixate (watch the farm house) together with retinal motion does determine relative depth, Retinal Motion, Pursuit & Depth